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507 
 
 
PROBLEM 13.CQ1 
Block A is traveling with a speed v0 on a smooth surface when the 
surface suddenly becomes rough with a coefficient of friction of µ 
causing the block to stop after a distance d. If block A were traveling 
twice as fast, that is, at a speed 2v0, how far will it travel on the rough 
surface before stopping? 
(a) d/2 
(b) d 
(c) d2 
(d) 2d 
(e) 4d 
 
SOLUTION 
Answer: (e) 
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508 
 
PROBLEM 13.1 
A 400-kg satellite was placed in a circular orbit 1500 km above the surface of the earth. At this elevation the 
acceleration of gravity is 6.43 m/s2. Determine the kinetic energy of the satellite, knowing that its orbital 
speed is 325.6 10 km/h.× 
 
SOLUTION 
Mass of satellite: 400 kgm = 
Velocity: 3 325.6 10 km/h 7.111 10 m/sv = × = × 
Kinetic energy: 2 3 2
1 1
(400 kg)(7.111 10 m/s)
2 2
T mv= = × 
 910.113 10 JT = × 10.11 GJT = W 
Note: Acceleration of gravity has no effect on the mass of the satellite. 
 
 
 
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509 
 
 
PROBLEM 13.2 
A 1-lb stone is dropped down the “bottomless pit” at 
Carlsbad Caverns and strikes the ground with a speed of 
95ft/s. Neglecting air resistance, determine (a) the kinetic 
energy of the stone as it strikes the ground and the height 
h from which it was dropped, (b) Solve Part a assuming 
that the same stone is dropped down a hole on the moon. 
(Acceleration of gravity on the moon = 5.31 ft/s2.) 
 
SOLUTION 
Mass of stone: 2
2
lb 1 lb
0.031056 lb s /ft
32.2 ft/s
W
m
g
= = = ⋅ 
Initial kinetic energy: 1 0 (rest)T = 
(a) Kinetic energy at ground strike: 
 2 22 2
1 1
(0.031056)(95) 140.14 ft lb
2 2
T mv= = = ⋅ 
 2 140.1 ft lbT = ⋅ W 
 Use work and energy: 1 1 2 2T U T→+ = 
 where 1 2
2
2
1
0
2
U wh mgh
mgh mv
→ = =
+ =
 
 
2 2
2 (95)
2 (2)(32.2)
v
h
g
= = 140.1 fth = W 
(b) On the moon: 25.31 ft/sg = 
 T1 and T2 will be the same, hence 2 140.1 ft lbT = ⋅ W 
 
2 2
2 (95)
2 (2)(5.31)
v
h
g
= = 850 fth = W 
 
 
 
 
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510 
 
 
PROBLEM 13.3 
A baseball player hits a 5.1-oz baseball with an initial velocity of 140 ft/s 
at an angle of 40° with the horizontal as shown. Determine (a) the kinetic 
energy of the ball immediately after it is hit, (b) the kinetic energy of the 
ball when it reaches its maximum height, (c) the maximum height above 
the ground reached by the ball. 
 
SOLUTION 
 Mass of baseball: 
1 lb
(5.1 oz) 0.31875 lb
16 oz
W § ·= =¨ ¸
© ¹
 
 2
2
0.31875 lb
0.009899 lb s /ft
32.2 ft/s
W
m
g
= = = ⋅ 
(a) Kinetic energy immediately after hit. 
 0 140 ft/sv v= = 
 2 21
1 1
(0.009899)(140)
2 2
T mv= = 1 97.0 ft lbT = ⋅ W 
(b) Kinetic energy at maximum height: 
 0 cos 40 140cos 40 107.246 ft/sv v= ° = ° = 
 2 22
1 1
(0.009899)(107.246)
2 2
T mv= = 2 56.9 ft lbT = ⋅ W 
 Principle of work and energy: 1 1 2 2T U T→+ = 
 1 2 2 1 40.082 ft lbU T T→ = − = − ⋅ 
 Work of weight: 1 2U Wd→ = − 
 Maximum height above impact point. 
 2 1
40.082 ft lb
125.7 ft
0.31875 lb
T T
d
W
− − ⋅= = =
− −
 125.7 ft W 
(c) Maximum height above ground: 
 125.7 ft 2 fth = + 127.7 fth = W 
 
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511 
 
 
PROBLEM 13.4 
A 500-kg communications satellite is in a circular geosynchronous orbit and completes one revolution about 
the earth in 23 h and 56 min at an altitude of 35800 km above the surface of the earth. Knowing that the radius 
of the earth is 6370 km, determine the kinetic energy of the satellite. 
 
SOLUTION 
Radius of earth: 6370 kmR = 
Radius of orbit: 66370 35800 42170 km 42.170 10 mr R h= + = + = = × 
Time one revolution: 23 h 56 mint = + 
 3(23 h)(3600 s/h) (56 min)(60 s/min) 86.160 10 st = + = × 
Speed: 
6
3
2 2 (42.170 10 )
3075.2 m/s
86.160 10
r
v
t
π π ×= = =
×
 
Kinetic energy: 2
1
2
T mv= 
 2 9
1
(500 kg)(3075.2 m/s) 2.3643 10 J
2
T = = × 
 2.36 GJT = W 
 
 
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512 
 
 
PROBLEM 13.5 
In an ore-mixing operation, a bucket full of ore is suspended 
from a traveling crane which moves along a stationary bridge. 
The bucket is to swing no more than 10 ft horizontally when 
the crane is brought to a sudden stop. Determine the maximum 
allowable speed v of the crane. 
 
SOLUTION 
Let position M be the position with bucket B directly below A, and position N be that of maximum swing 
where d = 10 ft. Let L be the length AB. 
Kinetic energies: 21 2
1
, 0
2
T mv T= = 
Work of the weight: 1 2U Wh mgh→ = − = − 
where h is the vertical projection of position N above position M 
From geometry (see figure), 
 
 
2 2
2 2
2 230 (30) (10)
1.7157 ft
y L d
h L y
L L d
= −
= −
= − −
= − −
=
 
 
 
 
Principle of work and energy: 1 1 2 2T U T→+ = 
 2
1
0
2
mv mgh− = 
 2 2 2 22 (2)(32.2 ft/s )(1.7157 ft) 110.49 ft /sv gh= = = 10.51 ft/sv = W 
 
 
 
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513 
 
PROBLEM 13.6 
In an ore-mixing operation, a bucket full of ore is suspended 
from a travelingcrane which moves along a stationary bridge. 
The crane is traveling at a speed of 10 ft/s when it is brought to 
a sudden stop. Determine the maximum horizontal distance 
through which the bucket will swing. 
 
SOLUTION 
Let position M be the position with bucket B directly below A, and position N be that of maximum swing 
where the horizontal distance is d. Let L be the length AB. 
Kinetic energies: 21 2
1
, 0
2
T mv T= = 
Work of the weight: 1 2U Wh mgh→ = − = − 
where h is the vertical projection of position N above position M. 
Principle of work and energy: 1 1 2 2T U T→+ = 
 
2
2 2
2
1
0
2
(10 ft/s)
1.5528 ft
2 (2)(32.2 ft/s )
mv mgh
v
h
g
− =
= = =
 
 
From geometry (see figure), 
 
 
2 2
2 2
2 2
( )
(30) (30 1.5528)
9.53 ft
d L y
L L h
= −
= − −
= − −
=
 
 
 
 9.53 ftd = W 
 
 
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514 
 
PROBLEM 13.7 
Determine the maximum theoretical speed that may be achieved over a distance of 110 m by a car starting 
from rest assuming there is no slipping. The coefficient of static friction between the tires and pavement is 0.75, 
and 60 percent of the weight of the car is distributed over its front wheels and 40 percent over its rear wheels. 
Assume (a) front-wheel drive, (b) rear-wheel drive. 
 
SOLUTION 
Let W be the weight and m the mass. W mg= 
(a) Front wheel drive: 0.60 0.60
0.75s
N W mg
µ
= =
=
 
 Maximum friction force without slipping: 
 
1 2
2
1 2 2
(0.75)(0.60 ) 0.45
0.45
1
0,
2
sF N W mg
U Fd mgd
T T mv
µ
→
= = =
= =
= =
 
 Principle of work and energy: 1 1 2 2T U T→+ = 
 
2
2
2 2 2 2
2
1
0 0.45
2
(2)(0.45 ) (2)(0.45)(9.81 m/s )(110 m) 971.19 m /s
mgd mv
v gd
+ =
= = =
 
 2 31.164 m/sv = 2 112.2 km/hv = W 
(b) Rear wheel drive: 0.40 0.40
0.75s
N W mg
µ
= =
=
 
 Maximum friction force without slipping: 
 
1 2
2
1 2 2
(0.75)(0.40 ) 0.30
0.30
1
0,
2
sF N W mg
U Fd mgd
T T mv
µ
→
= = =
= =
= =
 
 Principle of work and energy: 1 1 2 2T U T→+ = 
 22
2 2 2 2
2
1
0 0.30
2
(2)(0.30) (2)(0.30)(9.81 m/s )(110 m) 647.46 m /s
mgd mv
v gd
+ =
= = =
 
 2 25.445 m/sv = 2 91.6 km/hv = W�
Note: The car is treated as a particle in this problem. The weight distribution is assumed to be the same for 
static and dynamic conditions. Compare with sample Problem 16.1 where the vehicle is treated as a rigid 
body. 
 
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515 
 
 
 
PROBLEM 13.8 
Skid marks on a drag racetrack indicate that the rear (drive) 
wheels of a car slip for the first 20 m of the 400-m track. 
(a) Knowing that the coefficient of kinetic friction is 0.60, 
determine the speed of the car at the end of the first 20-m 
portion of the track if it starts from rest and the front wheels 
are just off the ground. (b) What is the maximum 
theoretical speed for the car at the finish line if, after 
skidding for 20 m, it is driven without the wheels slipping 
for the remainder of the race? Assume that while the car is 
rolling without slipping, 60 percent of the weight of the car 
is on the rear wheels and the coefficient of static friction is 
0.75. Ignore air resistance and rolling resistance. 
 
SOLUTION 
(a) For the first 20 m, the normal force at the real wheels is equal to the weight of the car. Since the wheels 
are skidding, the friction force is 
 k k kF N W mgµ µ µ= = = 
 Principle of work and energy: 1 1 2 2T U T→+ = 
 
2
2
2
2
2 2 2 2
2
1
0
2
1
0
2
2 (2)(0.6)(9.81 m/s )(20 m) 235.44 m /s
k
k
Fd mv
mgd mv
v gd
µ
µ
+ =
+ =
= = =
 
� � 2 15.34 m/sv = W 
(b) Assume that for the remainder of the race, sliding is impending and N = 0.6 W 
 (0.6 ) (0.75)(0.6 ) 0.45s sF N W mg mgµ µ= = = = 
 Principle of work and energy: 2 2 3 3T U T→+ = 
 2 22 3
1 1
(0.45 )
2 2
mv mg d mv′+ = 
 
2 2
3 2
2 2 2
2 2
(2)(0.45)
235.44 m /s (2)(0.45)(9.81 m/s )(400 m 20 m)
3590.5 m /s
v v gd ′=
= + −
=
 
 3 59.9 m/sv = W 
 
 
 
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516 
 
PROBLEM 13.9 
A package is projected up a 15° incline at A with an initial 
velocity of 8 m/s. Knowing that the coefficient of kinetic friction 
between the package and the incline is 0.12, determine (a) the 
maximum distance d that the package will move up the incline, 
(b) the velocity of the package as it returns to its original position. 
 
SOLUTION 
(a) Up the plane from A to B: 
 
2 21 1 (8 m/s) 32 0
2 2
( sin15 ) 0.12 N
A A B
A B k
W W
T mv T
g g
U W F d F Nµ−
= = = =
= − ° − = =
 
 0 cos15 0 cos15F N W N WΣ = − ° = = ° 
 
(sin15 0.12cos15 ) (0.3747)
: 32 (0.3743) 0
A B
A A B B
U W d Wd
W
T U T Wd
g
−
−
= − ° + ° = −
+ = − =
 
 
32
(9.81)(0.3747)
d = 8.71 md = W�
(b) Down the plane from B to A: (F reverses direction) 
 
2
2
1
0 8.71 m/s
2
( sin15 )
(sin15 0.12cos15 )(8.70 m/s)
1.245
1
0 1.245 
2
A A B
B A
B A
B B A A A
W
T v T d
g
U W F d
W
U W
W
T U T W v
g
−
−
−
= = =
= ° −
= ° − °
=
+ = + =
 
 
2 (2)(9.81)(1.245)
24.43
4.94 m/s
A
A
v
v
=
=
= 4.94 m/sA =v 15° W 
 
 
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517 
 
PROBLEM 13.10 
A 1.4 kg model rocket is launched vertically from rest with a constant thrust of 25 N until the rocket reaches 
an altitude of 15 m and the thrust ends. Neglecting air resistance, determine (a) the speed of the rocket when 
the thrust ends, (b) the maximum height reached by the rocket, (c) the speed of the rocket when it returns to 
the ground. 
 
SOLUTION 
Weight: (1.4)(9.81) 13.734 NW mg= = = 
(a) First stage: 1 0T = 
 1 2 (25 13.734)(15) 169.0 N mU → = − = ⋅ 
 1 1 2 2T U T→+ = 
 22 1 2
1
169.0 N m
2
T mv U →= = = ⋅ 
 1 22
2 (2) (169.0)
1.4
U
v
m
→= = 2 15.54 m/sv = W 
(b) Unpowered flight to maximum height h: 
 2 169.0 N mT = ⋅ 3 0T = 
 2 3 ( 15)U W h→ = − − 
 2 2 3 3
2( 15)
T U T
W h T
→+ =
− =
 
 2
169.0
15
13.734
T
h
W
− = = � 27.3 mh = W 
(c) Falling from maximum height: 
 
2
3 4 4
3 4
2
3 3 4 4 4
1
0
2
1
: 0
2
T T mv
U Wh mgh
T U T mgh mv
→
→
= =
= =
= = + =
 
 2 2 2 24 2 (2)(9.81 m/s )(27.3 m) 535.6 m /sv gh= = = 4 23.1 m/sv = W 
 
 
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518 
 
 
PROBLEM 13.11 
Packages are thrown down an incline at A with a velocity 
of 1 m/s. The packages slide along the surface ABC to a 
conveyor belt which moves with a velocity of 2 m/s. 
Knowing that 0.25kµ = between the packages and the 
surface ABC, determine the distance d if the packages are 
to arrive at C with a velocity of 2 m/s. 
 
SOLUTION 
On incline AB: cos30
0.25 cos 30
sin 30 
(sin 30 cos 30 )
AB
AB k AB
A B AB
k
N mg
F N mg
U mgd F d
mgd
µ
µ
→
= °
= = °
= ° −
= ° − °
 
On level surface BC: 7 mBC BC
BC k
B C k BC
N mg x
F mg
U mg x
µ
µ→
= =
=
= −
 
At A, 2
1
and 1 m/s
2A A A
T mv v= = 
At C, 2
1
and 2 m/s
2C C C
T mv v= = 
Assume that no energy is lost at the corner B. 
Work and energy. A A B B C CT U U T→ →+ + = 
 2 20
1 1
(sin 30 cos30 ) 
2 2A k k BC
mv mgd mg x mvµ µ+ ° − ° − = 
Dividing by m and solving for d, 
 
2 2
2 2
/2 /2
(sin 30 cos30 )
(2) /(2)(9.81) (0.25)(7) (1) /(2)(9.81)
sin 30 0.25cos30
C k BC A
k
v g x v g
d
µ
µ
ª º+ −¬ ¼=
° − °
+ −=
° − °
 
 6.71 md = W 
 
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519 
 
 
PROBLEM 13.12 
Packages are thrown down an incline at A with a velocity of 
1 m/s. The packages slide along the surface ABC to a conveyor 
belt which moves with a velocity of 2 m/s. Knowing that 
7.5 md = and 0.25kµ = between the packages and all 
surfaces, determine (a) the speed of the package at C, (b) the 
distance a package will slide on the conveyor belt before it 
comes to rest relative to the belt. 
 
SOLUTION 
(a) On incline AB: 
 
cos 30
0.25 cos 30
sin 30 
(sin 30 cos 30 )
AB
AB k AB
A B AB
k
N mg
F N mg
U mgd F d
mgd
µ
µ
→
= °
= = °
= ° −
= ° − °
 
 On level surface BC: 7 m
 
BC BC
BC k
B C k BC
N mg x
F mg
U mg x
µ
µ→
= =
=
= −
 
 At A, 2
1
and 1 m/s
2A A A
T mv v= = 
 At C, 2
1
and 2 m/s
2C C C
T mv v= = 
 Assume that no energy is lost at the corner B. 
 Work and energy. A A B B C CT U U T→ →+ + = 
 2 20
1 1
(sin 30 cos30 ) 
2 2A k k BC
mv mgd mg x mvµ µ+ ° − ° − = 
 Solving for 2Cv , 
 
2 2
2
2 (sin 30 cos30 ) 2 
(1) (2)(9.81)(7.5)(sin 30 0.25cos30 ) (2)(0.25)(9.81)(7)
C A k k BCv v gd g xµ µ= + ° − ° −
= + ° − ° −
 
 2 28.3811 m /s= 2.90 m/sCv = W�
(b) Box on belt: Let beltx be the distance moves by a package as it slides on the belt. 
 0y y
x k k
F ma N mg N mg
F N mgµ µ
Σ = − = =
= =
 
 At the end of sliding, belt 2 m/sv v= = 
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520 
PROBLEM 13.12 (Continued) 
 
 Principle of work and energy: 
 
2 2
belt belt
2 2
belt
belt
2
1 1
 
2 2
2 
8.3811 (2)
(2)(0.25)(9.81)
C k
C
k
mv mg x mv
v v
x
g
µ
µ
− =
−=
−= belt 0.893 mx = W�
 
 
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521 
 
 
PROBLEM 13.13 
Boxes are transported by a conveyor belt with a velocity v0 to a 
fixed incline at A where they slide and eventually fall off at B. 
Knowing that 0.40,kµ = determine the velocity of the conveyor 
belt if the boxes leave the incline at B with a velocity of 8 ft/s. 
 
SOLUTION 
Forces when box is on AB. 
 
0: cos15 0
cos15
yF N W
N W
Σ = − ° =
= °
 
Box is sliding on AB. 
 cos15f k kF N Wµ µ= = ° 
Distance 20 ftAB d= = 
Work of gravity force: ( ) sin15A B gU Wd− = ° 
Work of friction force: cos15f kF d Wdµ− = − ° 
Total work (sin15 sin15 )A B kU Wd µ→ = ° − ° 
Kinetic energy: 
2
0
2
1
2
1
2
A
B B
W
T v
g
W
T v
g
=
=
 
Principle of work and energy: A A B BT U T→+ = 
 2 20
1 1
(sin15 cos15 )
2 2k B
W W
v Wd v
g g
µ+ ° − ° = 
 
2 2
0
2
2 2
2 (sin15 cos15 )
(8) (2)(32.2)(20)[sin15 (0.40)(cos15 )]
228.29 ft /s
B kv v gd µ= − ° − °
= − ° − °
=
 
 0 15.11 ft/s=v 15° W�
 
 
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522 
 
 
PROBLEM 13.14 
Boxes are transported by a conveyor belt with a velocity v0 to a 
fixed incline at A where they slide and eventually fall off at B. 
Knowing that 0.40,kµ = determine the velocity of the conveyor 
belt if the boxes are to have zero velocity at B. 
 
SOLUTION 
Forces when box is on AB. 
 
0: cos15 0
cos15
yF N W
N W
Σ = − ° =
= °
 
Box is sliding on AB. 
 cos15f k kF N Wµ µ= = ° 
Distance 20 ftAB d= = 
Work of gravity force: ( ) sin15A B gU Wd− = ° 
Work of friction force: cos15f kF d Wdµ− = − ° 
Total work (sin15 cos15 )A B kU Wd µ− = ° − ° 
Kinetic energy: 
2
0
2
1
2
1
2
A
B B
W
T v
g
W
T v
g
=
=
 
Principle of work and energy: A A B BT U T−+ = 
 2 20
1 1
(sin15 cos15 )
2 2k B
W W
v Wd v
g g
µ+ ° − ° = 
 
2 2
0
2 2
2 (sin15 cos15 )
0 (2)(32.2)(20)[sin15 (0.40)(cos15 )]
164.29 ft /s
B kv v gd µ= − ° − °
= − ° − °
=
 
 0 12.81 ft/s=v 15° W 
 
 
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523 
 
 
PROBLEM 13.15 
A 1200-kg trailer is hitched to a 1400-kg car. The car and trailer are 
traveling at 72 km/h when the driver applies the brakes on both the car 
and the trailer. Knowing that the braking forces exerted on the car and 
the trailer are 5000 N and 4000 N, respectively, determine (a) the 
distance traveled by the car and trailer before they come to a stop, 
(b) the horizontal component of the force exerted by the trailer hitch 
on the car. 
 
SOLUTION 
Let position 1 be the initial state at velocity 1 72 km/h 20 m/sv = = and position 2 be at the end of braking 
2( 0).v = The braking forces and 5000 NCF = for the car and 4000 N for the trailer. 
(a) Car and trailer system. ( braking distance)d = 
 
2
1 1 2
1 2
1 1 2 2
1
( ) 0
2
( )
C T
C T
T m m v T
U F F d
T U T
→
→
= + =
= − +
+ =
 
 21
1
( ) ( ) 0
2 C T C T
m m v F F d+ − + = 
 
2 2
1( ) (2600)(20) 57.778
2( ) (2)(9000)
C T
C T
m m v
d
F F
+= = =
+
 57.8 md = W 
(b) Car considered separately.Let H be the horizontal pushing force that the trailer exerts on the car through the hitch. 
 
2
1 1 2
1 2
1 1 2 2
1
0
2
( )
C
C
T m v T
U H F d
T U T
→
→
= =
= −
+ =
 
 21
1
( ) 0
2 C C
m v H F d+ − = 
 
2 2
1 (1400)(20)5000
2 (2)(57.778)
C
C
m v
H F
d
= − = − 
 Trailer hitch force on car: 154 N=H W 
 
 
 
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524 
 
PROBLEM 13.16 
A trailer truck enters a 2 percent uphill grade traveling at 72 km/h and reaches a speed of 108 km/h in 300 m. 
The cab has a mass of 1800 kg and the trailer 5400 kg. Determine (a) the average force at the wheels of the 
cab, (b) the average force in the coupling between the cab and the trailer. 
 
 
SOLUTION 
Initial speed: 1 72 km/h 20 m/sv = = 
Final speed: 2 108 km/h 30 m/sv = = 
Vertical rise: (0.02)(300) 6.00 mh = = 
Distance traveled: 300 md = 
(a) Traction force. Use cab and trailer as a free body. 
 1800 5400 7200 kgm = + = 3(7200)(9.81) 70.632 10 NW mg= = = × 
 Work and energy: 1 1 2 2T U T→+ = 
2 2
1 2
1 1
2 2t
mv Wh F d mv− + = 
 2 2 2 3 21 1
1 1 1 1 1
(7200)(30) (70.632 10 )(6.00) (7200)(20)
2 300 2 2t
F mv Wh mv
d
ª º ª º= + − = + × −« » « »¬ ¼¬ ¼
 
 37.4126 10 N= × 7.41 kNtF = W 
(b) Coupling force .cF Use the trailer alone as a free body. 
 35400 kg (5400)(9.81) 52.974 10 Nm W mg= = = = × 
 Assume that the tangential force at the trailer wheels is zero. 
 Work and energy: 1 1 2 2T U T→+ = 
2 2
1 2
1 1
2 2c
mv Wh F d mv− + = 
 The plus sign before cF means that we have assumed that the coupling is in tension. 
 2 2 2 3 22 1
1 1 1 1 1 1
(5400)(30) (52.974 10 )(6.00) (5400)(20)
2 2 300 2 2c
F mv Wh mv
d
ª º ª º= + − = + × −« » « »¬ ¼¬ ¼
 
 35.5595 10 N= × 5.56 kN (tension)cF = W 
 
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525 
 
 
PROBLEM 13.17 
The subway train shown is traveling at a speed of 30 mi/h 
when the brakes are fully applied on the wheels of cars B 
and C, causing them to slide on the track, but are not 
applied on the wheels of car A. Knowing that the 
coefficient of kinetic friction is 0.35 between the wheels 
and the track, determine (a) the distance required to bring 
the train to a stop, (b) the force in each coupling. 
 
SOLUTION 
 
 
0.35 (0.35)(100 kips) 35 kips
(0.35)(80 kips) 28 kips
k B
C
F
F
µ = = =
= =
 
 1 30 mi/h 44 ft/sv = = 2 20 0v T= = 
(a) Entire train: 1 1 2 2T U T−+ = 
 2
2
1 (80 kips 100 kips 80 kips)
(44 ft/s) (28 kips 35 kips) 0
2 32.2 ft/s
x
+ + − + = 
� � 124.07 ftx = � 124.1 ftx = W�
(b) Force in each coupling: Recall that 124.07 ftx = 
 Car A: Assume ABF to be in tension 
 1 1 2 2
21 80 kips (44) (124.07 ft) 0
2 32.2
19.38 kips
AB
AB
T V T
F
F
−+ =
− =
= +
 
 19.38 kips (tension)ABF = W 
 Car C: 1 1 2 2T U T−+ = 
 2
1 80 kips
(44) ( 28 kips)(124.07 ft) 0
2 32.2
28 kips 19.38 kips
BC
BC
F
F
+ − =
− = −
 
 8.62 kipsBCF = + 8.62 kips (tension)BCF = W 
 
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526 
 
 
PROBLEM 13.18 
The subway train shown is traveling at a speed of 30 mi/h 
when the brakes are fully applied on the wheels of cars A, 
causing it to slide on the track, but are not applied on the 
wheels of cars A or B. Knowing that the coefficient of 
kinetic friction is 0.35 between the wheels and the track, 
determine (a) the distance required to bring the train to a 
stop, (b) the force in each coupling. 
 
SOLUTION 
(a) Entire train: 
 
 1
(0.35)(80 kips) 28 kips
30 mi/h 44 ft/s 
A AF N
v
µ= = =
= = ← 2 20 0v T= = 
 
1 1 2 2
2
2
1 (80 kips 100 kips 80 kips)
(44 ft/s) (28 kips) 0
2 32.2 ft/s
T V T
x
−+ =
+ + − = 
 279.1 ftx = 279 ftx = W�
(b) Force in each coupling: 
 Car A: Assume ABF to be in tension 
 1 1 2 2T V T−+ = 
 2
2
1 80 kips
(44 ft/s) (28 kips )(279.1 ft) 0
2 32.2 ft/s
ABF− + = 
 28 kips 8.62 kipsABF+ = + 
 19.38 kipsABF = − 19.38 kips (compression)ABF = W 
 Car C: 1 1 2 2T V T−+ = 
 2
2
1 80 kips
(44 ft/s) (279.1 ft) 0
2 32.2 ft/s
BCF+ = 
 8.617 kipsBCF = − 8.62 kips (compression)BCF = W 
 
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527 
 
 
PROBLEM 13.19 
Blocks A and B weigh 25 lbs and 10 lbs, respectively, and they are both 
at a height 6 ft above the ground when the system is released from rest. 
Just before hitting the ground block A is moving at a speed of 9 ft/s. 
Determine (a) the amount of energy dissipated in friction by the pulley, 
(b) the tension in each portion of the cord during the motion. 
 
SOLUTION 
By constraint of the cable block B moves up a distance h when block A moves down a distance h. ( 6 ft)h = 
Their speeds are equal. 
Let FA and FB be the tensions on the A and B sides, respectively, of the pulley. 
Masses: 2
2
25
0.7764 lb s /ft
32.2
10
0.31056 lb s /ft
32.2
A
A
B
B
W
M
g
W
M
g
= = = ⋅
= = = ⋅
 
Let position 1 be the initial position with both blocks a distance h above the ground and position 2 be just 
before block A hits the ground. 
Kinetic energies: 1 1( ) 0, ( ) 0A BT T= = 
 
2 2
2
2 2
2
1 1
( ) (0.7764)(9) 31.444 ft lb
2 2
1 1
( ) (0.31056)(9) 12.578 ft lb
2 2
A A
B B
T m v
T m v
= = = ⋅
= = = ⋅
 
Principle of work and energy: 1 1 2 2T U T→+ = 
Block A: 1 2 ( )A AU W F h→ = − 
 0 (25 )(6) 31.444AF+ − = 19.759 lbAF = 
Block B: 1 2 ( )B BU F W h→ = − 
 0 ( 10)(6) 12.578BF+ − = 12.096 lbBF = 
At the pulley FA moves a distance h down, and FB moves a distance h up. The work done is 
 1 2 ( ) (19.759 12.096)(6) 46.0 ft lbA BU F F h→ = − = − = ⋅ 
 
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528 
PROBLEM 13.19 (Continued) 
 
Since the pulley is assumed to be massless, it cannot acquire kinetic energy; hence, 
(a) Energy dissipated by the pulley: 46.0 ft lbpE = ⋅ W 
(b) Tension in each portion of the cord: :19.76 lbA W 
 :12.10 lbB W 
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529 
 
 
PROBLEM 13.20 
The system shown is at rest when a constant 30 lb force is 
applied to collar B. (a) If the force acts through the entire 
motion, determine the speed of collar B as it strikes the 
support at C. (b) After what distance d should the 30 lb 
force be removed if the collar is to reach support C with 
zero velocity? 
 
SOLUTION 
Let F be the cable tension and vB be the velocity of collar B when it strikes the support. Consider the collar B. 
Its movement is horizontal so only horizontal forces acting on B do work. Let d be the distance through which 
the 30 lb applied force moves. 
 
1 1 2 2
2
( ) ( ) ( )
1 18
0 30 (2 )(2)
2 32.2
B B B
B
T U T
d F v
→+ =
+ − =
 
 230 4 0.27950 Bd F v− = (1) 
Now consider the weight A. When the collar moves 2 ft to the left, the weight moves 4 ft up, since the cable 
length is constant. Also, 2 .A Bv v= 
 
1 1 2 2
2
2
( ) ( ) ( )
1
0 ( )(4)
2
1 6
4 (6)(4) (2 )
2 32.2
A A B
A
A A
B
T U T
W
F W v
g
F v
−+ =
+ − =
− =
 
 24 24 0.37267 BF v− = (2) 
Add Eqs. (1) and (2) to eliminate F. 
 230 24 0.65217 Bd v− = (3) 
(a) Case a: 2 ft, ?Bd v= = 
 2(30)(2) (24) 0.65217 Bv− = 
 2 2 255.2 ft /sBv = 7.43 ft/sBv = W�
(b) Case b: ?, 0.Bd v= = 
 30 24 0d − = 0.800 ftd = W 
 
 
 
 
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530 
 
PROBLEM 13.21 
Car B is towing car A at a constant speed of 10 m/s on an uphill 
grade when the brakes of car A are fully applied causing all four 
wheels to skid. The driver of car B does not change the throttle 
setting or change gears. The masses of the cars A and B are 
1400 kg and 1200 kg, respectively, and the coefficient of kinetic 
friction is 0.8. Neglecting air resistance and rolling resistance, 
determine (a) the distance traveled by the cars before they come 
to a stop, (b) the tension in the cable. 
 
SOLUTION 
 
 
0.8 AF N= 
 
 
 
 
 
 
 
 
 
Given: Car B tows car A at 10 m/s uphill. 
Car A brakes so 4 wheels skid. 0.8kµ = 
Car B continues in same gear and throttle setting. 
Find: (a) Distance d, traveled to stop 
 (b) Tension in cable 
(a) 1F = traction force (from equilibrium) 
 1 (1400 )sin 5 (1200 )sin 5F g g= ° + ° 
 2600(9.81)sin 5= ° 
For system: A B+ 
1 2 1[( 1400 sin 5 1200 sin 5 ) ]U F g g F d− = − ° − ° − 
 2 22 1
1 1
0 (2600)(10)
2 2A B
T T m v+= − = − = − 
Since 1( 1400 sin 5 1200 sin 5 ) 0F g g− ° − ° = 
0.8[1400(9.81)cos5 ] 130,000 N mFd d− = − ° = − ⋅ 
 11.88 md = W 
(b) Cable tension, T 
1 2 2 1[ 0.8 ](11.88)AU T N T T− = − = − 
21400( 0.8(1400)(9.81)cos5 )11.88 (10)
2
T − ° = − 
( 10945) 5892T − = − 
 5.053 kN= 
 5.05 kNT = W 
 
 
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531 
 
 
PROBLEM 13.22 
The system shown is at rest when a constant 250-N force is applied to 
block A. Neglecting the masses of the pulleys and the effect of friction 
in the pulleys and between block A and the horizontal surface, 
determine (a) the velocity of block B after block A has moved 2 m, (b) 
the tension in the cable. 
 
SOLUTION 
Constraint of cable: 
 
3 constant
3 0
3 0
A B
A B
A B
x y
x y
v v
+ =
∆ + ∆ =
+ =
 
Let F be the tension in the cable. 
Block A: 130 kg, 250 N, ( ) 0A Am P T= = = 
 
1 1 2 2
2
2
( ) ( ) ( )
1
0 ( )( )
2
1
0 (250 )(2) (30)(3 )
2
A A A
A A A
B
T U T
P F x m v
F v
→+ =
+ − ∆ =
+ − =
 
 2500 2 135 BF v− = (1) 
Block B: 25 kg, 245.25 NB B Bm W m g= = = 
 
1 1 2 2
2
2
( ) ( ) ( )
1
0 (3 )( )
2
2 1
(3 ) 245.25) (25)
3 2
B B B
B B B B
B
T U T
F W y m v
F v
→+ =
+ − −∆ =
§ ·− =¨ ¸
© ¹
 
 22 163.5 12.5 BF v− = (2) 
 
 
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532 
PROBLEM 13.22 (Continued) 
 
Add Eqs. (1) and (2) to eliminate F. 
 
2
2 2 2
500 163.5 147.5
2.2814 m /s
B
B
v
v
− =
=
 
(a) Velocity of B. 1.510 m/sB =v W 
(b) Tension in the cable. 
 From Eq. (2), 2 163.5 (12.5)(2.2814)F − = 96.0 NF = W 
 
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533 
 
 
PROBLEM 13.23 
The system shown is at rest when a constant 250-N force is applied to 
block A. Neglecting the masses of the pulleys and the effect of friction in 
the pulleys and assuming that the coefficients of friction between block A 
and the horizontal surface are 0.25sµ = and 0.20,kµ = determine (a) the 
velocity of block B after block A has moved 2 m, (b) the tension in the cable. 
 
SOLUTION 
Check the equilibrium position to see if the blocks move. Let F be the tension in the cable. 
Block B: 3 0
(25)(9.81)
81.75 N
3 3
B
B
F m g
m g
F
− =
= = =
 
Block A: 0: 0y A AF N m gΣ = − = 
 (30)(9.81) 294.3 NA AN m g= = = 
 0: 250 0x AF F FΣ = − − = 
 250 81.75 168.25 NAF = − = 
Available static friction force: (0.25)(294.3) 73.57 Ns ANµ = = 
Since ,A s AF Nµ> the blocks move. 
The friction force, FA, during sliding is 
 (0.20)(294.3) 58.86 NA k AF Nµ= = = 
Constraint of cable: 
 
3 constant
3 0
3 0
A B
A B
A B
x y
x y
v v
+ =
∆ + ∆ =
+ =
 
 
 
 
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534 
PROBLEM 13.23 (Continued) 
 
Block A: 1
1 1 2 2
30 kg, 250 N, ( ) 0.
( ) ( ) ( )
A A
A A A
m P T
T U T→
= = =
+ =
 
 
2
2
1
0 ( )( )
2
1
0 (250 58.86 )(2) (30)(3 )
2
A A A A
B
P F F x m v
F v
+ − − ∆ =
+ − − =
 
 2382.28 2 135 BF v− = (1) 
Block B: 25 kg, 245.25 NB B BM W m g= = = 
 
1 1 2 2
2
2
( ) ( ) ( )
1
0 (3 )( )
2
2 1
(3 245.25) (25)
3 2
B B B
B B B B
B
T U T
F W y m v
F v
→+ =
+ − −∆ =
§ ·− =¨ ¸
© ¹
 
 22 163.5 12.5 BF v− = (2) 
Add Eqs. (1) and (2) to eliminate F. 
 
2
2 2 2
382.28 163.5 147.5
1.48325 m /s
B
B
v
v− =
=
 
(a) Velocity of B: 1.218 m/sB =v W 
(b) Tension in the cable: 
 From Eq. (2), 2 163.5 (12.5)(1.48325)F − = 91.0 NF = W�
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535 
 
 
PROBLEM 13.24 
Two blocks A and B, of mass 4 kg and 5 kg, respectively, are connected 
by a cord which passes over pulleys as shown. A 3 kg collar C is placed 
on block A and the system is released from rest. After the blocks have 
moved 0.9 m, collar C is removed and blocks A and B continue to move. 
Determine the speed of block A just before it strikes the ground. 
 
SOLUTION 
Position to Position .M N 1 10 0v T= = 
At N before C is removed from the system 
 
2 2 2
2 2 2 2
1 2
2
1 2
1 2
1 1 2 2
1 1
( ) (12 kg) 6
2 2
( ) (0.9 m)
(4 3 5)( )(0.9 m) (2 kg)(9.81 m/s )(0.9 m)
17.658 J
:
A B C
A C B
T m m m v v v
U m m m g
U g
U
T U T
−
−
−
−
= + + = =
= + −
= + − =
=
+ =
 
 2 22 20 17.658 6 2.943v v+ = = 
At Position N, collar C is removed from the system. 
Position to Position .N O 22 2
1 9
( ) kg (2.943) 13.244 J
2 2A B
T m m v § ·′ = + = =¨ ¸
© ¹
 
 2 23 3 3
1 9
( )( )
2 2A B
T m m v v= + = 
 
2
2 3
2 2 3 3
2 2
3 3
( )( )(0.7 m) ( 1 kg)(9.81 m/s )(0.7 m) 6.867 J
13.244 6.867 4.5 1.417
A BU m m g
T U T
v v
′−
−
= − = − = −
′ + =
− = =
 
 3 1.190 m/sAv v= = 1.190 m/sAv = W 
 
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536 
 
PROBLEM 13.25 
Four packages, each weighing 6 lb, are held 
in place by friction on a conveyor which is 
disengaged from its drive motor. When the 
system is released from rest, package 1 
leaves the belt at A just as package 4 comes 
onto the inclined portion of the belt at B. 
Determine (a) the speed of package 2 as it 
leaves the belt at A, (b) the speed of 
package 3 as it leaves the belt at A. Neglect 
the mass of the belt and rollers. 
 
SOLUTION 
Slope angle: sin 
6 ft
23.6
15 ft
β β= = ° 
(a) Package falls off the belt and 2, 3, 4 move down 
 
2 2 2
2 2 2 22
6
2 ft.
3
1 3 6 lb
3 0.2795
2 2 32.2 ft/s
T mv v v
=
§ ·ª º= = =¨ ¸« » ¨ ¸¬ ¼ © ¹
 
 1 2 (3)( )( ) (3)(6 lb)(2 ft) 36 lb ftU W R− = = = ⋅ 
 1 1 2 2 2 2
2 20 36 0.2795 128.8
T U T
v v
−+ =
+ = =
 
 2 11.35 ft/s=v 23.6° W�
(b) Package 2 falls off the belt and its energy is lost to the system and 3 and 4 move down 2 ft. 
�
2
2 2 2
2
2 2 2
3 3 3 32
1 6 lb
(2) (128.8)
2 32 ft/s
24 lb ft
1 6 lb
(2) ( ) 0.18634
2 32.2 ft/s
T mv
T
T mv v v
§ ·ª º′ = = ¨ ¸« » ¨ ¸¬ ¼ © ¹
′ = ⋅
§ ·ª º= = =¨ ¸« » ¨ ¸¬ ¼ © ¹
�
� 2 3
2 2 3 3
(2)( )(2) (2)(6 lb)(2 ft) 24 lb ftU W
T U T
−
−
= = = ⋅
+ =
�
� 2 23 324 24 0.18634 257.6v v+ = = � 3 16.05 ft/s=v 23.6° W�
 
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537 
 
 
PROBLEM 13.26 
A 3-kg block rests on top of a 2-kg block supported by, but not attached to, a spring 
of constant 40 N/m. The upper block is suddenly removed. Determine (a) the 
maximum speed reached by the 2-kg block, (b) the maximum height reached by the 
2-kg block. 
 
SOLUTION 
Call blocks A and B. 2 kg, 3 kgA Bm m= = 
(a) Position 1: Block B has just been removed. 
 Spring force: ( )S A BF m m g k x= − + = − 
 Spring stretch: 
2
1
( ) (5 kg)(9.81 m/s )
1.22625 m
40 N/m
A Bm m gx
k
+= − = − = − 
Let position 2 be a later position while the spring still contacts block A. 
 Work of the force exerted by the spring: 
2
1
2
1
1 2
2 2 2
1 2
2 2 2
2 2
( )
1 1 1
2 2 2
1 1
(40)( 1.22625) (40) 30.074 20
2 2
x
e
x
x
x
U k x dx
k x k x k x
x x
→ = −
= − = −
= − − = −
³ 
 Work of the gravitational force: 1 2 2 1
2 2
( ) ( )
(2)(9.81)( 1.22625) 19.62 24.059
g AU m g x x
x x
→ = − −
= − + = − −
 
 Total work: 21 2 2 220 19.62 6.015U x x→ = − + + 
 Kinetic energies: 1
2 2 2
2 2 2 2
0
1 1
(2)
2 2A
T
T m v v v
=
= = =
 
 Principle of work and energy: 1 1 2 2T U T→+ = 
 2 22 2 20 20 19.62 6.015x x v+ − + = 
 Speed squared: 2 22 2 220 19.62 6.015v x x= − − + (1) 
 At maximum speed, 2
2
0
dv
dx
= 
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538 
PROBLEM 13.26 (Continued) 
 
 Differentiating Eq. (1), and setting equal to zero, 
 
2
2 2
2
2 40 19.62 0
19.62
0.4905 m
40
dv
v x
dx
x
= − = − =
= − = −
 
 Substituting into Eq. (1), 2 2 2 22 (20)( 0.4905) (19.62)( 0.4905) 6.015 10.827 m /sv = − − − − + = 
 Maximum speed: 2 3.29 m/sv = W 
(b) Position 3: Block A reaches maximum height. Assume that the block has separated from the spring. 
 Spring force is zero at separation. 
 Work of the force exerted by the spring: 
 
1
0 2 2
1 3 1
1 1
( ) (40)(1.22625) 30.074 J
2 2e x
U kxdx kx→ = − = = =³ 
 Work of the gravitational force: 
 1 3( ) (2)(9.81) 19.62 g AU m gh h h→ = − = − = − 
 Total work: 1 3 30.074 19.62 U h→ = − 
 At maximum height, 3 30, 0v T= = 
 Principle of work and energy: 1 1 3 3T U T→+ = 
 0 30.074 19.62 0h+ − = 
 Maximum height: 1.533 mh = W 
 
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539 
 
 
PROBLEM 13.27 
Solve Problem 13.26, assuming that the 2-kg block is attached to the spring. 
PROBLEM 13.26 A 3-kg block rests on top of a 2-kg block supported by, but not 
attached to, a spring of constant 40 N/m. The upper block is suddenly removed. 
Determine (a) the maximum speed reached by the 2-kg block, (b) the maximum 
height reached by the 2-kg block. 
 
 
SOLUTION 
Call blocks A and B. 2 kg, 3 kgA Bm m= = 
(a) Position 1: Block B has just been removed. 
 Spring force: 1( )S A BF m m g kx= − + = − 
 Spring stretch: 
2
1
( ) (5 kg)(9.81 m/s )
1.22625 m
40 N/m
A Bm m gx
k
+= − = − = − 
 Let position 2 be a later position. Note that the spring remains attached to block A. 
 Work of the force exerted by the spring: 
2
1
2
1
1 2
2 2 2
1 2
2 2 2
2 2
( )
1 1 1
2 2 2
1 1
(40)( 1.22625) (40) 30.074 20
2 2
x
e
x
x
x
U kxdx
kx kx kx
x x
→ = −
= − = −
= − − = −
³ 
 Work of the gravitational force: 1 2 2 1
2 2
( ) ( )
(2)(9.81)( 1.22625) 19.62 24.059
g AU m g x x
x x
→ = − −
= − + =− −
 
 Total work: 21 2 2 220 19.62 6.015U x x→ = − − + 
 Kinetic energies: 1
2 2 2
2 2 2 2
0
1 1
(2)
2 2A
T
T m v v v
=
= = =
 
 Principle of work and energy: 1 1 2 2T U T→+ = 
 2 22 2 20 20 19.62 6.015x x v+ − + = 
 Speed squared: 2 22 2 220 19.62 6.015v x x= − − + (1) 
 At maximum speed, 2
2
0
dv
dx
= 
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540 
PROBLEM 13.27 (Continued) 
 
 Differentiating Eq. (1) and setting equal to zero, 
 
2
2 2
2
2
2 40 19.62 0
19.62
0.4905 m
40
dv
v x
dx
x
= − = − =
= − = −
 
 Substituting into Eq. (1), 2 2 1 22 (20)( 0.4905) (19.62)( 0.4905) 6.015 10.827 m /sv = − − − − + = 
 Maximum speed: 2 3.29 m/sv = W 
(b) Maximum height occurs when v2 = 0. 
 Substituting into Eq. (1), 22 20 20 19.62 6.015x x= − − + 
 Solving the quadratic equation 
 2 1.22625 m and 0.24525 mx = − 
 Using the larger value, 2 0.24525 mx = 
 Maximum height: 2 1 0.24525 1.22625h x x= − = + 1.472 mh = W 
 
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541 
 
 
PROBLEM 13.28 
An 8-lb collar C slides on a horizontal rod between springs A and B. 
If the collar is pushed to the right until spring B is compressed 2 in. 
and released, determine the distance through which the collar will 
travel assuming (a) no friction between the collar and the rod, 
(b) a coefficient of friction 0.35.kµ = 
 
SOLUTION 
(a) 144 lb/ft
216 lb/ft
B
A
k
k
=
=
 
 
 Since the collar C leaves the spring at B and there is no friction, it must engage the spring at A. 
 
2/12
0 0
2
2
0 0
 
144 lb/ft 2 216 lb/ft
ft ( )
2 12 2
A B
y
A B B A
A B
T T
U k xdx k xdx
U y
−
−
= =
= −
§ ·§ · § ·= −¨ ¸¨ ¸ ¨ ¸
© ¹© ¹ © ¹
³ ³ 
 
2: 0 2 108 0
 0.1361 ft 1.633 in.
A A B BT U T y
y
−+ = + − =
= =
 
 Total distance 2 16 (6 1.633)d = + − − 
 13.63 in.d = W 
(b) Assume that C does not reach the spring at B because of friction. 
 
6 lb
(0.35)(8 lb) 2.80 lb
0
f
A D
N W
F
T T
= =
= =
= =
 
 
2/12
0
144 ( ) 2 2.80A D fU dx F y y− = × − = −³ 
 
0 2 2.80 0
0.714 ft 8.57 in.
A A D DT U T y
y
−+ = + − =
= =
 
 The collar must travel 16 6 2 12 in.− + = before it engages the spring at B. Since 8.57 in.,y = it stops 
before engaging the spring at B. 
 Total distance 8.57 in.d = W 
 
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542 
 
 
PROBLEM 13.29 
A 6-lb block is attached to a cable and to a spring as shown. The constant of the spring is 
k 8 lb/in.= and the tension in the cable is 3 lb. If the cable is cut, determine (a) the 
maximum displacement of the block, (b) the maximum speed of the block. 
 
SOLUTION 
8 lb/in. 96 lb/ftk = = 
 
 
1
2 2
1 1 2 2 2
0: ( ) 6 3 3 lb 
1 6 lb
0 0: 0.09317
2 32.2
y sF F C
v T T v v
Σ = = − =
§ ·= = = =¨ ¸
© ¹
 
For weight: 1 2 (6 lb) 6U x x− = = 
For spring: 21 2
0
(3 96 ) 3 48
x
U x dx x x− = − + = − −³ 
 
2 2
1 1 2 2 2
2 2
2
: 0 6 3 48 0.09317
 3 48 0.09317
T U T x x x v
x x v
−+ = + − − =
− =
 (1) 
(a) For 2, 0:mx v = 
23 48 0x x− = 
 
3 1
0, ft
48 16m
x x= = = 0.75 in.m =x W�
 
 
 
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543 
PROBLEM 13.29 (Continued) 
 
(b) For mv we see maximum of 
2
1 2 3 48U x x− = − 
 1 2
3 1
3 96 0 ft ft
96 32
dU
x x
dx
− = − = = = 
 Eq. (1): 
2
21 13 ft 48 ft 0.09317
32 32 m
v§ · § ·− =¨ ¸ ¨ ¸
© ¹ © ¹
 
 2 0.5031 0.7093 ft/sm mv v= = 8.51 in./sm =v 7 W 
 Note: 1 2U − for the spring may be computed using 6F x− curve 
 1 2
2
area
1
3 96
2
U
x x
− =
= +
 
 
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544 
 
 
PROBLEM 13.30 
A 10-kg block is attached to spring A and connected to spring B by a cord and 
pulley. The block is held in the position shown with both springs unstretched 
when the support is removed and the block is released with no initial velocity. 
Knowing that the constant of each spring is 2 kN/m, determine (a) the velocity of 
the block after it has moved down 50 mm, (b) the maximum velocity achieved by 
the block. 
 
SOLUTION 
 
 
 
(a) W = weight of the block 10 (9.81) 98.1 N= = 
 
1
2B A
x x= 
 2 21 2
1 1
( ) ( ) ( )
2 2A A A B B
U W x k x k x− = − − 
 (Gravity) (Spring A) (Spring B) 
 21 2
1
(98.1 N)(0.05 m) (2000 N/m)(0.05 m)
2
U − = − 
 2
1
(2000 N/m)(0.025 m)
2
− 
 2 21 2
1 1
( ) (10 kg)
2 2
U m v v− = = 
 2
1
4.905 2.5 0.625 (10)
2
v− − = 
 0.597 m/sv = W�
(b) Let distance moved down by the 10 kg blockx = 
 
2
2 2
1 2
1 1 1
( ) ( ) ( )
2 2 2 2A B
x
U W x k x k m v−
§ ·= − − =¨ ¸
© ¹
 
 2
1
( ) 0 ( ) (2 )
2 8
B
A
d k
m v W k x x
dx
ª º = = − −« »¬ ¼
 
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545 
PROBLEM 13.30 (Continued) 
 
 
2000
0 98.1 2000 ( ) (2 ) 98.1 (2000 250)
8
x x x= − − = − + 
 0.0436 m (43.6 mm)x = 
For 2
1
0.0436, 4.2772 1.9010 0.4752 (10)
2
x U v= = − − = 
 max 0.6166 m/sv = 
 max 0.617 m/sv = W 
 
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546 
 
 
PROBLEM 13.31 
A 5-kg collar A is at rest on top of, but not attached to, a spring 
with stiffness k1 = 400 N/m; when a constant 150-N force is 
applied to the cable. Knowing A has a speed of 1 m/s when the 
upper spring is compressed75 mm, determine the spring 
stiffness k2. Ignore friction and the mass of the pulley. 
 
SOLUTION 
Use the method of work and energy applied to the collar A. 
 1 1 2 2T U T→+ = 
Since collar is initially at rest, 1 0.T = 
In position 2, where the upper spring is compressed 75 mm and 2 1.00 m/s,v = the kinetic energy is 
 2 22 2
1 1
(5 kg)(1.00 m/s) 2.5 J
2 2
T mv= = = 
As the collar is raised from level A to level B, the work of the weight force is 
 1 2( )gU mgh→ = − 
where 25 kg, 9.81 m/sm g= = and 450 mm 0.450 mh = = 
Thus, 1 2( ) (5)(9.81)(0.450) 22.0725 JgU → = − = − 
In position 1, the force exerted by the lower spring is equal to the weight of collar A. 
 1 (5 kg)(9.81 m/s) 49.05 NF mg= = − = − 
As the collar moves up a distance x1, the spring force is 
 1 1 2F F k x= − 
until the collar separates from the spring at 
 1
1
49.05 N
0.122625 m 122.625 mm
400 N/mf
F
x
k
= = = = 
 
 
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547 
PROBLEM 13.31 (Continued) 
 
Work of the force exerted by the lower spring: 
 
1 2 1 1 1
0
2 2 2 2
1 1 1 1
2
( ) ( )
1 1 1
2 2 2
1
(400 N/m)(0.122625) 3.0074 J
2
fx
f f f f f
U F k x dx
F x kx k x k x k x
→ = −
= − = − =
= =
³
 
In position 2, the upper spring is compressed by 75 mm 0.075 m.y = = The work of the force exerted 
by this spring is 
 2 21 2 2 2 2 2
1 1
( ) (0.075) 0.0028125 k
2 2
U k y k→ = − = − = − 
Finally, we must calculate the work of the 150 N force applied to the cable. In position 1, the length AB is 
 2 21( ) (450) (400) 602.08 mmABl = + = 
In position 2, the length AB is 2( ) 400 mm.ABl = 
The displacement d of the 150 N force is 
 1 2( ) ( ) 202.08 mm 0.20208 mAB ABd l l= − = = 
The work of the 150 N force P is 
 1 2( ) (150 N)(0.20208 m) 30.312 JPU Pd→ = = = 
Total work: 1 2 2
2
22.0725 3.0074 0.0028125 30.312
11.247 0.0028125
U k
k
→ = − + − +
= −
 
Principle of work and energy: 1 1 2 2T U T→+ = 
 20 11.247 0.0028125 2.5k+ − = 
 2 3110 N/mk = 2 3110 N/mk = W 
 
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548 
 
 
PROBLEM 13.32 
A piston of mass m and cross-sectional area A is equilibrium 
under the pressure p at the center of a cylinder closed at both 
ends. Assuming that the piston is moved to the left a distance 
a/2 and released, and knowing that the pressure on each side of 
the piston varies inversely with the volume, determine the 
velocity of the piston as it again reaches the center of the 
cylinder. Neglect friction between the piston and the cylinder 
and express your answer in terms of m, a, p, and A. 
 
SOLUTION 
Pressures vary inversely as the volume 
 
 
(2 ) (2 )
L
L
R
R
p Aa pa
p
P Ax x
p Aa pa
p
P A a x a x
= =
= =
− −
 
Initially at M, 
1
0
2
 0
a
v x
T
= =
=
 
At N, 22
1
, 
2
x a T mv= = 
 
1 2
/2 /2
1 2 /2
1 2
2
2
1 2
2
1 1 2 2
1 1
( )
2
[ln ln (2 )]
3
ln ln ln ln
2 2
3 4
ln ln ln
4 3
4 1
0 ln
3 2
a a
L R
a a
a
a
U p p Adx paA dx
x a x
U paA x a x
a a
U paA a a
a
U paA a paA
T U T paA mv
−
−
−
−
−
ª º= − = −« »−¬ ¼
= + −
ª º§ · § ·= + − −« »¨ ¸ ¨ ¸
© ¹ © ¹¬ ¼
ª º § ·= − =« » ¨ ¸
© ¹¬ ¼
§ ·+ = + =¨ ¸
© ¹
³ ³
 
 
( )432 2 ln 0.5754paA paAv
m m
= = 0.759 paAv
m
= W 
 
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549 
 
PROBLEM 13.33 
An uncontrolled automobile traveling at 65 mph strikes squarely a highway crash cushion of the type shown 
in which the automobile is brought to rest by successively crushing steel barrels. The magnitude F of the force 
required to crush the barrels is shown as a function of the distance x the automobile has moved into the 
cushion. Knowing that the weight of the automobile is 2250 lb and neglecting the effect of friction, determine 
(a) the distance the automobile will move into the cushion before it comes to rest, (b) the maximum 
deceleration of the automobile. 
 
 
SOLUTION 
(a) 65 mi/h 95.3 ft/s= 
 Assume auto stops in 5 14 ft.d≤ ≤ 
 
1
2 2
1 1 2
1
2
2
1 2
1 1 2 2
95.33 ft/s
1 1 2250 lb
(95.3 ft/s)
2 2 32.2 ft/s
317,530 lb ft
317.63 k ft
0
0
 (18 k)(5 ft) (27 k)( 5)
 90 27 135
27 45 k ft
317.53 27 45
v
T mv
T
v
T
U d
d
d
T U T
d
−
−
=
§ ·= ¨ ¸
© ¹
= ⋅
= ⋅
=
=
= + −
= + −
= − ⋅
+ =
= −
 
 13.43 ftd = W 
 Assumption that 14 ftd ≤ is ok. 
(b) Maximum deceleration occurs when F is largest. For 13.43 ft, 27 k.d F= = Thus, DF ma= 
 
2
2250 lb
(27,000 lb) ( )
32.2 ft/s
Da
§ ·= ¨ ¸
© ¹
 2386 ft/sDa = W�
 
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550 
 
 
PROBLEM 13.34 
Two types of energy-absorbing fenders designed to be used on a 
pier are statically loaded. The force-deflection curve for each type 
of fender is given in the graph. Determine the maximum 
deflection of each fender when a 90-ton ship moving at 1 mi/h 
strikes the fender and is brought to rest. 
 
SOLUTION 
Weight: 31 (90 ton)(2000 lb/ton) 180 10 lbW = = × 
Mass: 
3
2180 10 5590 lb s /ft
32.2
W
m
g
×= = = ⋅ 
Speed: 1
5280 ft
1 mi/h 1.4667 ft/s
3600 s
v = = = 
Kinetic energy: 2 21 1
2
1 1
(5590)(1.4667)
2 2
6012 ft lb
0 (rest)
T mv
T
= =
= ⋅
=
 
Principle of work and energy: 1 1 2 2T U T→+ = 
 1 2
1 2
6012 0
6012 ft lb 72.15 kip in.
U
U
→
→
+ =
= − ⋅ = − ⋅
 
The area under the force-deflection curve up to the maximum deflection is equal to 72.15 kip in.⋅ 
Fender A: From the force-deflection curve max
max
60
5 kip/in.
12
F
F kx k
x
= = = = 
 2
0 0
1
Area
2
x x
fdx kx dx kx= = =³ ³ 
 2
1
(5) 72.51
2
x = 
 2 228.86 in.x = 5.37 in.x = W 
Fender B: We divide area under curve B into trapezoids 
 Partial area Total Area 
From 0x = to 2 in.:x = 1 (2 in.)(4 kips) 4 kip in.
2
= ⋅ 4 kip in.⋅ 
From 2 in.x = to 4 in.:x = 1 (2 in.)(4 10) 14 kip in.
2
+ = ⋅ 18 kip in.⋅ 
From 4 in.x = to 6 in.:x = 1 (2 in.)(10 18) 28 kip in.
2
+ = ⋅ 46 kip in.⋅ 
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551 
PROBLEM 13.34 (Continued) 
 
We still need 72.1546 26.15 kip in.U∆ = − = ⋅ 
Equation of straight line approximating curve B from 6 in.x = to 8 in.x = is 
 
18
 18 6
2 30 18
x F
F x
∆ −= = + ∆
−
 
 2
1
18 (6 ) 26.15 kip in.
2
( ) 6 8.716 0
1.209 in.
U x x x
x x
x
∆ = ∆ + ∆ ∆ = ⋅
∆ + ∆ − =
∆ =
 
 Thus: 6 in. 1.209 in. 7.209 in.x = + = 7.21 in.x = W 
 
 
 
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552 
 
PROBLEM 13.35 
Nonlinear springs are classified as hard or soft, depending upon the curvature of their force-deflection curve 
(see figure). If a delicate instrument having a mass of 5 kg is placed on a spring of length l so that its base is 
just touching the undeformed spring and then inadvertently released from that position, determine the 
maximum deflection mx of the spring and the maximum force mF exerted by the spring, assuming (a) a linear 
spring of constant k = 3 kN/m, (b) a hard, nonlinear spring, for which 2(3 kN/m)( 160 )F x x= + . 
 
 
SOLUTION 
(5 kg)
49.05 N
W mg g
W
= =
=
 
 
 
Since 1 2 1 1 2 2 1 20, yields 0T T T U T U− −= = + = = 
 1 2
0 0
49.05 0
m mx x
m mU Wx Fdx x Fdx− = − = − =³ ³ (1) 
(a) For (300 N/m)F kx x= = 
 Eq. (1): 
0
49.05 3000 0
mx
mx x dx− =³ 
 249.05 1500 0m mx x− = 
332.7 10 m 32.7 mmmx
−= × = W 
 33000 3000(32.7 10 )m mF x
−= = × 98.1 Nm =F W 
(b) For 2(3000 N/m) (1 160 )F x x= + 
 Eq. (1) 3
0
49.05 3000( 160 ) 0
mx
mx x x dx− + =³ 
 2 4
1
49.05 3000 40 0
2m m m
x x x§ ·− + =¨ ¸
© ¹
 (2) 
 Solve by trial: 330.44 10 mmx
−= × 30.4 mmmx = W 
 3 3 2(3000)(30.44 10 )[1 160(30.44 10 ) ]mF
− −= × + × 104.9 Nm =F W 
 
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553 
PROBLEM 13.36 
A rocket is fired vertically from the surface of the moon with a speed 0.v Derive a formula for the ratio /n uh h 
of heights reached with a speed v, if Newton’s law of gravitation is used to calculate nh and a uniform 
gravitational field is used to calculate .uh Express your answer in terms of the acceleration of gravity mg on 
the surface of the moon, the radius mR of the moon, and the speeds v and 0.v 
 
SOLUTION 
Newton’s law of gravitation 
 
2 2
1 0 2
2
1 2 2
2
1 2 2
2
1 2
1 1 2 2
2 2
0
1 1
2 2
( )
1 1
1 1
2 2
m n
m
m n
m
R h
m m
n n
R
R h
m m
R
m m
m m n
m
m m
m n
T mv T mv
mg R
U F dr F
r
dr
U mg R
r
U mg R
R R h
T U T
R
mv mg R mv
R h
+
−
+
−
−
−
= =
= − =
= −
§ ·
= −¨ ¸+© ¹
+ =
§ ·
+ − =¨ ¸+© ¹
³
³
 
 
( ) ( )
2 2
0
2 2
0
2
2
gm
m
v vn
m m
Rv v
h
g R
−
ª º− « »= « »−« »¬ ¼
 (1) 
Uniform gravitational field 
 
2 2
1 0 2
1 2
2 2
1 1 2 2 0
1
2
( ) ( )
1 1
2 2
m n
m
R h
u m m u m u
R
m u
T mv T mv
U F dr mg R h R mgh
T U T mv mg h mv
+
−
−
= =
= − = − + − = −
+ = − =
³ 
 
( )2 20
2u m
v v
h
g
−
= (2) 
Dividing (1) by (2) ( )2 20
(2 )
1
1
m m
n
v v
u
g R
h
h −
=
−
 W 
 
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554 
 
PROBLEM 13.37 
Express the acceleration of gravity hg at an altitude h above the surface of the earth in terms of the 
acceleration of gravity 0g at the surface of the earth, the altitude h and the radius R of the earth. Determine 
the percent error if the weight that an object has on the surface of earth is used as its weight at an altitude of 
(a) 1 km, (b) 1000 km. 
 
SOLUTION 
 
( )
2
2 2
 /
( ) 1
E E
h
h
R
GM m GM m R
F mg
h R
= =
+ +
 
At earth’s surface, (h = 0) 02
EGM m mg
R
= 
 
2
02 2
1
GME
E R
h
h
R
GM
g g
R § ·+¨ ¸
© ¹
= = 
Thus, 0
2
1
6370 km
h
h
R
g
g
R
§ ·
+¨ ¸
© ¹
=
=
 
At altitude h, “true” weight h TF mg W= = 
Assumed weight 0 0W mg= 
 
( )
( )
( )
0 0 0
0 0 0
0
0 2
0
22
0
Error
1 1
1
11
T h h
g
h
R
h hh RR
W W mg mg g g
E
W mg g
g
g
g E
g
− − −= = = =
−
+ ª º−= = = « »
+« »+ ¬ ¼
 
(a) h = 1 km: ( )216370
1
1100 100
1
P E
ª º−= = « »
+« »¬ ¼
 P = 0.0314% W 
(b) h = 1000 km: ( )210006370
1
1100 100
1
P E
ª º−= = « »
+« »¬ ¼
 P = 25.3% W 
 
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555 
 
 
PROBLEM 13.38 
A golf ball struck on earth rises to a maximum height of 
60 m and hits the ground 230 m away. How high will the 
same golf ball travel on the moon if the magnitude and 
direction of its velocity are the same as they were on earth 
immediately after the ball was hit? Assume that the ball is 
hit and lands at the same elevation in both cases and that the 
effect of the atmosphere on the earth is neglected, so that the 
trajectory in both cases is a parabola. The acceleration of 
gravity on the moon is 0.165 times that on earth. 
 
SOLUTION 
 
Solve for .mh 
At maximum height, the total velocity is the horizontal component of the velocity, which is constant and the 
same in both cases. 
 
2 2
1 2
1 2
1 2
1 1
2 2
Earth
U Moon
H
e e
m m
T mv T mv
U mg h
mg h
−
−
= =
= −
= −
 
Earth 2 2
1 1
2 2e e H
mv mg h mv− = 
Moon 2 2
1 1
2 2m m H
mv mg h mv− = 
Subtracting 
0
(60 m)
0.165
m e
e e m m
e m
e
m
e
h g
g h g h
h g
g
h
g
− + = =
§ ·
= ¨ ¸
© ¹
 364 mmh = W 
 
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556 
 
 
PROBLEM 13.39 
The sphere at A is given a downward velocity 0v of 
magnitude 5 m/s and swings in a vertical plane at the 
end of a rope of length 2 ml = attached to a support at O. 
Determine the angle θ at which the rope will break, 
knowing that it can withstand a maximum tension equal 
to twice the weight of the sphere. 
 
SOLUTION 
 
2 2
1 0
1
2
2
1 2
1 1
(5)
2 2
12.5 m
1
2
( )sin
T mv m
T
T mv
U mg l θ−
= =
=
=
=
 
 21 1 2 2
1
12.5 2 sin
2
T U T m mg mvθ−+ = + = 
 225 4 sing vθ+ = (1) 
Newton’s law at N. 
 
2 2
2
2 sin
2
4 2 sin
v v
mg mg m m
v g g
θ
θ
− = =
= −
A
 (2) 
Substitute for 2v from Eq. (2) into Eq. (1) 
 
25 4 sin 4 2 sin
(4)(9.81) 25
sin 0.2419
(6)(9.81)
g g gθ θ
θ
+ = −
−= = 14.00θ = ° W 
 
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557 
 
 
PROBLEM 13.40 
The sphere at A is given a downward velocity 0v and swings 
in a vertical circle of radius l and center O. Determine the 
smallest velocity 0v for which the sphere will reach Point B as 
it swings about Point O (a) if AO is a rope, (b) if AO is a 
slender rod of negligible mass. 
 
SOLUTION 
 
 
 
 
 
 
2
1 0
2
2
1 2
2 2
1 1 2 2 0
2 2
0
1
2
1
2
1 1
2 2
2
T mv
T mv
U mgl
T U T mv mgl mv
v v gl
−
−
=
=
= −
+ = − =
= +
 
Newton’s law at N 
(a) For minimum v, tension in the cord must be zero. 
 Thus, 2v gl= 
 2 20 2 3v v gl gl= + = 0 3v gl= W 
(b) Force in the rod can support the weight so that v can be zero. 
 Thus, 20 0 2v gl= + 0 2v gl= W 
 
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558 
 
 
PROBLEM 13.41 
A small sphere B of weight W is released from rest in the position shown 
and swings freely in a vertical plane, first about O and then about the peg A 
after the cord comes in contact with the peg. Determine the tension in the 
cord (a) just before the sphere comes in contact with the peg, (b) just after 
it comes in contact with the peg. 
 
SOLUTION 
Velocity of the sphere as the cord contacts A 
 
2
0 0
1
2
( )(1)
B B
C C
B C
B B C C
v T
T mv
U mg
T U T
−
−
= =
=
=
+ =
 
 2
1
0 1
2 C
mg mv+ = 
 2 (2)( )Cv g= 
Newton’s law 
(a) Cord rotates about Point ( )O R L= 
 
 
2
(cos60 ) C
v
T mg m
L
− ° = 
 
(2)
(cos 60 )
2
m g
T mg= ° + 
 
3
2
T mg= 1.5 T W= W 
 
 
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559 
PROBLEM 13.41 (Continued) 
 
(b) Cord rotates about 
2
L
A R§ ·=¨ ¸
© ¹
 
 
 
2
2
(cos 60 ) C
L
mv
T mg− ° = 
 
(2)( )
2 1
mg m g
T = + 
 
1 5
2
2 2
T mg mg§ ·= + =¨ ¸
© ¹
 2.5T W= W�
 
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560 
 
 
PROBLEM 13.42 
A roller coaster starts from rest at A, rolls down the track to B, 
describes a circular loop of 40-ft diameter, and moves up and 
down past Point E. Knowing that h = 60 ft and assuming no 
energy loss due to friction, determine (a) the force exerted by his 
seat on a 160-lb rider at B and D, (b) the minimum value of the 
radius of curvature at E if the roller coaster is not to leave the 
track at that point. 
 
SOLUTION 
Let yp be the vertical distance from Point A to any Point P on the track. Let position 1 be at A and position 
2 be at P. Apply the principle of work and energy. 
 1 0T = 
2
2
1
2 P
T mv= 
 1 2 PU mgy→ = 
 
2
1 1 2 2
2
1
: 0
2
2
P P
P P
T U T mgy mv
v gy
→+ = + =
=
 
Magnitude of normal acceleration at P: 
 
2 2
( ) P PP n
P P
v gy
a
ρ ρ
= = 
(a) Rider at Point B. 
 
60 ft
20 ft
(2 )(60)
6
20
B
B
n
y h
r
g
a g
ρ
= =
= =
= =
 
 :F maΣ = 
 
(6 )
7 7 (7)(160 lb)
B
B
N mg m g
N mg W
− =
= = =
 
 1120 lbB =N W 
 
 
 
 
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561 
PROBLEM 13.42 (Continued) 
 
 Rider at Point D. 
 
2 20 ft
20 ft
(2 )(20)
2
20
D
D
n
y h r
g
a g
ρ
= − =
=
= =
 
 :F maΣ = 
 
(2 )
160 lb
D
D
N mg m g
N mg W
+ =
= = =
 
 160 lbD =N W 
(b) Car at Point E. 40 ft
0
E
E
y h r
N
= − =
=
 
 :
2
n
E
E
F ma
gy
mg m
ρ
Σ =
= ⋅
 
 2E Eyρ = 80.0 ftρ = W 
 
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562 
 
 
PROBLEM 13.43 
In Problem 13.42, determine the range of values of h for which 
the roller coaster will not leave the track at D or E, knowing 
that the radius of curvature at E is 75 ft.ρ = Assume no 
energy loss due to friction. 
PROBLEM 13.42 A roller coaster starts from rest at A, rolls 
down the track to B, describes a circular loop of 40-ft diameter, 
and moves up and down past Point E. Knowing that h = 60 ft 
and assuming no energy loss due to friction, determine (a) the 
force exerted by his seat on a 160-lb rider at B and D, (b) the 
minimum value of the radius of curvature at E if the roller 
coaster is not to leave the track at that point. 
 
SOLUTION 
Let yp be the vertical distance from Point A to any Point P on the track. Let position 1 be at A and position 
2 be at P. Apply the principle of work and energy. 
 1 0T = 
2
2
1
2 P
T mv= 
 1 2 PU mg y→ = 
 
2
1 1 2 2
2
1
: 0
2
2
p P
P P
T U T mgy mv
v gy
→+ = + =
=
 
Magnitude of normal acceleration of P: 
 
2 2
( ) P PP n
P P
v gy
a
ρ ρ
= = 
The condition of loss of contact with the track at P is that the curvature of the path is equal to ρp and the 
normal contact force 0.PN = 
Car at Point D. 
 
20 ft
2
2 ( 2 )
( )
D
D
D n
r
y h r
g h r
a
r
ρ = =
= −
−=
 
 F maΣ = 
 
2 ( 2 )
2 5
D
D
g h r
N mg m
r
h r
N mg
r
−+ =
−=
 
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563 
PROBLEM 13.43 (Continued) 
 
For 0DN > 
 
2 5 0
5
50 ft
2
h r
h r
− >
> =
 
Car at Point E. 
 
 
 
75 ft
20 ft
2 ( 20)
( )
75
E
E
E n
y h r h
g h
a
ρ ρ= =
= − = −
−=
 
 F maΣ = 
 
2 ( 20)
75
115 2
75
E
E
mg h
N mg
h
N mg
−− = −
−=
 
For 0, 115 2 0
57.5 ft
EN h
h
> − >
<
 
Range of values for h: 50.0 ft 57.5 fth≤ ≤ W 
 
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564 
 
 
PROBLEM 13.44 
A small block slides at a speed v on a horizontal surface. Knowing 
that h = 0.9 m, determine the required speed of the block if it is to 
leave the cylindrical surface BCD when θ = 30°. 
 
SOLUTION 
At Point C where the block leaves the surface BCD the contact force is reduced to zero. Apply Newton’s 
second law at Point C. 
 
n-direction: 
2
cos Cn
mv
N mg ma
h
θ− = − = − 
With N = 0, we get 2 coscv gh θ= 
Apply the work-energy principle to the block sliding over the path BC. Let position 1 correspond to Point B 
and position 2 to C. 
 21
1
2 B
T mv= 22
1 1
cos
2 2C
T mv mgh θ= = 
 1 2 weight change in vertical distance
(1 cos )
U
mgh θ
→ = ×
= −
 
 
2
1 1 2 2
2
1 1
: (1 cos ) cos
2 2
cos 2 (1 cos ) (3cos 2)
B
B
T U T mv mg mgh
v gh gh gh
θ θ
θ θ θ
→+ = + − =
= − − = −
 
Data: 2
2 2 2
9.81 m/s , 0.9 m, 30 .
(9.81)(0.9)(3cos30 2) 5.2804 m /sB
g h
v
θ= = = °
= ° − =
 
 2.30 m/sBv = W 
 
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565 
 
 
PROBLEM 13.45 
A small block slides at a speed 8v = ft/s on a horizontal surface at a 
height 3h = ft above the ground. Determine (a) the angleθ at which it 
will leave the cylindrical surface BCD, (b) the distance x at which it 
will hit the ground. Neglect friction and air resistance. 
 
SOLUTION 
 
Block leaves surface at C when the normal force 0.N = 
 
2
cos
cos
n
C
mg ma
v
g
h
θ
θ
=
=
 
 2 cosCv gh gyθ= = (1) 
Work-energy principle. 
(a) 2 2
1 1
(8) 32
2 2B
T mv m m= = = 
 
21 ( ) ( )
2C C B C C
B B C C
T mv U W h g mg h y
T U T
−
−
= = − = −
+ =
 
 2
1
32 ( )
2 C
m mg h y mv+ − = 
 Use Eq. (1) 
1
32 ( )
2C C
g h y gy+ − = (2) 
 ( )32
3
2
3
32
2
(32 )
(32 (32.2)(3))
(32.2)
C
C
C
gh gy
gh
y
g
y
+ =
+=
+=
 
 2.6625 ftCy = (3) 
 
2.6625
cos cos 0.8875
3
C
C
y
y h
h
θ θ= = = = 27.4θ = ° W�
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566 
PROBLEM 13.45 (Continued) 
 
(b) From (1) and (3) 
 (32.2)(2.6625)
9.259 ft/s
C
C
C
v gy
v
v
=
=
=
 
At C: ( ) cos (9.259)(cos 27.4 ) 8.220 ft/sC x Cv v θ= = ° = 
 ( ) sin (9.259)(sin 27.4 ) 4.261 ft/sC y Cv v θ= − = − ° = 
 2 2
1
( ) 2.6625 4.261 16.1
2C C y
y y v t gt t t= + − = − − 
At E: 20: 0.2647 0.1654 0Ey t t= + − = 
 0.2953 st = 
At E: (sin ) ( ) (3)(sin 27.4 ) (8.220)(0.2953)C xx h v tθ= + = ° + 
 1.381 2.427 3.808 ftx = + = 
3.81 ftx = W 
 
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567 
 
 
PROBLEM 13.46 
A chair-lift is designed to transport 1000 skiers per hour from the base 
A to the summit B. The average mass of a skier is 70 kg and the average 
speed of the lift is 75 m/min. Determine (a) the average power required, 
(b) the required capacity of the motor if the mechanical efficiency is 
85 percent and if a 300 percent overload is to be allowed. 
 
SOLUTION 
Note: Solution is independent of speed. 
(a) Average power 
2(1000)(70 kg)(9.81 m/s )(300 m) N m
57,225
3600 s s
U
t
∆ ⋅= = =
∆
 
 Average power 57.2 kW= W 
(b) Maximum power required with 300% over load 
 
100 300
(57.225 kW) 229 kW
100
+= = 
 Required motor capacity (85% efficient) 
 Motor capacity 
229 kW
269 kW
0.85
= = W 
 
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568 
 
PROBLEM 13.47 
It takes 15 s to raise a 1200-kg car and the supporting 300-kg hydraulic 
car-lift platform to a height of 2.8 m. Determine (a) the average output 
power delivered by the hydraulic pump to lift the system, (b) the average 
power electric required, knowing that the overall conversion efficiency 
from electric to mechanical power for the system is 82 percent. 
 
SOLUTION 
(a) ( ) ( )( ) ( )( )( )P A A C L AP F v m m g v= = + 
 s/t (2.8 m)/(15 s) 0.18667 m/sAv = = = 
 2 3( ) [(1200 kg) (300 kg)](9.81 m/s )(0.18667 m/s)P AP = + 
 ( ) 2.747 kJ/sP AP = ( ) 2.75 kWP AP = W 
(b) ( ) ( )/ (2.75 kW)/(0.82)E A PP P η= = ( ) 3.35 kWE AP = W 
 
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569 
 
 
PROBLEM 13.48 
The velocity of the lift of Problem 13.47 increases uniformly from zero to 
its maximum value at mid-height 7.5 s and then decreases uniformly to 
zero in 7.5 s. Knowing that the peak power output of the hydraulic pump is 
6 kW when the velocity is maximum, determine the maximum life force 
provided by the pump. 
PROBLEM 13.47 It takes 15 s to raise a 1200-kg car and the supporting 
300-kg hydraulic car-lift platform to a height of 2.8 m. Determine 
(a) the average output power delivered by the hydraulic pump to lift the 
system, (b) the average power electric required, knowing that the overall 
conversion efficiency from electric to mechanical power for the system is 
82 percent. 
 
SOLUTION 
Newton’s law 
 
 
( ) (1200 300)
1500
C LMg M M g g
Mg g
= + = +
=
 
 1500 1500F F g aΣ = − = (1) 
Since motion is uniformly accelerated, a = constant 
Thus, from (1), F is constant and peak power occurs when the velocity is a maximum at 7.5 s. 
 max
7.5 s
v
a = 
 max
max
(6000 W) ( )( )
(6000)/
P F v
v F
= =
=
 
Thus, (6000)/(7.5)( )a F= (2) 
Substitute (2) into (1) 
 1500 (1500)(6000)/(7.5)( )F g F− = 
 2 2
(1500 kg)(6000 N m/s)
(1500 kg)(9.81 m/s ) 0
(7.5 s)
F F
⋅− − = 
 
2 614,715 1.2 10 0
14,800 N
F F
F
− − × =
= 14.8 kNF = W 
 
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570 
 
PROBLEM 13.49 
(a) A 120-lb woman rides a 15-lb bicycle up a 3-percent slope at a constant speed of 5 ft/s. How much power 
must be developed by the woman? (b) A 180-lb man on an 18-lb bicycle starts down the same slope and 
maintains a constant speed of 20 ft/s by braking. How much power is dissipated by the brakes? Ignore air 
resistance and rolling resistance. 
 
 
SOLUTION 
 
 
3
tan 1.718
100
θ θ= = ° 
(a) 15 120B WW W W= + = + 
 135 lbW = 
 ( sin )( )WP W vθ= ⋅ =W v 
 (135)(sin 1.718 )(5)WP = ° (a) 
 20.24 ft lb/sWP = ⋅ 
20.2 ft lb/sWP = ⋅ W 
(b) 18 180B mW W W= + = + 
 198 lbW = 
 Brakes must dissipate the power generated by the bike and the man going down the slope at 20 ft/s. 
 ( sin )( )BP W vθ= ⋅ =W v 
 (198)(sin 1.718)(20)BP = 
118.7 ft lb/sBP = ⋅ W�
�
�
�
�
 (b) 
 
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571 
 
PROBLEM 13.50 
A power specification formula is to be 
derived for electric motors which drive 
conveyor belts moving solid material at 
different rates to different heights and 
distances. Denoting the efficiency of the 
motors by η and neglecting the power 
needed to drive the belt itself, derive a 
formula (a) in the SI system of units for 
the power P in kW, in terms of the mass 
flow rate m in kg/h, the height b and 
horizontal distance l in meters, and (b) in 
U.S. customary units, for the power in 
hp, in terms of the material flow rate w 
in tons/h, and the height b and horizontal 
distance l in feet. 
 
SOLUTION 
(a) Material is lifted to a height b at a rate, 2( kg/h)( m/s ) [ (N/h)]m g mg= 
 Thus, 
 
[ (N/h)][ ( )]
N m/s
(3600 s/h) 3600
U mg b m mgb
t
∆ § ·= = ⋅¨ ¸∆ © ¹
 
 1000 N m/s 1 kW⋅ = 
 Thus, including motor efficiency, η 
 
(N m/s)
(kw)
1000 N m/s
(3600) ( )
kW
mgb
P
η
⋅=
⋅§ ·
¨ ¸
© ¹
 
 6(kW) 0.278 10
mgb
P
η
−= × W 
(b) 
[ (tons/h)(2000 lb/ton)][ (ft)]
3600 s/h
U W b
t
∆ =
∆
 
 ft lb/s; 1hp 550 ft lb/s
1.8
Wb= ⋅ = ⋅ 
 With ,η 1 hp 1(ft lb/s)
1.8 550 ft lb/s
Wb
hp
η
ª º ª ºª º= ⋅ « » « »« » ⋅¬ ¼ ¬ ¼¬ ¼
 
 
31.010 10 Wb
hp
η
−×= W 
 
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572 
 
 
PROBLEM 13.51 
In an automobile drag race, the rear (drive) wheels of 
a 1000 kg car skid for the first 20 m and roll with sliding 
impending during the remaining 380 m. The front wheels of 
the car are just off the ground for the first 20 m, and for the 
remainder of the race 80 percent of the weight is on the rear 
wheels. Knowing that the coefficients of friction are 
0.90sµ = and 0.68,kµ = determine the power developed by 
the car at the drive wheels (a) at the end of the 20-m portion 
of the race, (b) at the end of the race: Give your answer in kW 
and in hp. Ignore the effect of air resistance and rolling 
friction. 
 
SOLUTION 
(a) First 20 m. (Calculate velocity at 20 m.) Force generated by rear wheels ,kWµ= since car skids. 
 Thus, (0.68)(1000)( )sF g= 
 2(0.68)(1000 kg)(9.81 m/s ) 6670.8 NsF = = 
 Work and energy. 2 21 2 20 20
1
0, 500
2
T T mv v= = = 
 1 1 2 2T U T−+ = 
 1 2 (20 m)( ) (20 m)(6670.8 N)sU F− = = 
 1 2 133,420 JU − = 
 2200 133,420 500v+ = 
 220
133,420
266.83
500
v = = 
 20 16.335 m/sv = 
 Power 20( )( ) (6670.8 N)(16.335 m/s)sF v= = 
 Power 108,970 J/s 108.97 kJ/s= = 
 1 kJ/s 1 kW= 
 1 hp 0.7457 kW= Power 109.0 kJ/s 109.0 kW= = W 
 Power 
(109.0 kW)
146.2 hp
(0.7457 kW/hp)
= = W 
�
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573 
PROBLEM 13.51 (Continued) 
 
(b) End of race. (Calculate velocity at 400 m.) For remaining 380 m, with 80% of weight on rear wheels, 
the force generated at impending sliding is ( )(0.80)( )s mgµ 
 2(0.90)(0.80)(1000 kg)(9.81 m/s )IF = 
 7063.2 NIF = 
 Work and energy, from 20 m N to 28 m O. 
 2 16.335 m/sv = [from part (a)] 
 22
1
(1000 kg)(16.335 m/s)
2
T = 
 2 133,420 JT = 
 2 23 380 380
1
500
2
T mv v= = 
 2 3 ( )(380 m) (7063.2 N)(380 m)IU F− = = 
 2 3 2,684,000 JU − = 
 2 2 3 3T U T−+ = 
 230(133,420 J) (2,684,000 J) 500v+ = 
 30 75.066 m/sv = 
 Power 30( )( ) (7063.2 N)(75.066 m/s)IF v= = 
 530,200 J/s= 
 kW Power 530, 200 J 530 kW= = W 
 hp Power 
530 kW
711 hp
(0.7457 kW/hp)
= = W�
 
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574 
 
PROBLEM 13.52 
The frictional resistance of a ship is known to vary directly as the 1.75 power of the speed v of the ship. 
A single tugboat at full power can tow the ship at a constant speed of 4.5 km/h by exerting a constant force of 
300 kN. Determine (a) the power developed by the tugboat, (b) the maximum speed at which two tugboats, 
capable of delivering the same power, can tow the ship. 
 
SOLUTION 
(a) Power developed by tugboat at 4.5 km/h. 
 0
0
4.5 km/h 1.25 m/s
300 kN
v
F
= =
=
 
 0 0 0 (300 kN)(1.25 m/s)P F v= = 0 375 kWP = W 
(b) Maximum speed. 
 Power required to tow ship at speed v: 
 
1.75 1.75 2.75
0 0 0 0
0 0 0
v v v
F F P Fv F v F v
v v v
§ · § · § ·
= = = =¨ ¸ ¨ ¸ ¨ ¸
© ¹ © ¹ © ¹
 (1) 
 Since we have two tugboats, the available power is twice maximum power 0 0F v developed by one 
tugboat. 
 
2.75
0 0 0 0
0
2.75
1/2.75
0 0
0
2
2 (2) (1.2867)
v
F v F v
v
v
v v v
v
§ ·
= ¨ ¸
© ¹
§ ·
= = =¨ ¸
© ¹
 
 Recalling that 0 4.5 km/hv = 
 (4.5 km/h)(1.2867) 5.7902 km/hv = = 5.79 km/hv = W 
 
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575 
 
PROBLEM 13.53 
A train of total mass equal to 500 Mg starts from rest and accelerates uniformly to a speed of 90 km/h in 50 s. 
After reaching this speed, the train travels with a constant velocity. The track is horizontal and axle friction 
and rolling resistance result in a total force of 15 kN in a direction opposite to the direction of motion. 
Determine the power required as a function of time. 
 
SOLUTION 
Let FP be the driving force and FR be the resisting force due to axle friction and rolling resistance. 
Uniformly accelerated motion. ( 50 s):t < 
 0v v at= + 0 0v =At 50 ,t s= 90 km/h 25 m/sv = = 
 2
2
25 m/s 0 (50)
0.5 m/s
(0.5 m/s )
a
a
v t
= +
=
=
 
Newton’s second law: P RF F ma− = 
where 3
3
2
15 kN 15 10 N
500 Mg 500 10 kg
0.5 m/s
RF
m
a
= = ×
= = ×
=
 
 3 3
3
15 10 (500 10 )(0.5)
265 10 N 265 kN
P RF F ma= + = × + ×
= × =
 
Power: 3(265 10 )(0.5 )PF v t= × 
 (0 50s)< Power (132.5 kW/s)t= W 
Uniform motion. ( 50 s):t > 0a = 
 315 10 N; 25 m/sP RF F v= = × = 
Power: 3 3(15 10 )(25 m/s) 375 10PF v W= × = × 
 ( 50 s)t > Power 375 kW= W 
 
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576 
 
 
PROBLEM 13.54 
The elevator E has a weight of 6600 lbs when fully loaded and is 
connected as shown to a counterweight W of weight of 2200 lb. 
Determine the power in hp delivered by the motor (a) when the elevator 
is moving down at a constant speed of 1 ft/s, (b) when it has an upward 
velocity of 1 ft/s and a deceleration of 20.18 ft/s . 
 
SOLUTION 
(a) Acceleration 0= 
Counterweight 
 
Elevator 
 
Motor 
 
 0: 0y W WF T WΣ = − = 0: 2 6600 0C WF T TΣ = + − = 
 2200 lbWT = 2200 lbCT = 
 Kinematics: 2 , 2 , 2 2 ft/sE C E C C Ex x x x v v= = = =� � 
(2200 lb)(2 ft/s) 4400 lb ft/s 8.00 hpC CP T v= ⋅ = = ⋅ = 
8.00 hpP = W�
(b) 20.18 ft/sEa = , 1 ft/sEv = 
Counterweight 
 
Elevator 
 
 Counterweight: : ( )W W
W
F Ma T W a
g
Σ = − = 
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577 
PROBLEM 13.54 (Continued) 
 
2
2
(2200 lb)(0.18 ft/s )
(2200 lb)
(32.2 ft/s )W
T = + 
2212 lbWT = 
 Elevator 
 2 ( )
E
C W E E
W
F ma T T W a
g
−Σ = + − = 
2
2
(6600 lb)(0.18 ft/s )
2 ( 2212 lb) (6600 lb)
(32.2 ft/s )C
T = − + − 
2 4351 lbCT = 
2175.6 lbCT = 
2 ft/s (see part( ))Cv a= 
(2175.6 lb)(2 ft/s) 4351.2 lb ft/sC CP T v= ⋅ = = ⋅ 
 7.911 hp= 
� � � 7.91 hpP = W 
 
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578 
 
 
PROBLEM 13.CQ2 
Two small balls A and B with masses 2m and m respectively are released 
from rest at a height h above the ground. Neglecting air resistance, which of 
the following statements are true when the two balls hit the ground? 
(a) The kinetic energy of A is the same as the kinetic energy of B. 
(b) The kinetic energy of A is half the kinetic energy of B. 
(c) The kinetic energy of A is twice the kinetic energy of B. 
(d) The kinetic energy of A is four times the kinetic energy of B. 
 
SOLUTION 
Answer: (c) 
 
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579 
 
 
PROBLEM 13.CQ3 
Block A is released from rest and slides down the frictionless ramp to 
the loop. The maximum height h of the loop is the same as the initial 
height of the block. Will A make it completely around the loop without 
losing contact with the track? 
(a) Yes 
(b) No 
(c) need more information 
 
SOLUTION 
Answer: (b) In order for A to not maintain contact with the track, the normal force must remain greater than 
zero, which requires a non-zero speed at the top of the loop. 
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580 
 
PROBLEM 13.55 
A force P is slowly applied to a plate that is attached to two springs and causes a deflection 0.x In each of the 
two cases shown, derive an expression for the constant ,ek in terms of 1k and 2 ,k of the single spring 
equivalent to the given system, that is, of the single spring which will undergo the same deflection 0x when 
subjected to the same force P. 
 
 
SOLUTION 
System is in equilibrium in deflected 0x position. 
Case (a) Force in both springs is the same P= 
 
0 1 2
0
e
x x x
P
x
k
= +
=
 
 1
1
P
x
k
= 2
2
=
P
x
k
 
 Thus, 
1 2e
P P P
k k k
= + 
 
1 2
1 1 1
ek k k
= + 1 2
1 2
e
k k
k
k k
=
+
 W 
 
Case (b) Deflection in both springs is the same 0x= 
 
1 0 2 0
1 2 0
0
( )
e
P k x k x
P k k x
P k x
= +
= +
=
 
 Equating the two expressions for 
 1 2 0 0( ) eP k k x k x= + = 1 2ek k k= + W 
 
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581 
 
 
PROBLEM 13.56 
A loaded railroad car of mass m is rolling at a constant 
velocity v0 when it couples with a massless bumper 
system. Determine the maximum deflection of the bumper 
assuming the two springs are (a) in series (as shown), 
(b) in parallel. 
 
SOLUTION 
Let position A be at the beginning of contact and position B be at maximum deflection. 
 
2
0
2 2
1 1 2 2
1
2
0 (zero force in springs)
0 ( 0 at maximum deflection)
1 1
2 2
A
A
B
B
T mv
V
T v
V k x k x
=
=
= =
= +
 
 where x1 is deflection of spring k1 and x2 is that of spring k2. 
 Conservation of energy: A A B BT V T V+ = + 
 2 2 20 1 1 2 2
1 1 1
0 0
2 2 2
mv k x k x+ = + + 
 2 2 21 1 2 2 0k x k x mv+ = (1) 
(a) Springs are in series. 
 Let F be the force carried by the two springs. 
 Then, 1 2
1 2
and
F F
x x
k k
= = 
 Eq. (1) becomes 2 20
1 2
1 1
F mv
k k
§ ·
+ =¨ ¸
© ¹
 
 so that 0
1 2
1 1
/F v m
k k
§ ·
= +¨ ¸
© ¹
 
 The maximum deflection is 1 2
1 2
0
1 2 1 2
1 1
1 1 1 1
/
x x F
k k
v m
k k k k
δ § ·= + = +¨ ¸
© ¹
§ · § ·
= + +¨ ¸ ¨ ¸
© ¹ © ¹
 
 0
1 2
1 1
v m
k k
§ ·
= +¨ ¸
© ¹
 0 1 2 1 2( )/v m k k k kδ = + W 
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582 
PROBLEM13.56 (Continued) 
 
(b) Springs are in parallel. 
 1 2x x δ= = 
Eq. (1) becomes 2 21 2 0( )k k mvδ+ = 0
1 2
m
v
k k
δ =
+
 W 
 
 
 
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583 
 
 
PROBLEM 13.57 
A 600-g collar C may slide along a horizontal, semicircular rod 
ABD. The spring CE has an undeformed length of 250 mm and a 
spring constant of 135 N/m. Knowing that the collar is released 
from rest at A and neglecting friction, determine the speed of the 
collar (a) at B, (b) at D. 
 
SOLUTION 
First calculate the lengths of the spring when the collar is at positions A, B, and D. 
 
2 2 2
2 2 2
2 2 2
440 300 180 562.14 mm
240 300 20 384.71 mm
40 300 180 352.14 mm
A
B
D
l
l
l
= + + =
= + + =
= + + =
 
The elongations of springs are given by 0.e l l= − 
 
562.14 250 312.14 mm 0.31214 m
384.71 250 134.71 mm 0.13471 m
352.14 250 102.14 mm 0.10214 m
A
B
D
e
e
e
= − = =
= − = =
= − = =
 
Potential energies: 2
1
2
V ke= 
 
2
2
2
1
(135 N/m)(0.31214 m) 6.5767 J
2
1
(135 N/m)(0.13471 m) 1.2249 J
2
1
(135 N/m)(0.10214 m) 0.7042 J
2
A
B
D
V
V
V
= =
= =
= =
 
Since the semicircular rod ABC is horizontal, there is no change in gravitational potential energy. 
Mass of collar: 600 g 0.600 kgm = = 
Kinetic energies: 2 2
2 2
2 2
1
0.300 0
2
1
0.300
2
0.300
2
A A A
B B B
D D D
T mv v
T mv v
T mv v
= = =
= =
1= =
 
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584 
PROBLEM 13.57 (Continued) 
 
(a) Speed of collar at B. 
 Conservation of energy: A A B BT V T V+ = + 
 20 6.5767 0.300 1.2249Bv+ = + 
 2 2 217.839 m /sBv = 4.22 m/sBv = W 
(b) Speed of collar at D. 
 Conservation of energy: A A D DT V T V+ = + 
 200 6.5767 0.300 0.7042v+ = + 
 2 2 219.575 m /sDv = 4.42 m/sDv = W 
 
 
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585 
 
 
PROBLEM 13.58 
A 3-lb collar is attached to a spring and slides without friction 
along a circular rod in a horizontal plane. The spring has an 
undeformed length of 7 in. and a constant 1.5 lb/in.k = 
Knowing that the collar is in equilibrium at A and is given a 
slight push to get it moving, determine the velocity of the 
collar (a) as it passes through B, (b) as it passes through C. 
 
SOLUTION 
 
0 7 in., 20 in.DAL L= = 
2 2(8 6) 6 15.23 in.DBL = + + = 
8 in.DCL = 
20 7 13 in.DAL∆ = − = 
15.23 7 8.23 in.DBL∆ = − = �
8 7 1 in.DCL∆ = − = �
(a) 2 2
1 1
0, ( ) (1.5)(13) 126.75 lb in.
2 2A A DA
T V k L= = ∆ = = ⋅ 
 10.5625 lb ft= ⋅ 
2 21 1.5 
2B B B
T mv v
g
= = 
21 (1.5)(8.23) 50.8 lb in. 4.233 lb ft
2B
V = = ⋅ = ⋅ 
 
21.5 
: 0 10.5625 4.233
32.2
B
A A B B
v
T V T V+ = + + = + 11.66 ft/sBv = W�
 
 
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586 
PROBLEM 13.58 (Continued) 
 
(b) 2
1.5
0, 10.5625 lb ft, 
32.2A A C C
T V T v= = ⋅ = 
 2
1
(1.5)(1) 0.75 lb in. 0.0625 lb ft
2C
V = = ⋅ = ⋅ 
 2
1.5
: 0 10.5625 0.0625
32.2A A C C c
T V T V v+ = + + = + � 15.01 ft/sCv = W 
 
 
 
 
 
 
 
 
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587 
 
 
 
PROBLEM 13.59 
A 3-lb collar C may slide without friction along a horizontal rod. 
It is attached to three springs, each of constant 2 lb/in.k = and 
6 in. undeformed length. Knowing that the collar is released from 
rest in the position shown, determine the maximum speed it will 
reach in the ensuing motion. 
 
SOLUTION 
Maximum velocity occurs at E where collar is passing through position of equilibrium. 
 Position M 
 1 0T = 
Note: Undeformed length of springs is 6 in. = 0.5 ft. 
Spring AC: 2 2(1 ft) (0.5 ft) 1.1180 ftL = + = 
 1.1180 0.50 0.6180 ft∆ = − = 
Spring CD: 2 2(0.5 ft) (0.5 ft) 0.70711 ft
0.70711 0.50 0.20711 ft
L = + =
∆ = − =
 
Spring BD: 0.50 ft, 0L = ∆ = 
Potential energy. ( 2 lb/in. 24 lb/ft for each spring)k = = 
 
2 2 2 2
1
1
1 1 1
(24 lb/ft)[(0.6180 ft) (0.20711 ft) 0]
2 2 2
5.0983 lb ft
V k k
V
= Σ ∆ = Σ∆ = + +
= ⋅
 
 Position N 
2 2
2 2 22
3.0 lb 1 1
0.093168 slug; (0.093168 slug)
2 232.2 ft/s
m T mv v= = = = 
Spring AC: 2 2(0.5 ft) (0.5 ft) 0.7071067 ft
0.70711 0.50 0.20711 ft
L = + =
∆ = − =
 
Spring CD: 0.50 ft
0
L =
∆ =
 
Spring BC: 2 2(0.5 ft) (0.5 ft) 0.7071067 ft
0.70711 0.50 0.20711 ft
L = + =
∆ = − =
 
 
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588 
PROBLEM 13.59 (Continued) 
 
Potential energy. 2 22
2 2
2
1 1
2 2
1
(24 lb/ft)[(0.20711 ft) 0 (0.20711 ft) ] 1.0294 lb ft
2
V k k
V
= Σ ∆ = Σ∆
= + + = ⋅
 
Conservation of energy. 21 1 2 2 2
1
: 0 5.0983 lb ft (0.093168 slug) 1.0294 lb ft
2
T V T V v+ = + + ⋅ = + ⋅ 
 22 87.345v = 2 9.35 ft/s= ↔v W 
 
 
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589 
 
 
PROBLEM 13.60 
A 500-g collar can slide without friction on the curved rod BC in a 
horizontal plane. Knowing that the undeformed length of the spring 
is 80 mm and that 400k = kN/m, determine (a) the velocity that the 
collar should be given at A to reach B with zero velocity, (b) the 
velocity of the collar when it eventually reaches C. 
 
SOLUTION 
(a) Velocity at A: 
 
2 2
2
2
3 2
2 3 2
1 0.5
 kg
2 2
(0.25)
0.150 m 0.080 m
0.070 m
1
( )
2
1
(400 10 N/m)(0.070 m)2
980 J
0 0
0.200 m 0.080 m 0.120 m
1 1
( ) (400 10 N/m)(0.120 m)
2 2
28
A A A
A A
A
A
A A
A
A
B B
B
B B
B
T mv v
T v
L
L
V k L
V
V
v T
L
V k L
V
§ ·= = ¨ ¸
© ¹
=
∆ = −
∆ =
= ∆
= ×
=
= =
∆ = − =
= ∆ = ×
= 80 J
 
 Substitute into conservation of energy. 
 
2
2
2 2 2
 0.25 980 0 2880
(2880 980)
(0.25)
7600 m /s
A A B B A
A
A
T V T V v
v
v
+ = + + = +
−=
=
 
 87.2 m/sAv = W�
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590 
PROBLEM 13.60 (Continued) 
 
(b) Velocity at C: 
Since slope at B is positive, the component of the spring force ,PF parallel to the rod, causes the block 
to move back toward A. 
 
2 2 2
2 3 2
0, 2880 J [from part ( )]
1 (0.5 kg)
0.25
2 2
0.100 m 0.080 m 0.020 m
1 1
( ) (400 10 N/m)(0.020 m) 80.0 J
2 2
B B
C C C C
C
C C
T V a
T mv v v
L
V k L
= =
= = =
∆ = − =
= ∆ = × =
 
 Substitute into conservation of energy. 
 2
2 2 2
0 2880 0.25 80.0
11,200 m /s
B B C C C
C
T V T V v
v
+ = + + = +
=
 
 105.8 m/sCv = W�
 
 
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591 
 
 
PROBLEM 13.61 
An elastic cord is stretched between two Points A and B, located 800 mm apart in 
the same horizontal plane. When stretched directly between A and B, the tension is 
40 N. The cord is then stretched as shown until its midpoint C has moved through 
300 mm to ;C′ a force of 240 N is required to hold the cord at .C′ A 0.1 kg pellet 
is placed at ,C′ and the cord is released. Determine the speed of the pellet as it 
passes through C. 
 
SOLUTION 
Let A = undeformed length of cord. 
Position 1. 1Length 1.0 m; Elongation 1.0AC B x′ = = = − A 
 1 1
3
0: 2 240 N 0 200 N
5x
F F F§ ·Σ = − = =¨ ¸
© ¹
 
 
Position 2. 2
2
Length 0.8 m; Elongation 0.8
Given 40 N
ACB x
F
= = = −
=
A 
 1 2 2,F kx F k x= = 
 1 2 1 2
( )
200 40 [(1.0 ) (0.8 )] 0.2
F F k x x
k k
− = −
− = − − − =A A
 
 
160
800 N/m
0.2
k = = 
 
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592 
PROBLEM 13.61 (Continued) 
 
 11
200 N
0.25 m
800 N/m
F
x
k
= = = 
 22
40 N
0.05 m
800 N/m
F
x
k
= = = 
Position M: 2 21 1 1
1 1
0 (800 N/m)(0.25 m) 25.0 N m
2 2
T V kx= = = = ⋅ 
PositionN: 0.10 kgm = 
 
2 2 2
2 2 2 2
2 2
2 2
1 1
(0.1 kg) 0.05
2 2
1 1
(800 N/m)(0.05 m) 1 N m
2 2
T mv v v
V k x
= = =
= = = ⋅
 
Conservation of energy: 
 
1 1 2 2
2
20 25.0 N m 0.05 1.0 N m
T V T V
v
+ = +
+ ⋅ = + ⋅
 
 22 224.0 0.05 21.909 m/sv v= = 2 21.9 m/sv = W 
Note: The horizontal force applied at the midpoint of the cord is not proportional to the horizontal distance .C C′ 
A solution based on the work of the horizontal force would be rather involved. 
 
 
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593 
 
PROBLEM 13.62 
An elastic cable is to be designed for bungee jumping from a tower 
130 ft high. The specifications call for the cable to be 85 ft long 
when unstretched, and to stretch to a total length of 100 ft when a 
600-lb weight is attached to it and dropped from the tower. 
Determine (a) the required spring constant k of the cable, (b) how 
close to the ground a 186-lb man will come if he uses this cable to 
jump from the tower. 
 
SOLUTION 
(a) Conservation of energy: 
 1 1 10 0 100V T V W= = = 
 Datum at N: 1
4
(100 ft)(600 lb)
6 10 ft lb
V =
= × ⋅
 
 2 20 0V T= = 
 
2
2
1 1 2 2
4
1
0 (15 ft)
2
0 6 10 0 (112.5)
g eV V V k
T V T V
k
= + = +
+ = +
+ × = +
 
� 533 lb/ftk = W 
(b) From (a), 533 lb/ftk = 
 
1
1
2
0
186 lb
(186)(130 )
0
T
W
V d
T
=
=
= −
=
 
Datum: 22
2
2
1
0 (533)(130 85 )
2
(266.67)(45 )
g eV V V d
V d
= + = + − −
= −
 
 
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594 
PROBLEM 13.62 (Continued) 
 
 d = distance from the ground 1 1 2 2T V T V+ = + 
 
2
2
0 (186)(130 ) 0 (266.67)(45 )
266.7 23815 515827 0
d d
d d
+ − = + −
− + =
 
 
2 36.99 ft23815 (23815) 4(266.7)(515827)
52.3 ft(2)(266.7)
d
−
= =
B
 
 Discard 52.3 ft (since the cord acts in compression when rebound occurs). 
 37.0 ftd = W�
 
 
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595 
 
 
 
PROBLEM 13.63 
It is shown in mechanics of materials that the stiffness of an elastic cable is k = AE/L 
where A is the cross sectional area of the cable, E is the modulus of elasticity and L is 
the length of the cable. A winch is lowering a 4000-lb piece of machinery using at a 
constant speed of 3ft/s when the winch suddenly stops. Knowing that the steel cable 
has a diameter of 0.4 in., E = 29 × 106 lb/in2, and when the winch stops L = 30 ft, 
determine the maximum downward displacement of the piece of machinery from the 
point it was when the winch stopped. 
 
SOLUTION 
Mass of machinery: 2
4000
124.22 lb s /ft
32.2
W
m
g
= = = ⋅ 
Let position 1 be the state just before the winch stops and the gravitational potential Vg be equal to zero at this 
state. 
For the cable, 2 2 2(diameter) (0.4 in.) 0.12566 in
4 4
A
π π= = = 
 2 6 2 6(0.12556 in. )(29 10 lb/in ) 3.6442 10 lbAE = × = × 
For 30 ft,L = 
6
33.6442 10 lb 121.47 10 lb/ft
30 ft
AE
k
L
×= = = × 
Initial force in cable (equilibrium): 1 4000 lb.F W= = 
Elongation in position 1: 11 3
4000
0.03293 ft
121.47 10
F
x
k
= = =
×
 
Potential energy: 
2
2 1
1 1
1
2 2
F
V kx
k
= = 
 
2
1 3
(4000 lb)
65.860 ft lb
(2)(121.47 10 lb/ft)
V = = ⋅
×
 
Kinetic energy: 21 1
1
2
T mv= 
 2 21
1
(124.22 lb s /ft)(3 ft/s ) 558.99 ft lb
2
T = ⋅ = ⋅ 
 
 
 
 
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596 
PROBLEM 13.63 (Continued) 
 
Let position 2 be the position of maximum downward displacement. Let x2 be the elongation in this position. 
Potential energy: 22 2 2 1
1
( )
2
V kx W x x= − − 
 
3 2
2 2 2
3 2
2
1
(121.47 10 ) (4000)( 0.03293)
2
60.735 10 4000 131.72
V x x
x x
= × − −
= × − +
 
Kinetic energy: 2 20 (since 0)T v= = 
Principle of work and energy: 1 1 2 2T V T V+ = + 
 
3 2
2 2
3 2
2 2
2
558.99 65.860 60.735 10 4000 131.72
60.735 10 4000 493.13 0
0.12887 ft
x x
x x
x
+ = × − +
× − − =
=
 
Maximum displacement: 2 1 0.09594 ftx xδ = − = 1.151 in.δ = 
 
 
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597 
 
 
 
PROBLEM 13.64 
A 2-kg collar is attached to a spring and slides without friction in 
a vertical plane along the curved rod ABC. The spring is 
undeformed when the collar is at C and its constant is 600 N/m. 
If the collar is released at A with no initial velocity, determine its 
velocity (a) as it passes through B, (b) as it reaches C. 
 
SOLUTION 
Spring elongations: 
 
At , 250 mm 150 mm 100 mm 0.100 m
At , 200 mm 150 mm 50 mm 0.050 m
At , 0
A
B
C
A x
B x
C x
= − = =
= − = =
=
 
Potential energies for springs. 
 
2 2
2 2
1 1
( ) (600)(0.100) 3.00 J
2 2
1 1
( ) (600)(0.050) 0.75 J
2 2
( ) 0
A e A
B e B
C e
V kx
V kx
V
= = =
= = =
=
 
Gravitational potential energies: Choose the datum at level AOC. 
 ( ) ( ) 0A g C gV V= = 
 ( ) (2)(9.81)(0.200) 3.924 JB gV mg y= − = − = − 
Kinetic energies: 0AT = 
 
2 2
2 2
1
1.00
2
1
1.00
2
B B B
C C C
T mv v
T mv v
= =
= =
 
(a) Velocity as the collar passes through B. 
 Conservation of energy: A A B BT V T V+ = + 
 20 3.00 0 1.00 0.75 3.924Bv+ + = + − 
 2 2 26.174 m /sBv = 2.48 m/sB =v W 
 
 
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598 
PROBLEM 13.64 (Continued) 
 
(b) Velocity as the collar reaches C. 
 Conservation of energy: A A C CT V T V+ = + 
 20 3.00 0 1.00 0 0Cv+ + = + + 
 2 2 23.00 m /sCv = 1.732 m/sC =v W 
 
 
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599 
 
 
PROBLEM 13.65 
A 1-kg collar can slide along the rod shown. It is attached to an 
elastic cord anchored at F, which has an undeformed length of 
250 mm and a spring constant of 75 N/m. Knowing that the 
collar is released from rest at A and neglecting friction, 
determine the speed of the collar (a) at B, (b) at E. 
 
SOLUTION 
 2 2 2
2 2
2 2
(0.5) (0.4) (0.3)
0.70711 m
(0.4) (0.3)
0.5 m
(0.5) (0.3)
0.58309 m
AF
AF
BF
BF
FE
FE
e g
L
L
L
L
L
L
V V V
= + +
=
= +
=
= +
=
= +
 
(a) Speed at B: 0, 0A Av T= = 
 Point A: 
 2 0
1
( ) ( ) 0.70711 0.25
2A e AF AF AF
V k L L L L= ∆ ∆ = − = − 
 0.45711 mAFL∆ = 
 
2
2
1
( ) (75 N/m)(0.45711 m)
2
( ) 7.8355 N m
( ) ( )(0.4)
(1.0 kg)(9.81 m/s )(0.4 m)
3.9240 N m
( ) ( )
7.8355 3.9240
11.7595 N m
A e
A e
A g
A A e A g
V
V
V mg
V V V
=
= ⋅
=
=
= ⋅
= +
= +
= ⋅
 
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600 
PROBLEM 13.65 (Continued) 
 
 Point B: 2 2
2
1 1
(1.0 kg)
2 2
0.5
B B B
B B
T mv v
T v
= =
=
 
 2 0
1
( ) ( ) 0.5 0.25
2B e BF BF BF
V k L L L L= ∆ ∆ = − = − 
 0.25 mBFL∆ = 
 
2
2
1
( ) (75 N/m)(0.25 m) 2.3438 N m
2
( ) ( )(0.4) (1.0 kg)(9.81 m/s )(0.4 m) 3.9240 N m
( ) ( ) 2.3438 3.9240 6.2678 N m
B e
B g
B B e B g
V
V mg
V v V
= = ⋅
= = = ⋅
= + = + = ⋅
 
 
2
2
2 2 2
0 11.7595 0.5 6.2678
(5.49169)/(0.5)
10.983 m /s
A A B B
B
B
B
T V T V
v
v
v
+ = +
+ = +
=
= 3.31 m/sBv = W 
(b) Speed at E: 
 Point A: 0 11.7595 N mA AT V= = ⋅ (from part (a)) 
 Point E: 
 2 2 2
2
0
1 1
(1.0 kg) 0.5
2 2
1
( ) ( ) 0.5831 0.25
2
E E E E
E e FE FE FE
T mv v v
V k L L L L
= = =
= ∆ ∆ = − = −
 
 0.3331 mFEL∆ = 
 
2
2
2
2 2 2
1
( ) (75 N/m)(0.3331 m) 4.1607 N m
2
( ) 0 4.1607 N m
0 11.7595 0.5 4.1607
7.5988/0.5
15.1976 m /s
E e
E g E
A A E E E
E
E
V
V V
T V T V v
v
v
= = ⋅
= = ⋅
+ = + + = +
=
= 3.90 m/sEv = W �
 
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601 
 
 
PROBLEM 13.66 
A thin circular rod is supported in a vertical plane by a bracket 
at A. Attached to the bracket and loosely wound around the rod is a 
spring of constant 3 lb/ftk = and undeformed length equal to the 
arc of circle AB. An 8-oz collar C, not attached to the spring, can 
slide without friction along the rod. Knowing that the collar is 
released from rest at an angle θ with the vertical, determine 
(a) the smallest value of θ for which the collar will pass through D 
and reach Point A, (b) the velocity of the collar as it reaches Point A. 
 
SOLUTION 
(a) Smallest angle θ occurs when the velocity at D is close to zero. 
 
0 0
0 0
C D
C D
e g
v v
T T
V V V
= =
= =
= +
 
 Point C: 
 2
2
(1 ft)( ) ft
1
( ) ( )
2
3
( )
2
BC
C e BC
C e
L
V k L
V
θ θ
θ
∆ = =
= ∆
=
 12 in. 1 ftR = = 
 ( ) (1 cos )
8 oz
( ) (1 ft)(1 cos )
16 oz/lb
C g
C g
V WR
V
θ
θ
= −
§ ·= −¨ ¸
© ¹
 
 
2
1
( ) (1 cos )
2
3 1
( ) ( ) (1 cos )
2 2
C g
C C e C g
V
V V V
θ
θ θ
= −
= + = + −
 
 Point D: 
 
2
( ) 0 (spring is unattached)
( ) (2 ) (2)(0.5 lb)(1 ft) 1 lb ft
3 1
0 (1 cos ) 1
2 2
D e
D g
C C D D
V
V W R
T V T V θ θ
=
= = = ⋅
+ = + + + − =
 
 2(1.5) (0.5)cos 0.5θ θ− = 
 By trial, 0.7592 radθ = 43.5θ = ° W 
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602 
PROBLEM 13.66 (Continued) 
 
(b) Velocity at A: 
 Point D: 
 0 0 1 lb ft[see Part ( )]D D DV T V a= = = ⋅ �
 Point A: 
 
2 2
2
2
2
2 2 2
1 1 (0.5 lb)
2 2 (32.2 ft/s )
0.0077640
( ) ( ) (0.5 lb)(1 ft) 0.5 lb ft
0.0077640 0.5 0 1
64.4 ft /s
A A A
A A
A A g
A A D D
A
A
T mv v
T v
V V W R
T V T V
v
v
= =
=
= = = = ⋅
+ = +
+ = +
= 8.02 ft/sA =v W �
 
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603 
 
PROBLEM 13.67 
The system shown is in equilibrium when 0.φ = Knowing that 
initially 90φ = ° and that block C is given a slight nudge when 
the system is in that position, determine the velocity of the block 
as it passes through the equilibrium position 0.φ = Neglect the 
weight of the rod. 
 
SOLUTION 
Find the unstretched length of the spring. 
 
1
2 2
1.1
tan
0.3
1.3045 rad
74.745
(1.1) .3
1.140 ft
BD
BD
L
L
θ
θ
−=
=
= °
= +
=
 
Equilibrium (0.3)( sin ) (25)(2.1) 0
(25 lb)(2.1 ft)
(0.3 ft)(sin 74.745 )
181.39 lb
A s
s
s BD
M F
F
F k L
θΣ = − =
=
°
=
= ∆
 
 
181.39 lb (600 lb/ft)( )
0.30232 ft
BD
BD
L
L
= ∆
∆ =
 
Unstretched length 0
0 1.140 0.3023
0.83768 ft
BD BDL L L
L
= − ∆
= −
=
 
Spring elongation, ,BDL′∆ when 90 .φ = ° 
 
0(1.1 ft 0.3 ft)
1.4 ft 0.8377 ft
0.56232 ft
BD
BD
L L
L
′∆ = + −
′∆ = −
=
 
 
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604 
PROBLEM 13.67 (Continued) 
 
At M, ( 90 )φ = ° 1 1
1 1 1
0, 0
( ) ( )e g
v T
V V V
= =
= +
 
 
2
1
2
1
1
1
1
1
( ) ( )
2
1
( ) (600 lb/ft)(0.5623 ft)
2
( ) 94.86 lb ft
( ) (25 lb)(2.1 ft) 52.5 ft lb
94.86 52.5 42.36 ft lb
e BD
e
e
g
V k L
V
V
V
V
′= ∆
=
= ⋅
= − = − ⋅
= − = ⋅
 
At N, ( 0 )φ = ° 
 
2 2
2
2
2 2
2 2 2
2 2 2 22
1 1 2 2
2
2
2
2
1 1
( ) ( ) (600 lb/ft)(0.3023 ft)
2 2
( ) 27.42 lb ft
( ) 0 27.42 ft lb
1 1 25 lb
0.3882
2 2 32.2 ft/s
0 42.36 0.3882 27.42
(14.941)/(0.3882)
e BD
e
g
V k L
V
V V
T mv v v
T V T V
v
v
= ∆ =
= ⋅
= = ⋅
§ ·
= = =¨ ¸¨ ¸
© ¹
+ = +
+ = +
=
 
 2 2 22 38.48 ft /sv = 2 6.20 ft/sv = W�
 
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605 
 
 
PROBLEM 13.68 
A spring is used to stop a 50-kg package which is moving down a 
20º incline. The spring has a constant 30 kN/mk = and is held by 
cables so that it is initially compressed 50 mm. Knowing that the 
velocity of the package is 2 m/s when it is 8 m from the spring and 
neglecting friction, determine the maximum additional deformation 
of the spring in bringing the package to rest. 
 
SOLUTION 
Let position 1 be the starting position 8 m from the end of the spring when it is compressed 50 mm by the 
cable. Let position 2 be the position of maximum compression. Let x be the additional compression of the 
spring. Use the principle of conservation of energy. 1 1 2 2.T V T V+ = + 
Position 1: 2 21 1
1 1
2 3 2
1 1
1 1
(50)(2) 100 J
2 2
(50)(9.81)(8 sin 20 ) 1342.09 J
1 1
(30 10 )(0.050) 37.5 J
2 2
g
e
T mv
V mgh
V ke
= = =
= = ° =
= = × =
 
Position 2: 22 2 2
2 2
2 3 2 2
2 2
1
0 since 0.
2
(50)(9.81)( sin 20 ) 167.76 
1 1
(30 10 )(0.05 ) 37.5 1500 15000
2 2
g
e
T mv v
V mgh x x
V ke x x x
= = =
= = − ° = −
= = × + = + +
 
Principle of conservation of energy: 
 
2
2
100 1342.09 37.5 167.61 37.5 1500 15000
15,000 1332.24 1442.09 0
x x x
x x
+ + = − + + +
+ − =
 
Solving for x, 
 0.26882 and 0.357 64x = − 0.269 mx = W �
 
 
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606 
 
PROBLEM 13.69 
Solve Problem 13.68 assuming the kinetic coefficient of friction between 
the package and the incline is 0.2. 
PROBLEM 13.68 A spring is used to stop a 50-kg package which is 
moving down a 20° incline. The spring has a constant 30 kN/mk = 
and is held by cables so that it is initially compressed 50 mm. Knowing 
that the velocity of the package is 2 m/s when it is 8 m from the spring 
and neglecting friction, determine the maximum additional deformation 
of the spring in bringing the package to rest. 
 
SOLUTION 
Let position 1 be the starting position 8 m from the end of the spring when it is compressed 50 mm by the 
cable. Let position 2 be the position of maximum compression. Let x be the additional compression of the 
spring. Use the principle of work and energy. 1 1 1 2 2 2T V U T V→+ + = + 
Position 1. 2 21 1
1 1
2 3 2
1 1
1 1
(50)(2) 100 J
2 2
(50)(9.81)(8sin 20 ) 1342.09 J
1 1
(30 10 )(0.05) 37.5 J
2 2
g
e
T mv
V mgh
V ke
= = =
= = ° =
= = × =
 
Position 2. 22 2 2
2 2
2 3 2 2
2 2
1
0 since 0.
2
(50)(9.81)( sin 20 ) 167.76 
1 1
(30 10 )(0.05 ) 37.5 1500 15,000
2 2
g
e
T mv v
V mgh x x
V ke x x x
= = =
= = − ° = −
= = × + = + +
 
Work of the friction force. 
 0nFΣ = 
 
1 2
 cos 20 0
cos 20
(50)(9.81)cos 20
460.92 N 
(0.2)(460.92)
92.184
92.184(8 )
737.47 92.184
f k
f
N mg
N mg
F N
U F d
x
x
µ
→
− ° =
= °
= °
=
=
=
=
= −
= − +
= − −
 
 
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607 
PROBLEM 13.69 (Continued) 
 
Principle of work and energy: 1 1 1 2 2 2T V U T V−+ + = + 
 100 x+1342.09 + 37.5 − 737.47 − 92.184 
 2167.76 37.5 1500 15,000x x x= − + + + 
 215,000 1424.42 704.62 0x x+ − = 
Solving for x, 
 0.17440 and 0.26936x = − 0.1744 mx = W�
 
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608 
 
 
PROBLEM 13.70 
A section of track for a roller coaster consists of two circular 
arcs AB and CD joined by a straight portion BC. The radius of 
AB is 27 m and the radiusof CD is 72 m. The car and its 
occupants, of total mass 250 kg, reach Point A with practically 
no velocity and then drop freely along the track. Determine the 
normal force exerted by the track on the car as the car reaches 
point B. Ignore air resistance and rolling resistance. 
 
SOLUTION 
Calculate the speed of the car as it reaches Point B using the principle of conservation of energy as the car 
travels from position A to position B. 
Position A: 2
1
0, 0, 0 (datum)
2A A A A
v T mv V= = = = 
Position B: BV mgh= − 
where h is the decrease in elevation between A and B. 
 2
1
2B B
T mv= 
Conservation of energy: :A A B BT V T V+ = + 
 
2
2
2
2 2
1
0 0
2
2
(2)(9.81 m/s )(27 m)(1 cos 40 )
123.94 m /s
B
B
mv mgh
v gh
+ = −
=
= − °
=
 
Normal acceleration at B: 
 
2 2 2
2123.94 m /s( ) 4.59 m/s
27 m
B
B n
v
a
ρ
= = = 
 2( ) 4.59 m/sB n =a 50° 
Apply Newton’s second law to the car at B. 
 
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609 
PROBLEM 13.70 (Continued) 
 
 50 : cos40n n nF ma N mg ma°Σ = − ° = − 
 
2 2
cos 40 ( cos 40 )
(250 kg)[(9.81 m/s )cos 40 4.59 m/s ]
n nN mg ma m g a= ° − = ° −
= ° −
 
 1878.7 1147.5= − 731 NN = W 
 
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610 
 
 
PROBLEM 13.71 
A section of track for a roller coaster consists of two circular arcs 
AB and CD joined by a straight portion BC. The radius of AB is 
27 m and the radius of CD is 72 m. The car and its occupants, of 
total mass 250 kg, reach Point A with practically no velocity and 
then drop freely along the track. Determine the maximum and 
minimum values of the normal force exerted by the track on the 
car as the car travels from A to D. Ignore air resistance and 
rolling resistance. 
 
SOLUTION 
Calculate the speed of the car as it reaches Point P, any point on the roller coaster track. Apply the principle of 
conservation of energy. 
Position A: 2
1
0, 0, 0 (datum)
2A A A A
v T mv V= = = = 
Position P: PV mgh= − 
where h is the decrease in elevation along the track. 
 2
1
 
2P
T mv= 
Conservation of energy: A A P PT V T V+ = + 
 2
1
0 0
2
mv mgh+ = − 2 2v gh= (1) 
Calculate the normal force using Newton’s second law. Let θ be the slope angle of the track. 
 
 
 : cosn n nF ma N mg maθΣ = − = 
 cos nN mg maθ= + (2) 
Over portion AB of the track, (1 cos )h ρ θ= − 
and 
2
n
mv
a
ρ
= − 
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611 
PROBLEM 13.71 (Continued) 
 
where ρ is the radius of curvature. ( 27 m)ρ = 
 
2 (1 cos )
cos (3cos 2)
mg
N mg mg
ρ θθ θ
ρ
−= − = − 
At Point A ( 0)θ = (250)(9.81) 2452.5 NAN mg= = = 
At Point B ( 40 )θ = ° (2452.5)(3cos 40 2)BN = ° − 
 731 NBN = 
Over portion BC, 40 , 0 (straight track)naθ = ° = 
 
cos 40 2452.5cos40
1879 N
BC
BC
N mg
N
= ° = °
=
 
Over portion CD, max (1 cos )h h r θ= − − 
and 
2
n
mv
a
r
= 
where r is the radius of curvature. ( 72 m)r = 
 max
max
2
cos
cos 2 1 cos
2
3cos 2
mgh
N mg
r
h
mg mg
r
h
mg
r
θ
θ θ
θ
= +
§ ·= + − −¨ ¸
© ¹
§ ·= − +¨ ¸
© ¹
 
which is maximum at Point D, where 
 max
2
1D
h
N mg
r
§ ·= +¨ ¸
© ¹
 
Data: max 27 18 45 m, 72 mh r= + = = 
 
(2) (45)
(2452.5) 1 5520 N
72D
N ª º= + =« »¬ ¼
 
Summary: minimum (just above ):B 731 N W�
� maximum (at ):D 5520 N W 
 
 
 
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612 
 
 
PROBLEM 13.72 
A 1-lb collar is attached to a spring and slides without friction along 
a circular rod in a vertical plane. The spring has an undeformed 
length of 5 in. and a constant 10 lb/ft.k = Knowing that the collar is 
released from being held at A determine the speed of the collar and 
the normal force between the collar and the rod as the collar passes 
through B. 
 
SOLUTION 
For the collar, 2
1
0.031056 lb s /ft
32.2
W
m
g
= = = ⋅ 
For the spring, 010 lb/ft 5 in.k l= = 
At A: 
0
7 5 5 17 in.
12 in. 1 ft
A
∆
= + + =
− = =
A
A A
 
At B: 2 2
0
(7 5) 5 13 in.
2
1.8 in. ft
3
B
B
= + + =
− = =
A
A A
 
Velocity of the collar at B. 
Use the principle of conservation of energy. 
 A A B BT V T V+ = + 
Where 2
1
0
2A A
T mv= = 
 
2
0
2
2 2 2
2
0
2
2
2 2 2
1
( ) (0)
2
1
(10)(1) 0 5 ft lb
2
1 1
(0.031056) 0.015528
2 2
1
( )
2
1 2 5
(10) (1)
2 3 12
1.80556 ft lb
0 5 0.015528 1.80556
205.72 ft /s
A A
B B B B
B B
B
B
V k W
T mv v v
V k Wh
v
v
= − +
= + = ⋅
= = =
= − +
§ · § ·= + −¨ ¸ ¨ ¸
© ¹ © ¹
= ⋅
+ = =
=
A A
A A
 14.34 ft/sBv = W 
 
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613 
PROBLEM 13.72 (Continued) 
 
Forces at B. 
 
0
2
2
( ) (10) 6.6667 lb.
3
5
sin
13
5
5 in. ft
12
(0.031056)(205.72)
5/12
15.3332 lb
s B
B
n
F k
mv
ma
α
ρ
ρ
§ ·= − = =¨ ¸
© ¹
=
= =
=
=
=
A A
 
 : siny y s nF ma F W N maαΣ = − + = 
 
sin
5
15.3332 1 (6.6667)
13
13.769 lb
n sN ma W F
N
α= + −
§ ·= + − ¨ ¸
© ¹
= 13.77 lb=N W 
 
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614 
 
 
PROBLEM 13.73 
A 10-lb collar is attached to a spring and slides without 
friction along a fixed rod in a vertical plane. The spring 
has an undeformed length of 14 in. and a constant k = 4 
lb/in. Knowing that the collar is released from rest in the 
position shown, determine the force exerted by the rod on 
the collar at (a) Point A, (b) Point B. Both these points are 
on the curved portion of the rod. 
 
SOLUTION 
Mass of collar: 2
10
0.31056 lb s /ft
32.2
W
m
g
= = = ⋅ 
Let position 1 be the initial position shown, and calculate the potential energies of the spring for positions 1,A, and B. 0 14 in.l = 
 
2 2
1
1 1 0
2 2
1 1
(14 14) (14) 31.305 in.
31.305 14 17.305 in.
1 1
( ) (4)(17.305) 598.92 in lb 49.910 ft lb
2 2e
l
x l l
V kx
= + + =
= − = − =
= = = ⋅ = ⋅
 
 
2 2
0
2 2
(14) (14) 19.799 in.
19.799 14 5.799 in.
1 1
( ) (4)(5.799) 67.257 in lb 5.605 ft lb
2 2
A
A A
A e A
l
x l l
V kx
= + =
= − = − =
= = = ⋅ = ⋅
 
 0
2 2
14 14 28 in.
28 14 14 in.
1 1
( ) (4)(14) 392 in lb 32.667 ft lb
2 2
B
B B
B e B
l
x l l
V kx
= + =
= − = − =
= = = ⋅ = ⋅
 
Gravitational potential energies: Datum at level .A 
 1( ) 0 ( ) 0g A gV V= = 
 ( ) (10 lb)( 14 in.) 140 in lb 11.667 ft lbBV g Wy= = − = − ⋅ = − ⋅ 
 
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615 
PROBLEM 13.73 (Continued) 
 
Total potential energies: e gV V V= + 
 1 49.910 ft lb, 5.605 ft lb, 21.0 ft lbA BV V V= ⋅ = ⋅ = ⋅ 
Kinetic energies: 1 0T = 
 
2 2 2
2 2 2
1 1
(0.31056) 0.15528
2 2
1 1
(0.31056) 0.15528
2 2
A A A A
B B B B
T mv v v
T mv v v
= = =
= = =
 
Conservation of energy: 1 1 :A AT V T V+ = + 
 2 2 2 20 49.910 0.15528 5.605 285.32 ft /sA Av v+ = + = 
Conservation of energy: 1 1 B BT V T V+ = + 
 2 2 2 20 49.910 0.15528 21.0 186.18 ft /sB Bv v+ = + = 
Normal accelerations at A and B. 2 /na v ρ= 
 
2 2
2
14 in. 1.16667 ft
285.32 ft /s
( ) ( ) 244.56 ft/s
1.16667 ftA n A n
a
ρ = =
= =a 
 
2 2
2189.10 ft /s( ) ( ) 159.58 ft/s
1.16667 ftB n B n
a = =a 
Spring forces at A and B: F kx= 
 (4 lb/in.)(5.799 in.) 23.196 lbA AF = =F 45° 
 (4 lb/in.)(14 in.)BF = 56.0 lbB =F 
To determine the forces ( and )A BN N exerted by the rod on the collar, apply Newton’s second law. 
 (a) At Point A: 
 ( ) :A nF m aΣ = 
 
sin 45 ( )
10 23.196sin 45 (0.31056)(244.56)
A A A n
A
W N F m a
N
+ − ° =
+ − ° =
 
 82.4 lbA =N W 
 (b) At Point B: 
 ( ) :B nF m aΣ = 
 (0.31056)(159.58)BN = 49.6 lbB =N W 
 
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616 
 
 
PROBLEM 13.74 
An 8-oz package is projected upward with a velocity 
v0 by a spring at A; it moves around a frictionless 
loop and is deposited at C. For each of the two loops 
shown, determine (a) the smallest velocity v0 for 
which the package will reach C, (b) the corresponding 
force exerted by the package on the loop just before 
the package leaves the loop at C. 
 
SOLUTION 
Loop 1 
 
(a) The smallest velocity at B will occur when the force exerted by the 
 tube on the package is zero. 
 
2
 0 B
mv
F mg
r
Σ = + = 
2 2 1.5 ft(32.2 ft/s )Bv rg= = 
 
2 48.30Bv = 
 At A 
2
0
1
2A
T mv= 
A
0.5
 0 8 oz 0.5 lb 0.01553
32.2
V § ·= = � = =¨ ¸
© ¹ 
 At B 
21 1 (48.30) 24.15 m
2 2B B
T mv m= = = 
 (7.5 1.5) 9 9(0.5) 4.5 lb ftBV mg mg= + = = = ⋅ 
2
0
1
: (0.01553) 24.15(0.01553) 4.5
2A A B B
T V T V v+ = + = + 
� 20 0 627.82 25.056v v= = � 0 25.1 ft/sv = W�
 At C 
2 21 0.007765 7.5 7.5(0.5) 3.75
2C C C C
T mv v V mg= = = = = 
2 2
0: 0.007765 0.007765 3.75A A C C CT V T V v v+ = + = + 
2 20.007765(25.056) 3.75 0.007765 Cv− = 
 2 144.87Cv = 
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617 
 
 
 
 
 
Loop 2 
 
 
 PROBLEM 13.74 (Continued) 
 
(b) 
 
(144.87)
 : 0.01553
1.5n
F ma NΣ = = 
 1.49989N = 
 {Package in tube} 1.500 lbCN = W 
(a) At B, tube supports the package so, 
0Bv ≈ 
0, 0 (7.5 1.5)B B Bv T V mg= = = + 
 4.5 lb ft= ⋅ 
 A A B BT V T V+ = + 
21 (0.01553) 4.5 24.073
2 A A
v v= � = 
24.1 ft/sAv = W�
(b) At C 20.007765 , 7.5 3.75C C CT v V mg= = = 
2 2: 0.007765(24.073) 0.007765 3.75A A C C CT V T V v+ = + = + 
 2 96.573Cv = 
 
96.573
0.01553 0.99985
1.5C
N § ·= =¨ ¸
© ¹
 
{Package on tube} 1.000 lb CN = W 
 
 
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618 
 
 
PROBLEM 13.75 
If the package of Problem 13.74 is not to hit the 
horizontal surface at C with a speed greater than 10 ft/s, 
(a) show that this requirement can be satisfied only by 
the second loop, (b) determine the largest allowable 
initial velocity 0v when the second loop is used. 
 
SOLUTION 
 
 
 
 
 
 
 
 
(a) Loop 1 
 From Problem 13.74, at B 
 
2 2 2 48.3 ft /s 6.9498 ft/sB Bv gr v= = � = 
 2
1 1
(0.01553)(48.3) 0.37505
2 2B B
T mv= = = 
 (7.5 1.5) (0.5)(9) 4.5 lb ftBV mg= + = = ⋅ 
 2 2 2
1 1
(0.01553) 0.007765 
2 2C C C C
T mv v v= = = 
 7.5(0.5) 3.75 lb ftCV = = ⋅ 
2: 0.37505 4.5 0.007765 3.75B B C C CT V T V v+ = + + = + 
 2 144.887 12.039 ft/sC Cv v= � = 
12.04 ft/s 10 ft/s> � Loop (1) does not work W�
(b) Loop 2 at A 2 20 0
1
0.007765 
2A
T mv v= = 
 0AV = 
 At C assume 10 ft/sCv = 
2 21 0.007765(10) 0.7765
2C C
T mv= = = 
 7.5(0.5) 3.75Cv = = 
 20: 0.007765 0.7765 3.75A A C CT V T V v+ = + = + 
� �� 0 24.144v = � 0 24.1 ft/sv = W 
 
 
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619 
PROBLEM 13.76 
A small package of weight W is projected into a 
vertical return loop at A with a velocity v0. The package 
travels without friction along a circle of radius r and is 
deposited on a horizontal surface at C. For each of the 
two loops shown, determine (a) the smallest velocity v0 
for which the package will reach the horizontal surface 
at C, (b) the corresponding force exerted by the loop on 
the package as it passes Point B. 
 
SOLUTION 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Loop 1: 
(a) Newton’s second law at position C: 
 
2
2
ma:
c
c
F
v
mg m v gr
r
Σ =
= =
 
 Conservation of energy between position A and B. 
2
0
2
1
2
0
1 1
2 2
(2 ) 2
A
A
C C
C
T mv
V
T mv mgr
V mg r mgr
=
=
= =
= =
 
 
 
2
0
2
0
1 1
: 0 2
2 2
5
A A C CT V T V mv mgr mgr
v gr
+ = + + = +
=
 
 Smallest velocity 0:v 0 5v gr= W�
(b) Conservation of energy between positions A and B. 
 2
1
( ) ; ( )
2B B B
b T mv V mg r= = 
 2 2
1 1
: 0
2 2A A B B A B
T V T V mv mv mgr+ = + + = + 
 2
1 1
(5 ) 0
2 2 B
m gr mv mgr+ = + � 2 3Bv gr= 
�
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620 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 PROBLEM 13.76 (Continued) 
 
Newton’s second law at position B. 
2 3
3Bn
v gr
ma m m mg
r r
= = = 
: 3 eff BF F N mgΣ = Σ = 
Force exerted by the loop: 3BN W= W�
Loop 2: 
(a) At point C, 0cv = 
 Conservation of energy between positions A and C. 
 
21 0
2
(2 ) 2
C C
C
T mv
V mg r mgr
= =
= =
 
 20
2
0
:
1
0 0 2
2
4
A A C CT V T V
mv mgr
v gr
+ = +
+ = +
=
 
 Smallest velocity 0:v 0 4v gr= W 
(b) Conservation of energy between positions A and B. 
 2 20
1 1
: 0
2 2A A B B B
T V T V mv mv mgr+ − + + = + 
 2
1 1
(4 )
2 2 B
m gr mv mgr= + � 2 2Bv gr= 
 Newton’s second law at position B. 
 
2 2
2 Bn
v gr
ma m m mg
r r
= = = 
 eff : 2 F N mgΣ = Σ = 
 Force exerted by loop:� 2W=N W 
 
 
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621 
 
 
PROBLEM 13.77 
The 1 kg ball at A is suspended by an inextensible cord and given an initial 
horizontal velocity of 5 m/s. If 0.6 ml = and 0,Bx = determine yB so that 
the ball will enter the basket. 
 
SOLUTION 
Let position 1 be at A. 1 0v v= 
Let position 2 be the point described by the angle where the path of the ball changes from circular to 
parabolic. At position 2, the tension Q in the cord is zero. 
Relationship between 2v and θ based on 0.Q = Draw the free body diagram. 
 
 
2
20: sin n
mv
F Q mg ma
l
θΣ = + = = 
With 22 20, sin or sinQ v gl v glθ θ= = = (1) 
Relationship among 0 2,v v and θ based on conservation of energy. 
 
1 1 2 2
2 2
0 2
1 1
sin
2 2
T V T V
mv mgl mv mgl θ
+ = +
− = +
 
 2 20 2 2 (1 sin )v v gl θ− = + (2) 
 
 
 
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622 
PROBLEM 13.77 (Continued) 
 
Eliminating 2v from Eqs. (1) and (2), 
 
2
0
2 2
0
sin 2 (1 sin )
1 1 (5)
sin 2 2 0.74912
3 3 (9.81)(0.6)
48.514
v gl gl
v
gl
θ θ
θ
θ
− = +
ª º ª º
= − = − =« » « »
« » ¬ ¼¬ ¼
= °
 
From Eq. (1), 2 2 22
2
(9.81)(0.6)sin 48.514 4.4093 m /s
2.0998 m/s
v
v
= ° =
=
 
x and y coordinates at position 2. 
 2
2
cos 0.6cos 48.514 0.39746 m
sin 0.6sin 48.514 0.44947 m
x l
y l
θ
θ
= = ° =
= = ° =
 
Let 2t be the time when the ball is a position 2. 
Motion on the parabolic path. The horizontal motion is 
 
2
2 2
sin 2.0998 sin 48.514
1.5730 m/s
1.5730( )
x v
x x t t
θ= − = − °
= −
= − −
�
 
At Point B, 0Bx = 
 2 20 0.39746 1.5730( ) 0.25267 sB Bt t t t= − − − = 
The vertical motion is 
 22 2 2 2
1
cos ( ) ( )
2
y y v t t g t tθ= + − − − 
At Point B, 22 2 2 2
1
cos ( ) ( )
2B B B
y y v t t g t tθ= + − − − 
 
2
0.44947 (2.0998 cos 48.514 )(0.25267)
1
(9.81)(0.25267)
2
0.48779 m
By = + °
−
= 0.448 mBy = W�
 
 
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623 
 
 
PROBLEM 13.78* 
Packages are moved from Point A on the upper floor of a warehouse to 
Point B on the lower floor, 12 ft directly below A, by means of a chute, 
the centerline of which is in the shape of a helix of vertical axis y and 
radius R = 8 ft. The cross section of the chute is to be banked in such a 
way that each package, after being released at A with no velocity, will 
slide along the centerline of the chute without ever touching its edges. 
Neglecting friction, (a) express as a function of the elevation y of a 
given Point P of the centerline the angle φ formed by the normal to the 
surface of the chute at P and the principal normal of the centerline at 
that point, (b) determine the magnitude and direction of the force 
exerted by the chute on a 20-lb package as it reaches Point B. Hint: The 
principal normal to the helix at any Point P is horizontal and directed 
toward the y axis, and the radius of curvature of the helix is 
ρ = R[1 – (h/2πR)2]. 
 
SOLUTION 
(a) At Point A: 0 0A A
A
v T
V mgh
= =
=
 
 At any Point P: 2
1
2P
T mv= 
 2
2
1
0
2
2 ( )
P
A A P P
V Wy mgy
T V T V
mgh mv mgy
v g h y
= =
+ = +
+ = +
= −
 
 en along principal normal, horizontal and directed toward y axis 
 et tangent to centerline of the chute 
 eD along binormal 
 1 1
(12 ft)
tan tan 13.427
2 2 (8 ft)
0b
h
R
ma
β
π π
− −= = = °
=
 
 since 0ba = 
 Note: Friction is zero, 
 
2
: sin sin
: cos 0 cos
2 ( ) ( )
: 2
t t t t
b b b B
n n n
F ma mg ma a g
F ma N W N W
mv m g h y h y
F ma N W
e e e
β β
β β
Σ = = =
Σ = − = =
− −Σ = = = =
 
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624 
PROBLEM 13.78* (Continued) 
 
 The total normal force is the resultant of Nb and Nm, it lies in the b–m plane, and forms angle φ with 
 m axis. 
 
tan /
2( ( )
tan cos
tan ( /2( ))cos
b nN N
w h y
W
e
e h y
φ
φ β
φ β
=
−=
= −
 
 Given: 
2
2
2
1 (1 tan )
2 cos
h R
e R R
R
β
π β
ª º§ ·= + = + =« »¨ ¸
© ¹« »¬ ¼
 
 Thus, tan cos
2( ) 2( )cos
8 ft 4.112
tan
2(12 )cos13.427 12
e R
h y h y
y y
φ β
β
φ
= =
− −
= =
− ° −
 
 or cot 0.243(12 )yφ = − W 
(b) At Point B: y = 0 for x, y, z axes, we write, with W = 20 lb, 
 
2 2
2
2
sin cos sin (20 lb)cos 13.427 sin13.427 4.517 lb
cos cos (20 lb)cos 13.427 18.922 lb
2 2
/cos
(12 ft-0)
2(20 lb) cos 13.427 56.765 lb
8 ft
x b x
y b y
z n
z z
N N W N
N N W N
h y h y
N N w W
e R
N N
β β β
β β
β
= = = ° ° =
= = = ° =
− −= − = − = −
= − ° = −
 
 2 2 2(4.517) (18.922) ( 56.765)N = + + − 60.0 lbN = W 
 
4.517
cos
60
x
x
N
N
θ = = 85.7xθ = ° W 
 
18.922
cos
60
y
y
N
N
θ = = 71.6yθ = ° W 
 
56.742
cos
60
z
z
N
N
θ = = − 161.1zθ = ° W�
 
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625 
 
PROBLEM 13.79* 
Prove that a force F(x, y, z) is conservative if, andonly if, the following relations are satisfied: 
y yx xz z
F FF FF F
y x z y x z
∂ ∂∂ ∂∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
= = = 
 
SOLUTION 
For a conservative force, Equation (13.22) must be satisfied. 
 x y z
V V V
F F F
x y z
∂ ∂ ∂
∂ ∂ ∂
= − = − = − 
We now write 
2 2
yx
FF V V
y x y x y x
∂∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
= − = − 
Since 
2 2
:
V V
x y y x
∂ ∂
∂ ∂ ∂ ∂
= yx
FF
y x
∂∂
∂ ∂
= W 
We obtain in a similar way 
 y xz z
F FF F
z y x z
∂ ∂∂ ∂
∂ ∂ ∂ ∂
= = W 
 
 
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626 
 
PROBLEM 13.80 
The force ( )/yz zx xy xyz= + +F i j k acts on the particle ( , , )P x y z which moves in space. (a) Using the 
relation derived in Problem 13.79, show that this force is a conservative force. (b) Determine the potential 
function associated with F. 
 
SOLUTION 
(a) x y z
yz zx xy
F F F
xyz xyz xyz
= = = 
 
( ) ( )11
0 0
yyxx
FF
y y x x
∂∂ ∂∂
∂ ∂ ∂ ∂
= = = = 
 Thus, yx
FF
y x
∂∂
∂ ∂
= 
 The other two equations derived in Problem 13.79 are checked in a similar way. 
(b) Recall that , , x y z
V V V
F F F
x y z
∂ ∂ ∂
∂ ∂ ∂
= − = − = − 
 
1
 ln ( , )x
V
F V x f y z
x x
∂
∂
= = − = − + (1) 
 
1
 ln ( , )y
V
F V y g z x
y y
∂
∂
= = − = − + (2) 
 
1
 ln ( , )z
V
F V z h x y
z z
∂
∂
= = − = − + (3) 
 Equating (1) and (2) 
 ln ( , ) ln ( , )x f y z y g z x− + = − + 
 Thus, ( , ) ln ( )f y z y k z= − + (4) 
 ( , ) ln ( )g z x x k z= − + (5) 
 Equating (2) and (3) 
 ln ( , ) ln ( , )z h x y y g z x− + = − + 
 ( , ) ln ( )g z x z l x= − + 
 From (5), 
 ( , ) ln ( )g z x x k z= − + 
 
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627 
PROBLEM 13.80 (Continued) 
 
 Thus, 
 ( ) lnk z z= − 
 ( ) lnl x x= − 
 From (4), 
 ( , ) ln lnf y z y z= − − 
 Substitute for ( , )f y z in (1) 
 ln ln lnV x y z= − − − 
�� lnV xyz= − W�
 
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628 
 
 
PROBLEM 13.81* 
A force F acts on a particle P(x, y) which moves in the xy plane. 
Determine whether F is a conservative force and compute the work of F 
when P describes in a clockwise sense the path A, B, C, A including the 
quarter circle 2 2 2,x y a+ = if (a) ,ky=F i (b) ( ).k y x= +F i j 
 
SOLUTION 
(a) 0 0yxx y
FF
F ky F k
y x
∂∂
∂ ∂
= = = = 
 Thus, yx
FF
y x
∂∂
∂ ∂
≠ F is not conservative. 
 ( )
B C A
ABCA
A B B
ABCA
U d ky dy ky dx dy ky dx= ⋅ = ⋅ + ⋅ + + ⋅³ ³ ³ ³F r i j i i j i j 
 0,
B
A
=³ F is perpendicular to the path. 
 ( )
C C
B B
ky dx dy ky dx⋅ + =³ ³i i j 
 From B to C, the path is a quarter circle with origin at A. 
 Thus, 2 2 2
2 2
x y a
y a x
+ =
= −
 
 Along BC, 
2
2 2
0 4
C a
B
ka
kydx k a x dx
π= − =³ ³ 
 0 ( 0 on )
A
C
ky dx y CA⋅ = =³ i j 
 
2
0 0
4
B C A
ABCA
A B C
ka
U
π= + + = + +³ ³ ³ 
2
4ABCA
ka
U
π= W 
(b) yxx y
FF
F ky F kx k k
y x
∂∂
∂ ∂
= = = = 
 ,yx
FF
y x
∂∂
∂ ∂
= F is conservative. 
 Since ABCA is a closed loop and F is conservative, 0ABCAU = W�
 
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629 
PROBLEM 13.82* 
The potential function associated with a force P in space is known 
to be 2 2 2 1/2( , , ) ( ) .V x y z x y z= − + + (a) Determine the x, y, and z 
components of P. (b) Calculate the work done by P from O to D by 
integrating along the path OABD, and show that it is equal to the 
negative of the change in potential from O to D. 
 
SOLUTION 
(a) 
2 2 2 1/2
2 2 2 1/2[ ( ) ] ( )x
V x y z
P x x y z
x x
∂ ∂
∂ ∂
−− + += − = − = + + W 
 
2 2 2 1/2
2 2 2 1/2[ ( ) ] ( )y
V x y z
P y x y z
y y
∂ ∂
∂ ∂
−− + += − = − = + + W 
 
2 2 2 1/2
2 2 2 1/2[ ( ) ] ( )z
V x y z
P z x y z
z z
∂ ∂
∂ ∂
−− + += − = − = + + W 
(b) OABD OA AB BDU U U U= + + 
 O–A: Py and Px are perpendicular to O–A and do no work. 
 Also, on O–A 0 and 1zx y P= = = 
 Thus, 
0 0
a a
O A zU P dz dz a− = = =³ ³ 
 A–B: Pz and Py are perpendicular to A–B and do no work. 
 Also, on A–B 0,y z a= = and 
2 2 1/2( )
x
x
P
x a
=
+
 
 Thus, 
2 2 1/20 ( )
( 2 1)
a
A B
xdx
U
x a
a
− = +
= −
³ 
 B–D: Px and Pz are perpendicular to B–D and do no work. 
 On ,B D− 
2 2 1/2( 2 )
y
k a
z a
y
P
y a
=
=
=
+
 
 
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630 
PROBLEM 13.82* (Continued) 
 
 Thus, 2 2 1/2
2 2 1/2 00
2 2 1/2 2 1/2
( 2 )
( 2 )
( 2 ) (2 ) ( 3 2)
a a
BD
BD
OABD O A A B B D
y
U dy y a
y a
U a a a a
U U U U− − −
= = +
+
= + − = −
= + +
³ 
 ( 2 1) ( 3 2)a a a= + − + − 3OABDU a= W 
 2 2 2 1/2
( , , ) (0,0,0)
( ) 0
ODV V a a a V
a a a
∆ = −
= − + + − 3ODV a∆ = − W 
 Thus, OABD ODU V= −∆ 
 
 
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631 
 
 
PROBLEM 13.83* 
(a) Calculate the work done from D to O by the force P of 
Problem 13.82 by integrating along the diagonal of the cube. (b) Using 
the result obtained and the answer to part b of Problem 13.82, verify 
that the work done by a conservative force around the closed path 
OABDO is zero. 
PROBLEM 13.82 The potential function associated with a force P in 
space is known to be V(x, y, z) 2 2 2 1/2( ) .x y z= − + + (a) Determine the 
x, y, and z components of P. (b) Calculate the work done by P from O 
to D by integrating along the path OABD, and show that it is equal to 
the negative of the change in potential from O to D. 
 
SOLUTION 
From solution to (a) of Problem 13.82 
 
2 2 2 1/2( )
x y z
x y z
+ +=
+ +
i j k
P 
(a) 
D
OD
O
U d= ⋅³ P r 
 
2 2 2 1/2( )
x y z
d dx dy dz
x y z
x y z
= + +
= + +
+ +=
+ +
r i j k
r i j k
i j k
P
 
 Along the diagonal. x y z= =Thus, 
2 1/2
3
3
(3 )
x
dr
x
⋅ = =P 
 
0
3 3
a
O DU dx a− = =³ 3ODU a= W 
(b) OABDO OABD DOU U U= + 
 From Problem 13.82 
 3OABDU a= at left 
 The work done from D to O along the diagonal is the negative of the work done from O to D. 
 3DO ODU U a= − = − [see part (a)] 
 Thus, 
 3 3 0OABDOU a a= − = W�
 
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632 
 
PROBLEM 13.84* 
The force 2 2 2 3/2( )/( )x y z x y z= + + + +F i j k acts on the particle ( , , )P x y z which moves in space. (a) Using 
the relations derived in Problem 13.79, prove that F is a conservative force. (b) Determine the potential 
function V(x, y, z) associated with F. 
 
SOLUTION 
(a) 
2 2 2 3/2
2 2 2 1/2
( )
( )
x
y
x
F
x y z
y
F
x y z
=
+ +
=
+ +
 
 
( )
( )
3
2
2 2 2 5/2
3
2
2 2 2 5/2
(2 )
( )
2
( )
x
y
x yF
y x y z
y yF
x x y z
∂
∂
∂
∂
−
=
+ +
−
=
+ +
 
 Thus, yx
FF
y x
∂∂
∂ ∂
= 
 The other two equations derived in Problem 13.79 are checked in a similar fashion. 
(b) Recalling that 
2 2 2 3/2
2 2 2 1/2
, ,
( )
( ) ( , )
x y z
x
V V V
F F F
x y z
V x
F V dx
x x y z
V x y z f y z
∂ ∂ ∂
∂ ∂ ∂
∂
∂
−
= − = − = −
= − = −
+ +
= + + +
³
 
 Similarly integrating /V y∂ ∂ and /V z∂ ∂ shows that the unknown function ( , )f x y is a constant. 
�
2 2 2 1/2
1
( )
V
x y z
=
+ +
 W 
 
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633 
 
PROBLEM 13.85 
(a) Determine the kinetic energy per unit mass which a missile must have after being fired from the surface of 
the earth if it is to to reach an infinite distance from the earth. (b) What is the initial velocity of the missile 
(called the escape velocity)? Give your answers in SI units and show that the answer to part b is independent 
of the firing angle. 
 
SOLUTION 
At the surface of the earth, 29.81 m/sg = 
 61 6370 km 6.37 10 mr R= = = × 
Centric force at the surface of the earth, 
 2
2 6 2 12 3 2(9.81)(6.37 10 ) 398.06 10 m /s
GMm
F mg
R
GM gR
= =
= = × = ×
 
Let position 1 be on the surface of the earth 1( )r R= and position 2 be at 2 .r OD= Apply the conservation of 
energy principle. 
 
1 1 2 2
2 2
1 2
1 2
1 2
1 2 2
1 1
2 2
T V T V
GMm GMm
mv mv
r r
GMm GMm
T T
R
T T TGM
gR
m m R m
+ = +
− = +
= + −
∞
= + = +
 
For the escape condition set 2 0
T
m
= 
 2 6 6 2 21 (9.81 m/s )(6.37 10 m) 62.49 10 m /s
T
gR
m
= = × = × 
(a) 1 62.5 MJ/kg
T
m
= W 
 2esc
1
2
mv gr= 
(b) 
esc
6 3
esc
2
(2)(9.81)(6.37 10 ) 11.18 10 m/s
v gR
v
=
= × = × esc 11.18 km/sv = W 
Note that the escape condition depends only on the speed in position 1 and is independent of the direction of 
the velocity (firing angle). 
 
 
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634 
 
PROBLEM 13.86 
A satellite describes an elliptic orbit of minimum altitude 606 km 
above the surface of the earth. The semimajor and semiminor axes 
are 17,440 km and 13,950 km, respectively. Knowing that the speed 
of the satellite at Point C is 4.78 km/s, determine (a) the speed at 
Point A, the perigee, (b) the speed at Point B, the apogee. 
 
SOLUTION 
 
6
2 2 6
6370 606 6976 km 6.976 10 m
(17440 6976) (13950) 17438.4 km 17.4384 10 m
A
C
r
r
= + = = ×
= − + = = ×
 
 
6(2)(17440) 6976 27904 km 27.904 10 mBr = − = = × 
For earth, 
66370 km 6.37 10 mR = = × 
 
2 2 6 2 12 3 2(9.81 m/s )(6.370 10 ) 398.06 10 m /sGM gR= = × = × 
 4.78 km/s 4780 m/sCv = = 
(a) Speed at Point A: Use conservation of energy. 
 2 21 1
2 2
A A C C
A C
A C
T V T V
GMm GMm
mv mv
r r
+ = +
− = − 
 
2 2
2 12
6 6
1 1
2
1 1
(4780) (2)(398.06 10 )
6.976 10 17.4384 10
A C
A C
v v GM
r r
§ ·
= + −¨ ¸
© ¹
ª º= + × −« »× ׬ ¼
 
 
6 2 291.318 10 m /s= × 
 
39.556 10 m/sAV = × 9.56 km/sAv = W 
 
 
 
 
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635 
PROBLEM 13.86 (Continued) 
 
(b) Speed at Point B: Use conservation of energy. 
 
2 2
2 2
2 12
6 6
6 2 2
1 1
2 2
1 1
2
1 1
(4780) (2)(398.06 10 )
27.904 10 17.4384 10
5.7258 10 m /s
B B C C
B C
B C
B C
B C
T V T V
GMm GMm
mv mv
r r
v v GM
r r
+ = +
− = −
§ ·
= + −¨ ¸
© ¹
ª º= + × −« »× ׬ ¼
= ×
 
 
32.39 10 m/sBv = × 2.39 km/sBv = W 
 
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636 
 
PROBLEM 13.87 
While describing a circular orbit 200 mi above the earth a space vehicle launches a 6000-lb communications 
satellite. Determine (a) the additional energy required to place the satellite in a geosynchronous orbit at an 
altitude of 22,000 mi above the surface of the earth, (b) the energy required to place the satellite in the same 
orbit by launching it from the surface of the earth, excluding the energy needed to overcome air resistance. 
(A geosynchronous orbit is a circular orbit in which the satellite appears stationary with respect to the ground). 
 
SOLUTION 
 
 
 
 
 
 
 
Geosynchronous orbit 
 61 3960 200 4160 mi 21.965 10 ftr = + = = × 
 62 3960 22,000 25,960 mi 137.07 10 ftr = + = = × 
Total energy 2
1
2
GMm
E T V mv
r
= + = − 
 mass of earthM = 
 mass of satellitem = 
Newton’s second law 
2
2
2:n
GMm mv GM
F ma v
r rr
= = � = 
 2
1
2 2
GM GMm
T mv m V
r r
= = = − 
 
1 1
2 2
GMm GMm GMm
E T V
r r r
= + = − = − 
 
2 2
2 1 1 where ( )
2 2
E E
E
gR m R W
GM gR E W mg
r r
= = − = − = 
 
6 2 181 (6000)(20.9088 10 ft) 1.3115 10
ft lb
2
E
r r
× ×= − = − ⋅ 
Geosynchronous orbit at 62 137.07 10 ftr = × 
18
9
6
1.3115 10
9.5681 10 ft lb
137.07 10Gs
E
− ×= = − × ⋅
×
 
(a) At 200 mi, 61 21.965 10 ftr = × 
18
10
200 6
1.3115 10
5.9709 10
21.965 10
E
×= − = − ×
×
 
 10300 200 5.0141 10GsE E E∆ = − = × 
 9300 50.1 10 ft lbE∆ = × ⋅ W 
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637 
PROBLEM 13.87 (Continued) 
 
(b) Launch from earth 
 At launch pad 
2
E
E E
E E
GMm gR m
E WR
R R
−= − = = − 
 116000(3960 5280) 1.25453 10EE = − × = − × 
 9 99.5681 10 125.453 10E Gs EE E E∆ = − = − × + × 
 9115.9 10 ft lb
E
E∆ = × ⋅ W 
 
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638 
 
PROBLEM 13.88 
A lunar excursion module (LEM) was used in the Apollo moon-landing missions to save fuel by making it 
unnecessary to launch the entire Apollo spacecraft from the moon’s surface on its return trip to earth. Check 
the effectiveness of this approach by computing the energy per pound required for a spacecraft (as weighed on 
the earth) to escape the moon’s gravitational field if the spacecraft starts from (a) the moon’s surface, 
(b) a circular orbit 50 mi above the moon’s surface. Neglect the effect of the earth’s gravitational field. (The 
radius of the moon is 1081 mi and its mass is 0.0123 times the mass of the earth.) 
 
SOLUTION 
Note: moon earth0.0123GM GM= 
By Eq. 12.30, 2moon 0.0123 EGM gR= 
At ∞ distance from moon: 2 2, Assume 0r v= ∞ = 
 2 2 2
0
0 0
0
M
E T V
GM m
= +
= −
∞
= −
=
 
(a) On surface of moon: 61081 mi 5.7077 10 ftMR = = × 
 1 10 0v T= = 
63960 mi 20.909 10 ftER = = × 
 1
M
M
GM m
V
R
= − 
2
1 1 1
0.0123 
0 E
M
gR m
E T V
R
= + = − 
 
2 6 2
1 6
(0.0123)(32.2 ft/s )(20.909 10 ft)
(5.7077 10 ft)
m
E
×= −
×
 
 WE = Weight of LEM on the earth 
 6 2 21
6
2 2
1 2
2 1
3
( 30.336 10 ft /s )
30.336 10
ft /s
32.2 ft/s
0 (942.1 10 ft lb/lb)
E
E
E
W
E m m
g
E W
E E E
W
= − × =
§ ·×= −¨ ¸¨ ¸
© ¹
∆ = −
= + × ⋅
 
 Energy per pound: 3942 10 ft lb/lb
E
E
W
∆ = × ⋅ W 
 
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639 
PROBLEM 13.88 (Continued) 
 
(b) 1
6
1
50 mi
(1081 mi 50 mi) 1131 mi 5.9717 10 ft
Mr R
r
= +
= + = = ×
 
 Newton’s second law: 
 
2
1
2
11
: Mn
GM m v
F ma m
rr
= = 
 
2 2
1 1 1
1 1
1
1
1 1 1
1 1
2
1
1 1
2 6 2
1 6
6 2 2
3
1 2
1 1
2 2
1
2
0.0123 1 1
2 2
1 (0.0123)(32.2 ft/s )(20.909 10 ft)
2 5.9717 10 ft
(14.498 10 ft /s )
450.2 10 ft lb/lb
(32.2 ft/s )
M
M
M M
M E
E
E
GMGMm
v T mv m
r r
GM m
V
r
GM m GM m
E T V
r r
GM m gR m
E
r r
m
E
W
E W
E
= = =
= −
= + = −
= − = −
×= −
×
×= = × ⋅
∆ 32 1 0 450.2 10 ft lb/lb EE E W= − = + × ⋅
 
 Energy per pound: 3450 10 ft lb/lb
E
E
W
∆ = × ⋅ W�
 
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640 
 
 
PROBLEM 13.89 
Knowing that the velocity of an experimental space probe 
fired from the earth has a magnitude 32.5 Mm/hAv = at 
Point A, determine the speed of the probe as it passes 
through Point B. 
 
SOLUTION 
 66370 4300 10740 km 10.670 10 mA Ar R h= + = + = = × 
 66370 12700 19070 km 19.070 10 mBr = + = = × 
 2 6 2 12 3 2(9.81)(6.370 10 ) 398.06 10 m /sGM gR= = × = × 
 332.5 Mm/h 9.0278 10 m/sAv = = × 
 
Use conservation of energy. 
 B B A AT V T V+ = + 
 
2 2
2 2
3 2 12
6 6
6 2 2
1 1
2 2
1 1
2
1 1
(9.0278 10 ) (2)(398.06 10 )
19.070 10 10.670 10
48.635 10 m /s
B A
B A
B A
B A
GMm GMm
mv mv
r r
v v GM
r r
− = −
§ ·
= + −¨ ¸
© ¹
ª º= × + × −« »× ׬ ¼
= ×
 
 36.97 10 m/sBv = × 25.1 Mm/hBv = W 
 
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641 
 
 
PROBLEM 13.90 
A spacecraft is describing a circular orbit at an altitude of 1500 km 
above the surface of the earth. As it passes through Point A, its speed is 
reduced by 40 percent and it enters an elliptic crash trajectory with the 
apogee at Point A. Neglecting air resistance, determine the speed of the 
spacecraft when it reaches the earth’s surface at Point B. 
 
SOLUTION 
Circular orbit velocity 
2
2
2 ,
Cv GM GM gR
r r
= = 
 
2 2 6 2
2
6 6
(9.81 m/s )(6.370 10 m)
(6.370 10 m 1.500 10 m)C
GM gR
v
r r
×= = =
× + ×
 
 2 6 2 250.579 10 m /sCv = × 
 7112 m/sCv = 
Velocity reduced to 60% of Cv gives 4267 m/s.Av = 
Conservation of energy: 
 A A B BT V T V+ = + 
 
1
2
m 2A
GM m
v − 1
2A
m
r
= 2B
GM m
v −
Br
 
6 2 2 6 2
3 2
6 6
1 9.81(6.370 10 ) 9.81(6.370 10 )
(4.267 10 )
2 2(7.870 10 ) (6.370 10 )
Bv× ×× − = −
× ×
 
 36.48 10 m/sBv = × 6.48 km/sBv = W 
 
 
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642 
 
PROBLEM 13.91 
Observations show that a celestial body traveling at 61.2 10 mi/h× appears to be describing about Point B a 
circle of radius equal to 60 light years. Point B is suspected of being a very dense concentration of mass called 
a black hole. Determine the ratio MB/MS of the mass at B to the mass of the sun. (The mass of the sun is 
330,000 times the mass of the earth, and a light year is the distance traveled by light in one year at a velocity 
of 186,300 mi/s.) 
 
SOLUTION 
 
One light year is the distance traveled by light in one year. 
Speed of light = 186,300 mi/s 
 
18
(60 yr)(186,300 mi/s)(5280 ft/mi)(365 days/yr)(24 h/day)(3600 s/h)
1.8612 10 ft
r
r
=
= ×
 
Newton’s second law 
 
2
2
2
2
earth earth
2 2
15 3 2
(32.2 ft/s )(3960 mi 5280 ft/mi)
14.077 10 (ft /s )
B
B
GM m v
F m
rr
rv
M
G
GM gR
= =
=
=
= ×
= ×
 
 sun sun earth330,000 : 330,000 EM M GM GM= = 
 15sun
21 3 2
21
sun
22
sun
21
(330,000)(14.077 10 )
4.645 10 ft /s
4.645 10
4.645 10
B
GM
G
M
rv Mrv
M
G
= ×
= ×
×=
= =
×
 
 
18 6 2
21
sun
(1.8612 10 )(1.76 10 )
4.645 10
BM
M
× ×=
×
 9
sun
1.241 10B
M
M
= × W 
 
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643 
 
PROBLEM 13.92 
(a)Show that, by setting r R y= + in the right-hand member of Eq. (13.17′) and expanding that member in a 
power series in y/R, the expression in Eq. (13.16) for the potential energy Vg due to gravity is a first-order 
approximation for the expression given in Eq. (13.17′). (b) Using the same expansion, derive a second-order 
approximation for Vg. 
 
SOLUTION 
 
2 2
1 2
setting :
1
( 1) ( 1)( 2)
1 1
1 1 2
g g y
R
g
WR WR WR
V r R y V
r R y
y y y
V WR WR
R R R
−
= − = + = − = −
+ +
ª º− − −§ · § ·= − + = − + + +« »¨ ¸ ¨ ¸⋅© ¹ © ¹« »¬ ¼
"
 
We add the constant WR, which is equivalent to changing the datum from to :r r R= ∞ = 
 
2
g
y y
V WR
R R
ª º§ ·= − +« »¨ ¸
© ¹« »¬ ¼
" 
(a) First order approximation: 
 g
y
V WR Wy
R
§ ·= =¨ ¸
© ¹
 W 
 [Equation 13.16] 
(b) Second order approximation: 
2
g
y y
V WR
R R
ª º§ ·= −« »¨ ¸
© ¹« »¬ ¼
 
 
2
g
Wy
V Wy
R
= − W 
 
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644 
 
 
PROBLEM 13.93 
Collar A has a mass of 3 kg and is attached to a spring of constant 
1200 N/m and of undeformed length equal to 0.5 m. The system is 
set in motion with r = 0.3 m, vθ = 2 m/s, and vr = 0. Neglecting the 
mass of the rod and the effect of friction, determine the radial and 
transverse components of the velocity of the collar when r = 0.6 m. 
 
SOLUTION 
Let position 1 be the initial position. 
 
1
1 1 1
1 1 0
0.3 m
( ) 0, ( ) 2 m/s, 2 m/s
(0.3 0.5) 0.2 m
r
r
v v v
x r l
θ
=
= = =
= − = − = −
 
Let position 2 be when 0.6 m.r = 
 
2
2 2 2
2 2 0
0.6 m
( ) ?, ( ) ?, ?
(0.6 0.5) 0.1 m
r
r
v v v
x r l
θ
=
= = =
= − = − =
 
Conservation of angular momentum: 1 1 2 2
1 1
2
2
( ) ( )
( ) (0.3)(2)
( ) 1.000 m/s
0.6
r m v v m v
r v
v
r
θ θ
θ
θ
=
= = =
 
Conservation of energy: 
( )
1 1 2 2
2 2 2 2
1 1 2 2
2 2 2 2
2 1 1 2
2 2 2 2 2
2 2 2 2 2
2 2
1 1 1 1
2 2 2 2
1200
(2) (0.2) (0.1) 16 m /s
3
( ) ( ) 16 1 15 m /sr
T V T V
mv kx mv kx
k
v v x x
m
v v vθ
+ = +
+ = +
= + −
ª º= + − =¬ ¼
= − = − =
 
 3.87 m/srv = ± 3.87 m/srv = ± W�
 1.000 m/svθ = W�
 
 
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645 
 
 
PROBLEM 13.94 
Collar A has a mass of 3 kg and is attached to a spring of constant 
1200 N/m and of undeformed length equal to 0.5 m. The system is set 
in motion with r = 0.3 m, vθ = 2 m/s, and vr = 0. Neglecting the mass 
of the rod and the effect of friction, determine (a) the maximum 
distance between the origin and the collar, (b) the corresponding 
speed. (Hint: Solve the equation obtained for r by trial and error.) 
 
 
SOLUTION 
Let position 1 be the initial position. 
 
1
1 1 1
1 1 0
2 2
1
2 2
1 1
0.3 m
( ) 0, ( ) 2 m/s, 2 m/s
0.3 0.5 0.2 m
1 1
(3)(2) 6 J
2 2
1 1
(1200)( 0.2) 24 J
2 2
r
r
v v v
x r l
T mv
V kx
θ
=
= = =
= − = − = −
= = =
= = − =
 
Let position 2 be when r is maximum. 2( ) 0rv = 
 
2
2
2 2 2
2 2 2 2
2 2
2 2
( 0.5)
1 1
(3)( ) 1.5( )
2 2
1 1
(1200)( 0.5)
2 2
m
m
m
r r
x r
T mv v v
V kx r
θ θ
=
= −
= = =
= = −
 
 2600( 0.5)mr= − 
Conservation of angular momentum: 1 1 2 2
1
2
2
( ) ( )
(0.3) 0.6
( ) ( ), (2)
m m
r m v v m v
r
v v
r r r
θ θ
θ θ
=
= = =
 
Conservation of energy: 1 1 2 2
2 2
2
2
2
2
2
6 24 1.5( ) 600( 0.5)
0.6
30 (1.5) 600( 0.5)
0.54
( ) 600( 0.5) 30 0
m
m
m
m m
m
T V T V
v r
r
r
f r r
r
θ
+ = +
+ = + −
§ ·
= + −¨ ¸
© ¹
= + − − =
 
 
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646 
PROBLEM 13.94 (Continued) 
 
Solve for mr by trial and error. 
(m)mr 0.5 1.0 0.8 0.7 0.72 0.71 
( )mf r –27.8 120.5 24.8 –4.9 0.080 –2.469
 
 
(0.01)(0.08)
0.72 0.7197 m
2.467 0.08m
r = − =
+
 
(a) Maximum distance. 0.720 mmr = W 
(b) Corresponding speed. 
 2
0.6
( ) 0.8337 m/s
0.7197
vθ = = 
 2( ) 0rv = 2 0.834 m/sv = W�
 
 
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647 
 
 
PROBLEM 13.95 
A 4-lb collar A and a 1.5-lb collar B can slide without friction 
on a frame, consisting of the horizontal rod OE and the vertical 
rod CD, which is free to rotate about CD. The two collars are 
connected by a cord running over a pulley that is attached to 
the frame at O. At the instant shown, the velocity vA of collar 
A has a magnitude of 6 ft/s and a stop prevents collar B from 
moving. If the stop is suddenly removed, determine (a) the 
velocity of collar A when it is 8 in. from O, (b) the velocity of 
collar A when collar B comes to rest. (Assume that collar B 
does not hit O, that collar A does not come off rod OE, and 
that the mass of the frame is negligible.) 
 
SOLUTION 
Masses: 2
2
4
0.12422 lb s /ft
32.2
1.5
0.04658 lb s /ft
32.2
A
B
m
m
= = ⋅
= = ⋅
 
Constraint of the cord. Let r be the radial distance to the center of collar A and y be the distance that collar B 
moves up from its initial level. ; ry r y v= ∆ =� 
(a) Let position 1 be the initial position just after the stop at B is removed and position 2 be when the collar 
is 8 in. (0.66667 ft) from O. 
 
1 1
2
2
4 in. 0.33333 ft ( ) 0
8 in. 0.66667 ft
8 4 4 in. 0.33333 ft
rr v
r
r y
= = =
= =
∆ = = − = =
 
 Potential energy: 1 0,V = 
 2 2 (1.5)(0.33333) 0.5 ft lbBV W y= = = ⋅ 
 Conservation of angular momentum of collar A: 
 1 1 2 2( ) ( )A Am r v m r vθ θ= 
 1 12
2
( ) (0.33333)(6)
( ) 3 ft/s
0.66667
r v
v
r
θ
θ = = = 
 Conservation of energy: 1 1 2 2T V T V+ = + 
 2 2 2 2 2 21 1 1 2 2 2
1 1 1 1
[( ) ( ) ] [( ) ( ) ]
2 2 2 2A r B A r B
m v v m y m v v m y Vθ θ+ + = + + +� � 
 2 2 2 21 2 2 2
1 1 1
[0 ( ) ] 0 0 [( ) ( ) ] ( ) 0.5
2 2 2A A r B r
m v m v v m vθ θ+ + + = + + + 
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648 
PROBLEM 13.95 (Continued) 
 
 2 2 2 22 2
1 1 1
(0.12422)(6) (0.12422)[( ) (3) ] (0.04658)( ) 0.5
2 2 2r r
v v= + + + 
 2 22 22.236 0.06211( ) 0.559 0.02329( ) 0.5r rv v= + + + 
 220.0854( ) 1.177rv = 
 2 22( ) 13.78 ft /srv = 2( ) 3.71 ft/srv = W 
 2( ) 3.00 ft/svθ = W 
 4.77 ft/sv =W�
(b) Let position 3 be when collar B comes to rest. 
 3 3 3 30.33333, ( ) 0, 0ry r v y= − = =� 
 Conservation of angular momentum of collar A. 
 
1 1 3 3
1 1
3
3 3 3
( ) ( )
( ) (0.33333)(6) 2
( )
A Am r v m r v
r v
v
r r r
θ θ
θ
θ
=
= = =
 
 Conservation of energy: 1 1 3 3T V T V+ = + 
 2 2 2 2 2 21 1 1 3 3 3 3
1 1 1 1
[( ) ( ) ] [( ) ( ) ]
2 2 2 2A r B A r B B
m v v m y m v v m y w yθ θ+ + = + + +� � 
 
2
2
3
3
1 1 2
(0.12422)[0 (6) ] 0 (0.12422) 0 0 (1.5)( 0.33333)
2 2
r
r
ª º§ ·« »+ + = + + + −¨ ¸
« »© ¹¬ ¼
 
 32
3
0.24844
2.236 1.5 0.5r
r
= + − 
 3 23 31.5 2.736 0.24844 0r r− + = 
 Solving the cubic equation for r3, 
 3 1.7712 ft, 0.2805 ft, 0.33333 ftr = − 
 Since 3 1 0.33333 ft, the required root isr r> = 
 3 1.7712 ftr = 
 Corresponding velocity of collar A: 
 3( ) 0rv = W 
 3
3
2 2
( )
1.7712
v
rθ
= = 3( ) 1.129 ft/svθ = W�
� � � 3 1.129 ft/sv = W 
 
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649 
 
 
PROBLEM 13.96 
A 1.5-lb ball that can slide on a horizontal frictionless surface is attached to a 
fixed Point O by means of an elastic cord of constant 1 lb/in.k = and 
undeformed length 2 ft. The ball is placed at Point A, 3 ft from O, and given an 
initial velocity 0v perpendicular to OA. Determine (a) the smallest allowable 
value of the initial speed 0v if the cord is not to become slack, (b) the closest 
distance d that the ball will come to Point O if it is given half the initial speed 
found in part a. 
 
SOLUTION 
Let L1 be the initial stretched length of the cord and L2 the length of the closest approach to Point O if the cord 
does not become slack. Let position 1 be the initial state and position 2 be that of closest approach to Point O. 
The only horizontal force acting on the ball is the conservative central force due to the elastic cord. At the point 
of closest approach the velocity of the ball is perpendicular to the cord. 
Conservation of angular momentum: 1 1 2 2
1 0
1 0 2 2 2
2
 or 
r mv r mv
L v
L mv L mv v
L
=
= =
 
Conservation of energy: 1 1 2 2T V T V+ = + 
 2 2 2 21 1 0 2 2 0
2 2 2 2
1 2 1 0 2 0
2
2 2 2 21
0 0 1 0 2 02
2
1 1 1 1
( ) ( )
2 2 2 2
[( ) ( ) ]
[( ) ( ) ]
mv k L L mv k L L
k
v v L L L L
m
L k
v v L L L L
mL
+ − = + −
− = − − + −
− = − − + −
 
Data: 0 12 ft, 3 ftL L= = 
 2 0 2 ftL L= = for zero tension in the cord at the point of closest approach. 
 
2
1 lb/in. 12 lb/ft
/ 1.5/32.2 0.04658 lb s /ft
k
m W g
= =
= = = ⋅
 
 
2
2 2 2
0 02
2
0
(3) 12
[(3 2) (2 2) ]
0.04658(2)
1.25 257.6
v v
v
− = − − + −
− = −
 
(a) 2 2 20 206.1 ft /sv = 0 14.36 ft/sv = W 
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650 
PROBLEM 13.96 (Continued) 
 
(b) Let 210 2 (14.36 ft/s) 7.18 ft/sv = = so that the cord is slack in the position of closest approach to Point O. 
Let position 1 be the initial position and position 2 be position of closest approach with the cord being 
slack. 
 Conservation of energy: 1 1 2 2T V T V+ = + 
 2 2 20 1 0 2
1 1 1
( )
2 2 2
mv k L L mv+ − = 
 
2 2 2
2 0 0
2 2 2 2
2
( )
12
(7.18) (3 2) 309.17 ft /s
0.04658
17.583 ft/s
k
v v L L
m
v
= + −
= + − =
=
 
 Conservation of angular momentum: 
 
1 1 2 2
1 1 1 0
2
2 2
sin
sin
r mv r mv
r v L v
r d
v v
φ
φ
=
= = =
 
 
(3)(7.18)
17.583
d = 1.225 ftd = W�
 
 
 
 
 
 
 
 
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651 
 
 
PROBLEM 13.97 
A 1.5-lb ball that can slide on a horizontal frictionless surface is attached to a 
fixed Point O by means of an elastic cord of constant 1 lb/in.k = and undeformed 
length 2 ft. The ball is placed at Point A, 3 ft from O, and given an initial velocity 
0v perpendicular to OA, allowing the ball to come within a distance 9 in.d = of 
Point O after the cord has become slack. Determine (a) the initial speed 0v of the 
ball, (b) its maximum speed. 
 
SOLUTION 
Let L1 be the initial stretched length of the cord. Let position 1 be the initial position. Let position 2 be the 
position of closest approach to point after the cord has become slack. While the cord is slack there are no 
horizontal forces acting on the ball, so the velocity remains constant. While the cord is stretched, the only 
horizontal force acting on the ball is the conservative central force due to the elastic cord. At the point of 
closest approach the velocity of the ball is perpendicular to the radius vector. 
Conservation of angular momentum: 1 1 2 2
1
1 0 2 2 0or
r mv r mv
L
L v d v v v
d
=
= =
 
Conservation of energy: 1 1 2 2T V T V+ = + 
 
2 2 2
0 1 0 2
2 2 2
0 2 1 0
2
2 1
0 0 1 02
1 1 1
( ) 0
2 2 2
( )
( )
mv k L L mv
k
v v L L
m
L k
v v L L
md
+ − = +
− = − −
§ ·− = − −¨ ¸
© ¹
 
Data: 0 1
2
2 ft, 3 ft, 9 in. 0.75 ft
1 lb/in. 12 lb/ft
/ 1.5/32.2 0.04658 lb s /ft
L L d
k
m W g
= = = =
= =
= = = ⋅
 
 
2
2 20
0
2
0
3 12
(3 2)
0.75 0.04658
15 257.6
v
v
v
§ ·− = − −¨ ¸
© ¹
− = −
 
(a) 2 2 20 17.17 ft /sv = 0 4.14 ft/sv = W 
(b) Maximum speed. 02
3
0.75m
v
v v= = 16.58 ft/smv = W 
 
 
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652 
 
PROBLEM 13.98 
Using the principles of conservation of energy and conservation of angular momentum, solve part a of Sample 
Problem 12.9. 
 
SOLUTION 
 
0
6
0
0
6
3
3
6370 km
500 km 6370 km
6870 km 6.87 10 m
36,900 km/h
36.9 10 m
3.6 10 s
10.25 10 m/s
R
r
r
v
=
= +
= = ×
=
×=
×
= ×
 
Conservation of angular momentum: 
 
0 0 1 0 min 1 max
6
30
0
1 1
, ,
6.870 10
(10.25 10 )
A
A
r mv r mv r r r r
r
v v
r r′
= = =
§ ·§ · ×= = ר ¸¨ ¸ ¨ ¸© ¹ © ¹
 
 
9
1
70.418 10
Av r′
×= (1) 
Conservation of energy: 
Point A: 
 
3
0
2 3 2
0
6
0
2 2 6 2
12 3 2
6
0
12 3 2
6
6
10.25 10 m/s
1 1
(10.25 10 )
2 2
( )(52.53 10 )(J)
(9.81 m/s )(6.37 10 m)
398 10 m /s
6.87 10 m
(398 10 m /s )
(6.87 10 m)
57.93 10 (J)
A
A
A
A
v
T mv m
T m
GMm
V
r
GM gR
GM
r
m
V
m
= ×
= = ×
= ×
= −
= = ×
= ×
= ×
×= −
×
= − ×
 
 
 
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653 
PROBLEM 13.98 (Continued) 
 
Point A′: 
 2
1
12
1
6
1
2
398 10
(J)
52.53 10
A A
A
A A A A
T mv
GMm
V
r
m
r
T V T V
m
′ ′
′
′ ′
=
= −
×= −
+ = +
× 657.93 10 m− × 1
2
m=
12
2 398 10
A
m
v ′
×−
1r
 
Substituting for Av ′ from (1) 
 
9 2 12
6
2
11
21 12
6
2
11
(70.418 10 ) 398 10
5.402 10
(2)( )
(2.4793 10 ) 398 10
5.402 10
rr
rr
× ×− × = −
× ×− × = −
 
 6 2 12 211 1(5.402 10 ) (398 10 ) 2.4793 10 0r r× − × + × = 
 6 61 66.7 10 m, 6.87 10 mr = × × max 66,700 kmr = W 
 
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654 
 
 
PROBLEM 13.99 
Solve sample Problem 13.8, assuming that the elastic cord is 
replaced by a central force F of magnitude (80/r2) N directed 
toward O. 
PROBLEM 13.8 Skid marks on a drag racetrack indicate 
that the rear (drive) wheels of a car slip for the first 20 m of 
the 400-m track.(a) Knowing that the coefficient of kinetic 
friction is 0.60, determine the speed of the car at the end of 
the first 20-m portion of the track if it starts from rest and the 
front wheels are just off the ground. (b) What is the maximum 
theoretical speed for the car at the finish line if, after skidding 
for 20 m, it is driven without the wheels slipping for the 
remainder of the race? Assume that while the car is rolling 
without slipping, 60 percent of the weight of the car is on the 
rear wheels and the coefficient of static friction is 0.75. 
Ignore air resistance and rolling resistance. 
 
SOLUTION 
(a) The force exerted on the sphere passes through O. Angular momentum about O is conserved. 
 Minimum velocity is at B, where the distance from O is maximum. 
 Maximum velocity is at C, where distance from O is minimum. 
 
 sin 60A A m mr mv r mv° = 
 (0.5 m)(0.6 kg)(20 m/s)sin 60 (0.6 kg)m mr v° = 
 
8.66
m
m
v
r
= (1) 
 Conservation of energy: 
 At Point A, 2 2
2
1 1
(0.6 kg)(20 m/s) 120 J
2 2
80 80
,
80
160 J
0.5
A A
A
T mv
V Fdr dr
rr
V
= = =
−= = =
−= = −
³ ³
 
 At Point B, 2 2 2
1 1
(0.6 kg) 0.3
2 2B m m m
T mv v v= = = 
 
 
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655 
PROBLEM 13.99 (Continued) 
 
 and Point C : 
80
B
m
A A B B
V
r
T V T V
−=
+ = +
 
 2
80
120 160 0.3 m
m
v
r
− = − (2) 
 Substitute (1) into (2) 
 
2
2
8.66 80
40 (0.3)
2 0.5625 0
0.339 m and 1.661 m
m m
m m
m m
r r
r r
r r
§ ·
− = −¨ ¸
© ¹
− + =
′ = =
 
 max 1.661 mr = W 
 min 0.339 mr = W 
(b) Substitute mr′ and mr from results of part (a) into (1) to get corresponding maximum and minimum 
values of the speed. 
 
8.66
25.6 m/s
0.339m
v′ = = max 25.6 m/sv = W 
 
8.66
5.21 m/s
1.661m
v = = min 5.21 m/sv = W 
 
 
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656 
 
 
PROBLEM 13.100 
A spacecraft is describing an elliptic orbit of minimum 
altitude 2400Ah = km and maximum altitude 9600Bh = km 
above the surface of the earth. Determine the speed of the 
spacecraft at A. 
 
SOLUTION 
 
6370 km 2400 km
8770 km
6370 km 9600 km
15,970 km
A
A
B
r
r
r
= +
=
= +
=
 
Conservation of momentum: A A B Br mv r mv= 
 
8770
0.5492
15,970
A
B A A A
B
r
v v v v
r
= = = (1) 
Conservation of energy: 2 2
1 1
2 2A A A B B BA B
GMm GMm
T mv V T mv V
r r
− −= = = = 
 
2 2 3 2 12 3 2
12
6
3
12
3
(9.81 m/s )(6370 10 m) 398.1 10 m /s
(398.1 10 )
45.39 10
8770 10
(398.1 10 )
24.93 
(15,970 10 )
A
B
GM gR
m
V m
m
V m
= = × = ×
− ×= = − ×
×
− ×= = −
×
 
 
:
1
 
2
A A B BT V T V
m
+ = +
2 645.39 10Av m− ×
1
2
m= 2 624.93 10Bv m− × (2) 
Substituting for Bv in (2) from (1) 
 
2 2 6
2 6 2 2
[1 (0.5492) ] 40.92 10
58.59 10 m /s
A
A
v
v
− = ×
= ×
 
 37.65 10 m/sAv = × 
327.6 10 km/hAv = × W�
 
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657 
 
PROBLEM 13.101 
While describing a circular orbit, 185 mi above the surface of the 
earth, a space shuttle ejects at Point A an inertial upper stage (IUS) 
carrying a communication satellite to be placed in a geosynchronous 
orbit (see Problem 13.87) at an altitude of 22,230 mi above the 
surface of the earth. Determine (a) the velocity of the IUS relative 
to the shuttle after its engine has been fired at A, (b) the increase in 
velocity required at B to plae the satellite in its final orbit. 
 
SOLUTION 
For earth, 6
2
2 6 2 15 3 2
3960 mi 20.909 10 ft
32.2 ft
(32.2)(20.909 10 ) 14.077 10 ft /s
R
g
GM gR
= = ×
=
= = × = ×
 
Speed on a circular orbits of radius r, rA, and rB. 
 
2
2
2
nF ma
GMm mv
rr
GM GM
v v
r r
=
=
= =
 
 
6
15
3
circ 6
6
15
3
circ 6
3960 185 4145 mi 21.886 10 ft
14.077 10
( ) 25.362 10 ft/s
21.886 10
3960 22230 26190 mi 138.283 10 ft
14.077 10
( ) 10.089 10 ft/s
138.283 10
A
A
B
B
r
v
r
v
= + = = ×
×= = ×
×
= + = = ×
×= = ×
×
 
Calculate speeds at A and B for path AB. 
Conservation of angular momentum: sin sinA A A B B Amr v mr vφ φ= 
 
6
6
sin 90 21.886 10
0.15816
sin 90 138.283 10
A A A
B A
B
r v v
v v
r
° ×= = =
° ×
 
 
 
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658 
PROBLEM 13.101 (Continued) 
 
Conservation of energy: A A B BT V T V+ = + 
 
2 2
2 2
1 1
2 2
2 ( )1 1
2
A B
A B
B A
A B
A B A B
GMm GMm
mv mv
r r
GM r r
v v GM
r r r r
+ = −
§ · −− = − =¨ ¸
© ¹
 
 
15 6
2 2
6 6
2 9
3
6 3
(2)(14.077 10 )(116.397 10 )
(0.15816 )
(21.886 10 )(138.283 10 )
0.97499 1.082796 10
33.325 10 ft/s
(0.15816)(33.325 10 ) 5.271 10 ft/s
A A
A
A
B
v v
v
v
v
× ×− =
× ×
= ×
= ×
= × = ×
 
(a) Increase in speed at A: 
 3 3 333.325 10 25.362 10 7.963 10 ft/sAv∆ = × − × = × 7960 ft/sAv∆ = W 
(b) Increase in speed at B: 
 3 3 310.089 10 5.271 10 4.818 10 ft/sBv∆ = × − × = × 4820 ft/sBv∆ = W 
 
 
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659 
 
 
PROBLEM 13.102 
A spacecraft approaching the planet Saturn reaches Point A with a 
velocity Av of magnitude 
368.8 10 ft/s.× It is to be placed in an elliptic 
orbit about Saturn so that it will be able to periodically examine Tethys, 
one of Saturn’s moons. Tethys is in a circular orbit of radius 3183 10 mi× 
about the center of Saturn, traveling at a speed of 337.2 10 ft/s.× 
Determine (a) the decrease in speed required by the spacecraft at A to 
achieve the desired orbit, (b) the speed of the spacecraft when it reaches 
the orbit of Tethys at B. 
 
SOLUTION 
�
�
�
�
 
(a) 
 
6
6
607.2 10 ft
966.2 10 ft
A
B
r
r
= ×
= ×
 
Av′ = speed of spacecraft in the elliptical orbit after its speed has been 
decreased. 
Elliptical orbit between A and B. 
Conservation of energy 
Point A: 2
sat
1
2A A
A
A
T mv
GM m
V
r
′=
−=
 
saM = Mass of Saturn, determine saGM from the speed of Tethys in 
its circular orbit. 
(Eq. 12.44) 2satcirc sat circB
GM
v GM r v
r
= = 
 
6 2 3 2
sat
18 3 2
18 3 2
6
9
(966.2 10 ft )(37.2 10 ft/s)
1.337 10 ft /s
(1.337 10 ft /s )
(607.2 10 ft)
2.202 10
A
GM
m
V
m
= × ×
= ×
×= −
×
= − ×
 
�
 
 
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660 
PROBLEM 13.102 (Continued) 
 
Point B: 
18 3 2
2 sat
6
9
1 (1.337 10 ft /s )
2 (966.2 10 ft)
1.384 10
B B B
B
B
GM m m
T mv V
r
V
− ×= = = −
×
= ×
 
 
; 
1
2
A A B BT V T V
m
+ = +
2 92.202 10Av m′ − ×
1
2
m= 2 91.384 10Bv m− ×
 
 2 2 91.636 10A Bv v′ − = × 
Conservation of angular momentum: 
 
6
6
2 2 9
607.2 10
0.6284
966.2 10
[1 (0.6284) ] 1.636 10
52,005 ft/s
A
A A B B B A A A
B
A
A
r
r mv r mv v v v v
r
v
v
×′ ′ ′ ′= = = =
×
′ − = ×
′ =
 
(a) 68,800 52,005A A Av v v′∆ = − = − 16,795 ft/sAv∆ = W 
(b) (0.6284)(52,005)AB A
B
r
v v
r
′= = 32,700 ft/sBv = W
 
 
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661 
 
 
PROBLEM 13.103 
A spacecraft traveling along a parabolic path toward the planet Jupiter 
is expected to reach Point A with a velocity vA of magnitude 26.9 km/s. 
Its engines will then be fired to slow it down, placing it into an elliptic 
orbit which will bring it to within 100 × 103 km of Jupiter. Determine the 
decrease in speed v∆ at Point A which will place the spacecraft into the 
required orbit. The mass of Jupiter is 319 times the mass of the earth. 
 
SOLUTION 
Conservation of energy. 
Point A: 
 
2
2
6
2 6 2
15 3 2
6
15 3 2
6
6
1
( )
2
 319 319
6.37 10 m
(319)(9.81 m/s )(6.37 10 m)
126.98 10 m /s
350 10 m
(126.98 10 m /s )
(350 10 m)
(362.8 10 )
A A A
J
A
A
J E E
E
J
J
A
A
A
T m v v
GM m
V
r
GM GM gR
R
GM
GM
r
m
V
V m
= − ∆
−=
= =
= ×
= ×
= ×
= ×
− ×=
×
= − ×
 
Point B: 
 
2
15 3 2
6
6
2 6 2 6
1
2
(126.98 10 m /s )
(100 10 )
(1269.8 10 )
1 1
( ) 362.8 10 1269.8 10 m
2 2
B B
J
B
B
B
A A B B
A A B
T mv
GM m m
V
r m
V m
T V T V
m v v m mv
=
− − ×= =
×
= − ×
+ = +
− ∆ − × = − ×
 
 2 2 6( ) 1814 10A A Bv v v− ∆ − = − × (1) 
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662 
PROBLEM 13.103 (Continued) 
 
Conservation of angular momentum. 
 
6
6
350 10 m
100 10 m
( )
A
B
A A A B B
r
r
r m v v r mv
= ×
= ×
− ∆ =
 
 
( )
350
( )
100
A
B A A
B
A A
r
v v v
r
v v
§ ·
= − ∆¨ ¸
© ¹
§ ·= − ∆¨ ¸
© ¹
 (2) 
Substitute Bv in (2) into (1) 
 
2 2 6
2 6
3
( ) [1 (3.5) ] 1814 10
( ) 161.24 10
( ) 12.698 10 m/s
A A
A A
A A
v v
v v
v v
− ∆ − = − ×
− ∆ = ×
− ∆ = ×B
 
(Take positive root; negative root reverses flight direction.) 
 326.9 10 m/sAv = × (given) 
 3 3(26.9 10 m/s 12.698 10 m/s)Av∆ = × − × 
 14.20 km/sAv∆ = W 
 
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663 
 
PROBLEM 13.104 
As a first approximation to the analysis of a space flight from the 
earth to Mars, it is assumed that the orbits of the earth and Mars are 
circular and coplanar. The mean distances from the sun to the earth 
and to Mars are 6149.6 10× km and 6227.8 10× km, respectively. 
To place the spacecraft into an elliptical transfer orbit at Point A, its 
speed is increased over a short interval of time to Av which is faster 
than the earth’s orbital speed. When the spacecraft reaches Point B on 
the elliptical transfer orbit, its speed Bv is increased to the orbital 
speed of Mars. Knowing that the mass of the sun is 3332.8 10× 
times the mass of the earth, determine the increase in velocity required 
(a) at A, (b) at B. 
 
SOLUTION 
mass of the sunM = 
3 2 6 2 20 3 2332.8(10) (9.81 m/s )(6.37 10 m) 1.3247(10) m /sGM = × = 
Circular orbits 9Earth 29.758 m/s149.6(10)E
GM
v = = 
 9Mars 24.115 m/s227.8(10)M
GM
v = = 
Conservation of angular momentum 
Elliptical orbit (149.6) (227.8)A Bv v= 
Conservation of energy 
 2 29 9
1 1
2 2149.6(10) 227.8(10)A B
GM GM
v v− = − 
 
(227.8)
1.52273
(149.6)A B B
v v v= = 
 
20 20
2 2 2
9 9
1 1.3247(10) 1 1.3247(10)
(1.52273)
2 2149.6(10) 227.8(10)B B
v v− = − 
2 80.65935 3.0398(10)Bv = 
 2 84.6102(10)Bv = 
 21, 471 m/s, 32,695 m/sB Av v= = 
(a) Increase at A, 32.695 29.758 2.94 km/sA Ev v− = − = W�
(b) Increase at B, 24.115 21.471 2.64 km/sB Mv v− = − = W 
 
 
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664 
 
PROBLEM 13.105 
The optimal way of transferring a space vehicle from an inner circular 
orbit to an outer coplanar circular orbit is to fire its engines as it passes 
through A to increase its speed and place it in an elliptic transfer orbit. 
Another increase in speed as it passes through B will place it in the 
desired circular orbit. For a vehicle in a circular orbit about the earth at 
an altitude 1 200h = mi, which is to be transferred to a circular orbit 
at an altitude 2 500h =mi, determine (a) the required increases in speed 
at A and at B, (b) the total energy per unit mass required to execute the 
transfer. 
 
SOLUTION 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Elliptical orbit between A and B 
Conservation of angular momentum 
A A B Bmr v mr v= 
7.170
6.690
B
A B B
A
r
v v v
r
= = 
66370 km 320 km 6690 km, 6.690 10 mA Ar r= + = = × 
 1.0718A Bv v= (1) 
66370 km 800 km 7170 km, 7.170 10 mB Br r= + = = × 
 6(6370 km) 6.37 10 mR = = × 
Conservation of energy 
2 2 6 2 12 3 2(9.81 m/s )(6.37 10 m) 398.060 10 m /sGM gR= = × = × 
Point A: 
12
2
6
1 (398.060 10 )
 
2 (6.690 10 )A A A A
GMm m
T mv V
r
×= = − = −
×
 
659.501 10 mAV = × 
Point B: 
12
2
6
1 (398.060 10 )
 
2 (7.170 10 )B B B B
GMm m
T mv V
r
×= = − = −
×
 
655.5 10 mBV = × 
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665 
 
 
 
 
PROBLEM 13.105 (Continued) 
 
A A B BT V T V+ = + 
2 6 2 61 159.501 10 m 55.5 10 m
2 2A B
mv mv− × = − × 
 2 2 68.002 10A Bv v− = × 
From (1) 2 2 61.0718 [(1.0718) 1] 8.002 10A B Bv v v= − = × 
2 6 2 253.79 10 m /s , 7334 m/sB Bv v= × = 
(1.0718)(7334 m/s) 7861 m/sAv = = 
Circular orbit at A and B 
(Equation 12.44) 
12
6
398.060 10
( ) 7714 m/s
6.690 10A C A
GM
v
r
×= = =
×
 
12
6
398.060 10
( ) 7451 m/s
7.170 10B C B
GM
v
r
×= = =
×
 
(a) Increases in speed at A and B 
 ( ) 7861 7714 147 m/sA A A Cv v v∆ = − = − = W 
 ( ) 7451 7334 117 m/sB B C Bv v v∆ = − = − = W 
(b) Total energy per unit mass 
2 2 2 21/ [( ) ( ) ( ) ( ) ]
2 A A C B C B
E m v v v v= − + − 
2 2 2 21/ [(7861) (7714) (7451) (7334) ]
2
E m = − + − 
 6/ 2.01 10 J/kgE m = × W 
 
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666 
 
 
PROBLEM 13.106 
During a flyby of the earth, the velocity of a spacecraft is 10.4 km/s as it 
reaches its minimum altitude of 990 km above the surface at Point A. At Point B 
the spacecraft is observed to have an altitude of 8350 km. Determine (a) the 
magnitude of the velocity at Point B, (b) the angle .Bφ 
 
SOLUTION 
At A: 4 6 6[1.04(10) m/s][6.37(10) m 0.990(10) m]Ah vr= = + 
 9 276.544(10) m /sAh = 
 2
1 1
( )
2A A
GM
T V v
m r
+ = − 
 
6 2
4 2
6 6
1 (9.81)[6.37(10) ]
[1.04(10) ] 0
2 [6.37(10) 0.990(10) ]
= − ≅
+
 
(Parabolic orbit) 
At B: 2
1 1
( ) 0
2B B B B
GM
T V v
m r
+ = − = 
 
6 2
2
6 6
1 (9.81)[6.37(10 )]
2 [6.37(10) 8.35(10) ]B
v =
+
 
 2 654.084(10)Bv = 
(a) 7.35 km/sBv = W 
9sin 76.544(10)B B B Bh v rφ= = 
 
9
6 6 6
76.544(10)
sin
7.35(10 )[6.37(10) 8.35(10) ]B
φ =
+
 
 0.707483= 
(b) 45.0Bφ = °W 
 
 
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667 
 
 
PROBLEM 13.107 
A space platform is in a circular orbit about the earth at an altitude of 
300 km. As the platform passes through A, a rocket carrying a 
communications satellite is launched from the platform with a 
relative velocity of magnitude 3.44 km/s in a direction tangent to the 
orbit of the platform. This was intended to place the rocket in an 
elliptic transfer orbit bringing it to Point B, where the rocket would 
again be fired to place the satellite in a geosynchronous orbit of 
radius 42,140 km. After launching, it was discovered that the relative 
velocity imparted to the rocket was too large. Determine the angle γ 
at which the rocket will cross the intended orbit at Point C. 
 
SOLUTION 
 
6
6
2
6 2
12 3 2
6370 km
6370 km 300 km
6.67 10 m
42.14 10 m
(9.81 m/s)(6.37 10 m)
398.1 10 m /s
A
A
C
R
r
r
r
GM gR
GM
GM
=
= +
= ×
= ×
=
= ×
= ×
 
For any circular orbit: 
 circ
2
n n
mv
F ma
r
= = 
 
circ
2
2
circ
n
mvGMm
F m
rr
GM
v
r
= =
=
 
Velocity at A: 
 
12 3 3
3
circ 6
3 3 3
circ
(398.1 10 m /s )
( ) 7.726 10 m/s
(6.67 10 m)
( ) ( ) 7.726 10 3.44 10 11.165 10 m/s
A
A
A A A R
GM
v
r
v v v
×= = = ×
×
= + = × + × = ×
 
 
 
 
 
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668 
PROBLEM 13.107 (Continued) 
 
Velocity at C: 
Conservation of energy: 
1
2
A A C CT V T V
m
+ = +
2
A
GM m
v − 1
2A
m
r
= 2C
GM m
v −
Cr
 
 
2 2
3 2 12
6 6
2 6 6
6 2 2
3
1 1
2
1 1
(11.165 10 ) 2(398.1 10 )
42.14 10 6.67 10
124.67 10 100.48 10
24.19 10 m /s
4.919 10 m/s
C A
C A
C
C
v v GM
r r
v
v
§ ·
= + −¨ ¸
© ¹
§ ·= × + × −¨ ¸× ש ¹
= × − ×
= ×
= ×
 
Conservation of angular momentum: 
 
6 3
6 3
cos
cos
(6.67 10 )(11.165 10 )
(42.14 10 )(4.919 10 )
cos 0.35926
A A C C
A A
C C
r mv r mv
r v
r v
γ
γ
γ
=
=
× ×=
× ×
=
 
 68.9γ = ° W 
 
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669 
 
 
PROBLEM 13.108 
A satellite is projected into space with a velocity v0 at a distance r0 from the 
center of the earth by the last stage of its launching rocket. The velocity v0 
was designed to send the satellite into a circular orbit of radius r0. However, 
owing to a malfunction of control, the satellite is not projected horizontally 
but at an angle α with the horizontal and, as a result, is propelled into an 
elliptic orbit. Determine the maximum and minimum values of the distance 
from the center of the earth to the satellite. 
 
SOLUTION 
 
 
 
 
 
 
For circular orbit of radius 0r 
 
2
0
2
00
2
0
0
n
vGMm
F ma m
rr
GM
v
r
= =
=
 
But 0v forms an angle α with the intended circular path. 
For elliptic orbit. 
Conservation of angular momentum: 
 0 0 cos A Ar mv r mvα = 
 0 0cosA
A
r
v v
r
α§ ·= ¨ ¸
© ¹
 (1) 
Conservation of energy: 
 
2 2
0
0
2 2 0
0
0
1 1
2 2
2
1
A
A
A
A
GMm GMm
mv mv
r r
rGM
v v
r r
− = −
§ ·
− = −¨ ¸
© ¹
 
Substitute for Av from (1) 
 
2
2 20 0
0
0
2
1 cos 1
A A
r rGM
v
r r r
α
ª º§ · § ·« »− = −¨ ¸ ¨ ¸
« »© ¹ © ¹¬ ¼
 
But 20
0
,
GM
v
r
= thus 
2
20 01 cos 2 1
A A
r r
r r
α§ · § ·− = −¨ ¸ ¨ ¸
© ¹ © ¹
 
 
2
2 0 0cos 2 1 0
A A
r r
r r
α § · § ·− + =¨ ¸ ¨ ¸
© ¹ © ¹
 
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670 
PROBLEM 13.108 (Continued) 
 
Solving for 0
A
r
r
 
 
2
0
2 2
0 0
2 4 4cos 1 sin
2cos 1 sin
(1 sin )(1 sin )
(1 sin )
1 sin
A
A
r
r
r r r
α α
α α
α α α
α
+ ± − ±= =
−
+ −= =
±
B
 
 also valid for Point A′ 
Thus, 
 max 0(1 sin )r rα= + min 0(1 sin )r rα= − W 
 
 
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671 
 
 
PROBLEM 13.109 
Upon the LEM’s return to the command module, the Apollo spacecraft of 
Problem 13.88 was turned around so that the LEM faced to the rear. The 
LEM was then cast adrift with a velocity of 200 m/s relative to the command 
module. Determine the magnitude and direction (angle φ formed with the 
vertical OC) of the velocity vC of the LEM just before it crashed at C on the 
moon’s surface. 
 
SOLUTION 
Command module in circular orbit 
 
6
2
moon earth
6 2 12 3 2
1740 140 1880 km 1.88 10 m
0.0123 0.0123 
0.0123(9.81)(6.37 10 ) 4.896 10 m /s
Br
GM GM gR
= + = = ×
= =
= × = ×
 
 1740 kmR = 
 
2
0
2
12
0 6
4.896 10
1.88 10
m
n
BB
B
GM m mv
F ma
rr
GMm
v
r
Σ = =
×= =
×
 
 0 1614 m/s 1614 200 1414 m/sBv v= = − = 
Conservation of energy between B and C: 
 
2 2
2 2
12 3 2 6
2 2
6 6
1 1
2 2
2
1
(4.896 10 m /s ) 1.88 10
(1414 m/s) 2 1
(1.88 10 m) 1.74 10
m m
B C C
B C
B
C B
B
C
GM m GM m
mv mv r R
r r
rGMm
v v
r R
v
− = − =
§ ·= + −¨ ¸
© ¹
§ ·× ×= + −¨ ¸¨ ¸× ש ¹
 
 2 6 6 6 2 21.999 10 0.4191 10 2.418 10 m /sCv = × + × = × 1555 m/sCv = W 
Conservation of angular momentum: 
 6
6
sin
(1.88 10 m)(1414 m/s)
sin 0.98249
(1.74 10 m)(1555 m/s)
B B C
B B
C C
r mv Rmv
r v
r v
φ
φ
=
×= = =
×
 
 79.26φ = ° 79.3φ = ° W 
 
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672 
 
 
PROBLEM 13.110 
A space vehicle is in a circular orbit at an altitude of 225 mi above the 
earth. To return to earth, it decreases its speed as it passes through A by 
firing its engine for a short interval of time in a direction opposite to the 
direction of its motion. Knowing that the velocity of the space vehicle 
should form an angle 60°Bφ = with the vertical as it reaches Point B at 
an altitude of 40 mi, determine (a) the required speed of the vehicle as 
it leaves its circular orbit at A, (b) its speed at Point B. 
 
SOLUTION 
(a) 
3
3
3
2 2 3 2
15 3 2
3960 mi 225 mi 4185 mi
4185 mi 5280 ft/mi 22,097 10 ft
3960 mi 40 mi 4000 mi
4000 5280 21,120 10 ft
3960 mi 20,909 10 ft
(32.2 ft/s )(20,909 10 ft)
14.077 10 ft /s
A
A
B
B
r
r
r
r
R
GM gR
GM
= + =
= × = ×
= + =
= × = ×
= = ×
= = ×
= ×
 
Conservation of energy: 
 
2
15
3
6
2
15
3
6
1
2
14.077 10
22,097 10
637.1 10
1
2
14.077 10
21,120 10
666.5 10
1
2
A A
A
A
B B
B
B
A A B B
T mv
GMm
V
r
m
m
T mv
GMm
V
r
m
m
T V T V
m
=
−=
− ×=
×
= − ×
=
−=
− ×=
×
= − ×
+ = +
2 6637.1 10Av m− ×
1
2
m= 2 6666.5 10Bv m− ×
 
 2 2 658.94 10A Bv v= − × (1) 
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673 
PROBLEM 13.110 (Continued) 
 
Conservation of angular momentum: 
 
sin
( ) 4185 1
( )(sin ) 4000 sin 60
A A B B B
A A
B A
B B
r mv r mv
r v
v v
r
φ
φ
=
§ ·= = ¨ ¸°© ¹
 
 1.208B Av v= (2) 
Substitute Bv from (2) in (1) 
 
2 2 6
2 2 6
2 6 2 2
(1.208 ) 58.94 10
[(1.208) 1] 58.94 10
128.27 10 ft /s
A A
A
A
v v
v
v
= − ×
− = ×
= ×
 
(a) 311.32 10 ft/sAv = × W 
(b) From (2) 
 6
3
1.208
1.208(11.32 10 )
13.68 10 ft/s
B Av v=
= ×
= ×
 
 313.68 10 ft/sBv = × W 
 
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674 
 
PROBLEM 13.111* 
In Problem 13.110, the speed of the space vehicle was decreased as it passed through A by firing its engine in 
a direction opposite to the direction of motion. An alternative strategy for taking the space vehicle out of its 
circular orbit would be to turn it around so that its engine would point away from the earth and then give it an 
incremental velocity A∆v toward the center O of the earth. This would likely require a smaller expenditure of 
energy when firing the engine at A, but might result in too fast a descent at B. Assuming this strategy is used 
with only 50 percent of the energy expenditure used in Problem 13.109, determine the resulting values of Bφ 
and .Bv 
 
SOLUTION 
 
6
6
2 2 2
15 3 2
3960 mi 225 mi
4185 mi 22.097 10 ft
3960 mi 40 mi 4000 mi
21.120 10 ft
(32.2 ft/s )[(3960)(5280) ft ]
14.077 10 ft /s
A
A
B
B
r
r
r
r
GM gR
GM
= +
= = ×
= + =
= ×
= =
= ×
 
Velocity in circular orbit at 225 m altitude: 
 
Newton’s second law 
 
2
circ
2
15
circ 6
3
( )
:
14.077 10
( )
22.097 10
25.24 10 ft/s
A
n
AA
A
A
m vGMm
F ma
rr
GM
v
r
= =
×= =
×
= ×
 
Energy expenditure: 
From Problem 13.110, 311.32 10 ft/sAv = × 
Energy, 2 2109 circ
3 2 3 2
109
6
109
6
110 109
1 1
( )
2 2
1 1
(25.24 10 ) (11.32 10 )
2 2
254.46 10 ft lb
(254.46 10 )
(0.50) ft lb
2
A AE m v mv
E m m
E m
m
E E
∆ = −
∆ = × − ×
∆ = × ⋅
×∆ = ∆ = ⋅
 
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675 
PROBLEM 13.111* (Continued) 
 
Thus, additional kinetic energy at A is 
 
6
2
110
1 (254.46 10 )
( ) ft lb
2 2A
m
m v E
×∆ = ∆ = ⋅ (1) 
Conservation of energy between A and B: 
 2 2circ
1
[( ) ( ) ]
2A A A A A
GMm
T m v v V
r
−= + ∆ = 
 2
1
2B B B A
GMm
T mv V
r
−= = 
 A A B BT V T V+ = + 
 
6 15 15
3 2 2
6 6
1 254.46 10 14.077 10 1 14.077 10
(25.24 10 )
2 2 222.097 10 21.120 10
B
m m m
m mv
× × ×× + − = −
× ×
 
 
2 6 6 6 6
2 3
637.06 10 254.46 10 1274.1 10 1333 10
950.4 10
B
B
v
v
= × + × − × + ×
= ×
 
 330.88 10 ft/sBv = × W�
Conservation of angular momentum between A and B: 
 circ( ) sinA AB B Br m v r mv φ= 
 
3
circ
3
( ) (4185) (25.24 10 )
sin 0.8565
( ) (4000) (30.88 10 )
AA
B
B B
vr
r v
φ § · ×= = =¨ ¸ ש ¹
 
 58.9Bφ = ° W 
 
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676 
 
 
PROBLEM 13.112 
Show that the values vA and vP of the speed of an earth satellite 
at the apogee A and the perigee P of an elliptic orbit are defined 
by the relations 
2 22 2P A
A P
A P A A P P
r rGM GM
v v
r r r r r r
= =
+ +
 
where M is the mass of the earth, and Ar and Pr represent, 
respectively, the maximum and minimum distances of the orbit 
to the center of the earth. 
 
SOLUTION 
Conservation of angular momentum: 
 A A P Pr mv r mv= 
 PA P
A
r
v v
r
= (1) 
Conservation of energy: 
 2 2
1 1
2 2P AP A
GMm GMm
mv mv
r r
− = − (2) 
Substituting for Av from (1) into (2) 
 
2
2 22 2P
P P
P A A
rGM GM
v v
r r r
§ ·
− = −¨ ¸
© ¹
 
 
2
2 1 11 2P P
A P A
r
v GM
r r r
§ ·§ · § ·¨ ¸− = −¨ ¸ ¨ ¸¨ ¸© ¹ © ¹© ¹
 
 
2 2
2
2
2A P A PP
A PA
r r r r
v GM
r rr
− −= 
with 2 2 ( )( )A P A P A Pr r r r r r− = − + 
 2
2 A
P
A P P
rGM
v
r r r
§ ·
= ¨ ¸+ © ¹
 (3)ҏ W 
Exchanging subscripts P and A 
 2
2
 Q.E.D.PA
A P A
rGM
v
r r r
§ ·
= ¨ ¸+ © ¹
 W 
 
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677 
 
 
PROBLEM 13.113 
Show that the total energy E of an earth satellite of mass m 
describing an elliptic orbit is /( ),A PE GMm r r= − + where M is 
the mass of the earth, and Ar and Pr represent, respectively, 
the maximum and minimum distances of the orbit to the center 
of the earth. (Recall that the gravitational potential energy of a 
satellite was defined as being zero at an infinite distance from 
the earth.) 
 
SOLUTION 
See solution to Problem 13.112 (above) for derivation of Equation (3). 
 2
2
( )
A
P
A P P
rGM
v
r r r
=
+
 
Total energy at Point P is 
 
2
0
1
2
1 2
2 ( )
1
( )
( )
( )
P P P
P
A
A P P
A
P A P P
A A P
P A P
GMm
E T V mv
r
rGMm GMm
r r r r
r
GMm
r r r r
r r r
GMm
r r r
= + = −
ª º
= −« »+¬ ¼
ª º
= −« »+¬ ¼
− −=
+
 
 
A P
GMm
E
r r
= −
+
 W 
Note: Recall that gravitational potential of a satellite is defined as being zero at an infinite distance from the 
earth.�
 
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678 
 
PROBLEM 13.114* 
A space probe describes a circular orbit of radius nR with a velocity v0 about a planet of radius R and center O. 
Show that (a) in order for the probe to leave its orbit and hit the planet at an angle θ with the vertical, its 
velocity must be reduced to α v0, where 
2 2
2( 1)
sin
sin
n
n
α θ
θ
−=
−
 
(b) the probe will not hit the planet if α is larger than 2 /(1 ).n+ 
 
SOLUTION 
(a) Conservation of energy: 
 At A: 20
1
( )
2A
A
T m v
GMm
V
nR
α=
= −
 
 At B: 2
1
2B
B
T mv
GMm
V
R
=
= −
 
 M = mass of planet 
 m = mass of probe 
 A A B BT V T V+ = + 
 2 20
1 1
( )
2 2
GMm GMm
m v mv
nR R
α − = − (1) 
 Conservation of angular momentum: 
 0 sinnR m v Rmvα θ= 
 0
sin
n v
v
α
θ
= (2) 
 Replacing v in (1) by (2) 
 
2
2 0
0
2 2
( )
sin
n vGM GM
v
nR R
αα
θ
§ ·− = −¨ ¸
© ¹
 (3) 
 
 
 
 
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679 
PROBLEM 13.114* (Continued) 
 
 For any circular orbit. 
 
2
n
v
a
r
= 
 Newton’s second law 
 
circ
2
2
circ
( )m vGMm
rr
GM
v
r
− =
=
 
 For ,r nR= 0 circ
GM
v v
nR
= = 
 Substitute for v0 in (3) 
 
2 2
2
2
2
2
2
2 2
2
2 2 2 2
2 2
sin
1 2(1 )
sin
2(1 )(sin ) 2( 1)sin
(sin ) ( sin )
GM GM n GM GM
nR nR nR R
n
n
n n
n n
αα
θ
α
θ
θ θα
θ θ
§ ·− = −¨ ¸
© ¹
ª º
− = −« »
¬ ¼
− −= =
− −
 
 
2 2
2( 1)
sin
sin
n
n
α θ
θ
−=
−
 Q.E.D. W 
(b) Probe will just miss the planet if 90 ,θ > ° 
 
2 2
2( 1) 2
sin 90
1sin 90
n
nn
α −= ° =
+− °
 W�
 Note: 2 1 ( 1)( 1)n n n− = − + �
 
 
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680 
 
PROBLEM 13.115 
A missile is fired from the ground with an initial velocity 0v forming an angle 0φ with the vertical. If the 
missile is to reach a maximum altitude equal to ,Rα where R is the radius of the earth, (a) show that the 
required angle 0φ is defined by the relation 
2
esc
0
0
sin (1 ) 1
1
v
v
αφ α
α
§ ·
= + − ¨ ¸+ © ¹
 
where escv is the escape velocity, (b) determine the range of allowable values of 0.v 
 
SOLUTION 
(a) Ar R= 
 Conservation of angular momentum: 
 0 0sin B BRmv r mvφ = 
 (1 )Br R R Rα α= + = + 
 0 0 0 0
sin sin
(1 ) (1 )B
Rv v
v
R
φ φ
α α
= =
+ +
 (1) 
 Conservation of energy: 
 2 20
1 1
 
2 2 (1 )A A B B B
GMm GMm
T V T V mv mv
R Rα
+ = + − = −
+
 
 2 20
2 1 2
1
1 1B
GMm GMm
v v
R R
α
α α
§ · § ·− = − =¨ ¸ ¨ ¸+ +© ¹ © ¹
 
 Substitute for Bv from (1) 
 
2
2 0
0 2
sin 2
1
1(1 )
GMm
v
R
φ α
αα
§ · § ·− =¨ ¸ ¨ ¸¨ ¸ ++ © ¹© ¹
 
 From Equation (12.43): 2esc
2GM
v
R
= 
 
2
2 20
0 esc2
sin
1
1(1 )
v v
φ α
αα
§ · § ·− =¨ ¸ ¨ ¸¨ ¸ ++ © ¹© ¹
 
 
22
0 esc
2
0
sin
1
1(1 )
v
v
φ α
αα
§ ·
= − ¨ ¸ ++ © ¹
 (2) 
 
2
esc
0
0
sin (1 ) 1
1
v
v
αφ α
α
§ ·
= + − ¨ ¸+ © ¹
 Q.E.D. 
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681 
PROBLEM 13.115 (Continued) 
 
(b) Allowable values of 0v (for which maximum altitude = αR) 
 2 00 sin 1φ< < 
 For 0sin 0,φ = from (2) 
 
2
esc
0
0 esc
0 1
1
1
v
v
v v
α
α
α
α
§ ·
= − ¨ ¸ +© ¹
=
+
 
 For 0sin 1,φ = from (2) 
 
2
esc
2
0
2 2
esc
0
0 esc
1
1
1(1 )
1 1 1 2 1 2
1
1 (1 ) 1
1
2
v
v
v
v
v v
α
αα
α α αα
α α α α α
α
α
§ ·
= − ¨ ¸ ++ © ¹
§ · + + − +§ ·= + − = =¨ ¸ ¨ ¸+ + +© ¹© ¹
+=
+
 
 esc 0 esc
1
1 2
v v v
α α
αα
+< <
+ +
 W 
 
 
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682 
 
 
PROBLEM 13.116 
A spacecraft of mass m describes a circular orbit of radius 1r around 
the earth. (a) Show that the additional energy E∆ which must be 
imparted to the spacecraft to transfer it to a circular orbit of larger 
radius 2r is 
2 1
1 2
( )
2
GMm r r
E
r r
−∆ = 
where M is the mass of the earth. (b) Further show that if the transfer 
from one circular orbit to the other is executed by placing the 
spacecraft on a transitional semielliptic path AB, the amounts of 
energy AE∆ and BE∆ which must be imparted at A and B are, 
respectively, proportional to 2r and 1 :r 
 2
1 2
A
r
E E
r r
∆ = ∆
+
 1
1 2
B
r
E E
r r
∆ = ∆
+
 
 
SOLUTION 
(a) For a circular orbit of radius r 
 
2
2
:n
GMm v
F ma m
rr
= = 
 2
GM
v
r
= 
 2
1 1
2 2
GMm GMm
E T V mv
r r
= + = − = − (1) 
 Thus E∆ required to pass from circular orbit of radius 1r to circular orbit of radius 2r is 
 1 2
1 2
1 1
2 2
GMm GMm
E E E
r r
∆ = − = − + 
 2 1
1 2
( )
2
GMm r r
E
r r
−∆ = Q.E.D. (2) 
(b) For an elliptic orbit, we recall Equation (3) derived in 
 Problem 13.113 1(with )Pv v= 
 2 21
1 2 1
2
( )
rGm
v
r r r
=
+
 
 
 
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683 
PROBLEM 13.116 (Continued) 
 
 At Point A: Initially spacecraft is in a circular orbit of radius 1.r 
 
2
circ
1
2
circ circ
1
1 1
2 2
GM
v
r
GM
T mv m
r
=
= =
 
 After the spacecraft engines are fired and it is placed on a semi-elliptic path AB, we recall 
 2 21
1 2 1
2
( )
rGM
v
r r r
= ⋅
+
 
 and 2 21 1
1 1 2
21 1
2 2 ( )
GMr
T mv m
r r r
= =
+
 
 At Point A, the increase in energy is 
 
2
1 circ
1 1 2 1
2 1 2 2 1
1 1 2 1 1 2
2 2 1
1 2 1 2
21 1
2 ( ) 2
(2 ) ( )
2 ( ) 2 ( )
( )
2
A
A
A
GMr GM
E T T m m
r r r r
GMm r r r GMm r r
E
r r r r r r
r GMm r r
E
r r r r
∆ = − = −
+
− − −∆ = =
+ +
ª º−∆ = « »+ ¬ ¼
 
 Recall Equation (2): 2
1 2( )
A
r
E E
r r
∆ = ∆
+
 Q.E.D. 
 A similar derivation at Point B yields, 
 1
1 2( )
B
r
E E
r r
∆ = ∆
+
 Q.E.D.�
 
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684 
 
PROBLEM 13.117* 
Using the answers obtained in Problem 13.108, show that the intended circular orbit and the resulting elliptic 
orbit intersect at the ends of the minor axis of the elliptic orbit. 
PROBLEM 13.108 A satellite is projected into space with a velocity v0 at a distance r0 from the center of the 
earth by the last stage of its launching rocket. The velocity v0 was designed to send the satellite into a circular 
orbit of radius r0. However, owing to a malfunction of control, the satellite is not projected horizontally but at 
an angle α with the horizontal and, as a result, is propelled into an elliptic orbit. Determine the maximum and 
minimum values of the distance from the center of the earth to the satellite. 
 
SOLUTION 
 
 
 
 
 
�
 
If the point of intersection 0P of the circular and elliptic orbits is at an end of 
the minor axis, then 0v is parallel to the major axis. This will be the case 
only if 090 ,α θ+ ° = that is if 0cos sin .θ α= − We must therefore prove that 
 0cos sinθ α= − (1) 
We recall from Equation (12.39): 
 
2
1
cos
GM
C
r h
θ= + (2) 
When 0,θ = min min 0and (1 sin )r r r r α= = − 
 
2
0
1
(1 sin )
GM
C
r hα
= +
−
 (3) 
For 180 ,θ = ° max 0 (1 sin )r r r α= = + 
 
2
0
1
(1 sin )
GM
C
r hα
= −
+
 (4) 
Adding (3) and (4) and dividing by 2: 
 
2
0
2
0
1 1 1
2 1 sin 1 sin
1
cos
GM
rh
r
α α
α
§ ·= +¨ ¸− +© ¹
=
 
Subtracting (4) from (3) and dividing by 2: 
 
2
0 0
2
0
1 1 1 1 2sin
2 1 sin 1 sin 2 1 sin
sin
cos
C
r r
C
r
α
α α α
α
α
§ ·§ ·= − = ¨ ¸¨ ¸− + −© ¹ © ¹
=
�
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685 
PROBLEM 13.117* (Continued) 
 
Substituting for 2
GM
h
 and C into Equation (2) 
 
2
0
1 1
(1 sin cos )
cosr r
α θ
α
= + (5) 
Letting 0 0 and r r θ θ= = in Equation (5), we have 
 
2
0
2
0
2
cos 1 sin cos
cos 1
cos
sin
sin
sin
sin
α α θ
αθ
α
α
α
α
= +
−=
= −
= −
 
This proves the validity of Equation (1) and thus P0 is an end of the minor axis of the elliptic orbit.�
 
 
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686 
 
PROBLEM 13.118* 
(a) Express in terms of minr and maxv the angular momentum per unit mass, h, and the total energy per unit 
mass, E/m, of a space vehicle moving under the gravitational attraction of a planet of mass M (Figure 13.15). 
(b) Eliminating maxv between the equations obtained, derive the formula 
 
2
2
min
1 2
1 1
GM E h
r m GMh
ª º§ ·« »= + + ¨ ¸« »© ¹¬ ¼
 
(c) Show that the eccentricity ε of the trajectory of the vehicle can be expressed as 
 
2
2
1
E h
m GM
ε § ·= + ¨ ¸
© ¹
 
(d ) Further show that the trajectory of the vehicle is a hyperbola, an ellipse, or a parabola, depending on 
whether E is positive, negative, or zero. 
 
SOLUTION 
(a) Point A: 
 Angular momentum per unit mass. 
 
 
0
min max
H
h
m
r mv
m
=
=
 
 min maxh r v= (1) W 
 Energy per unit mass 
 2 2max max
min min
1
( )
1 1 1
2 2
E
T V
m m
E GMm GM
mv v
m m r r
= +
§ ·
= − = −¨ ¸
© ¹
 (2) W�
(b) From Eq. (1): max min/v h r= substituting into (2) 
 
2
2
minmin
1
2
E h GM
m rr
= − 
 
2
2 2
min min
2
1 2 1
0
E
GM m
r rh h
§ ·
¨ ¸§ · © ¹− ⋅ − =¨ ¸
© ¹
 
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687 
PROBLEM 13.118* (Continued) 
 
 Solving the quadratic: 
( )2
2 2 2
min
21 EmGM GM
r h h h
§ ·= + +¨ ¸
© ¹
 
 Rearranging 
 
2
2
min
1 2
1 1
GM E h
r m GMh
ª § ·«= + + ¨¸« © ¹¬
 (3) W�
(c) Eccentricity of the trajectory: 
 Eq. (12.39′) 
2
1
(1 cos )
GM
r h
ε θ= + 
 When min0, cos 1 and r rθ θ= = = 
 Thus, 
 
2
min
1
(1 )
GM
r h
ε= + (4) 
 Comparing (3) and (4), 
2
2
1
E h
m GM
ε § ·= + ¨ ¸
© ¹
 (5) 
(d ) Recalling discussion in section 12.12 and in view of Eq. (5) 
 1. Hyperbola if 1,ε > that is, if 0E > W�
 2. Parabola if 1,ε = that is, if 0E = W�
� 3. Ellipse if 1,ε < that is, if 0E < W 
 Note: For circular orbit 0ε = and 
 
2 2
2
1 0 or ,
2
E h GM m
E
m GM h
§ · § ·+ = = −¨ ¸ ¨ ¸
© ¹ © ¹
 
 but for circular orbit 2 2 2 2and ,
GM
v h v r GMr
r
= = = 
 thus 
21 ( ) 1
2 2
GM GMm
E m
GMr r
= − = − �
 
 
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688 
 
PROBLEM 13.CQ4 
A large insect impacts the front windshield of a sports car traveling down a road. Which of the following 
statements is true during the collision? 
(a) The car exerts a greater force on the insect than the insect exerts on the car. 
(b) The insect exerts a greater force on the car than the car exerts on the insect. 
(c) The car exerts a force on the insect, but the insect does not exert a force on the car. 
(d) The car exerts the same force on the insect as the insect exerts on the car. 
(e) Neither exerts a force on the other; the insect gets smashed simply because it gets in the way of the car. 
 
SOLUTION 
Answer: (d) This is Newton’s 3rd Law. 
 
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689 
 
 
PROBLEM 13.CQ5 
The expected damages associated with two types of perfectly plastic collisions are to 
be compared. In the first case, two identical cars traveling at the same speed impact 
each other head on. In the second case, the car impacts a massive concrete wall. In 
which case would you expect the car to be more damaged? 
(a) Case 1 
(b) Case 2 
(c) The same damage in each case 
 
SOLUTION 
Answer: (c) In both cases the car will come to a complete stop, so the applied impulse will be the same. 
 
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690 
 
 
PROBLEM 13.F1 
The initial velocity of the block in position A is 30 ft/s. The coefficient 
of kinetic friction between the block and the plane is µk = 0.30. Draw 
impulse-momentum diagrams that could be used to determine the time 
it takes for the block to reach B with zero velocity, if θ = 20°. 
 
 
SOLUTION 
Answer: 
 
 
 
 
 
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691 
 
 
PROBLEM 13.F2 
A 4-lb collar which can slide on a frictionless vertical rod is acted upon 
by a force P which varies in magnitude as shown. Knowing that the 
collar is initially at rest, draw impulse-momentum diagrams that could 
be used to determine its velocity at t = 3 s. 
 
 
SOLUTION 
Answer: 
 
 
Where
2
1
3
0
t
t
Pdt
=
=ò is the area under the curve.
 
 
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692 
 
 
PROBLEM 13.F3 
The 15-kg suitcase A has been propped up against one end 
of a 40-kg luggage carrier B and is prevented from sliding 
down by other luggage. When the luggage is unloaded and 
the last heavy trunk is removed from the carrier, the 
suitcase is free to slide down, causing the 40-kg carrier to 
move to the left with a velocity vB of magnitude 0.8 m/s. 
Neglecting friction, draw impulse-momentum diagrams 
that could be used to determine (a) the velocity of A as it 
rolls on the carrier and (b) the velocity of the carrier after the 
suitcase hits the right side of the carrier without bouncing 
back. 
 
SOLUTION 
Answer: 
(a) 
 
(b) 
 
 
 
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693 
 
 
PROBLEM 13.F4 
Car A was traveling west at a speed of 15 m/s and car B was traveling north 
at an unknown speed when they slammed into each other at an intersection. 
Upon investigation it was found that after the crash the two cars got stuck 
and skidded off at an angle of 50° north of east. Knowing the masses of A 
and B are mA and mB respectively, draw impulse-momentum diagrams that 
could be used to determine the velocity of B before impact. 
 
 
 
SOLUTION 
Answer: 
 
 
 
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694 
 
 
PROBLEM 13.F5 
Two identical spheres A and B, each of mass m, are attached to an 
inextensible inelastic cord of length L and are resting at a distance a from 
each other on a frictionless horizontal surface. Sphere B is given a velocity 
v0 in a direction perpendicular to line AB and moves it without friction until it 
reaches B' where the cord becomes taut. Draw impulse-momentum diagrams 
that could be used to determine the magnitude of the velocity of each sphere 
immediately after the cord has become taut. 
 
 
SOLUTION 
Answer: 
 
Where 
y yA B
v v since the cord is inextensible. 
 
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695 
 
PROBLEM 13.119 
A 35,000 Mg ocean liner has an initial velocity of 4 km/h. Neglecting the frictional resistance of the water, 
determine the time required to bring the liner to rest by using a single tugboat which exerts a constant force of 
150 kN. 
 
SOLUTION 
 635,000 35 10 kgm Mg= = × 
 3150 10 NF = × 
 1 4 km/hr 1.1111 m/sv = = 
 
 
1
6 3
0
(35 10 kg)(1.1111 m/s) (150 10 N) 0
259.26 s
mv Ft
t
t
− =
× − × =
= 4 min19 st = W 
 
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696 
 
PROBLEM 13.120 
A 2500-lb automobile is moving at a speed of 60 mi/h when the brakes are fully applied, causing all four 
wheels to skid. Determine the time required to stop the automobile (a) on dry pavement (µ k = 0.75), (b) on an 
icy road (µ k = 0.10). 
 
SOLUTION 
 
 1 60 mph 88 ft/sv = = 
 
1
1 1 1
0k
k k k
mv Wt
mv mv v
t
W mg g
µ
µ µ µ
− =
= = =
 
(a) For 0.75kµ = 
 
2
88 ft/s
(0.75)(32.2 ft/s )
t = 3.64 st = W�
(b) For 0.10kµ = 
 
2
88 ft/s
(0.10)(32.2 ft/s )
t = 27.3 st = W�
 
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697 
 
PROBLEM 13.121 
A sailboat weighing 980 lb with its occupants is running down 
wind at 8 mi/h when its spinnaker is raised to increase its speed. 
Determine the net force provided by the spinnaker over the 10-s 
interval that it takes for the boat to reach a speed of 12 mi/h. 
 
SOLUTION 
 1 1 2
2
8 mi/h 11.73 ft/s 10 sec
12 mi/h 17.60 ft/s
v t
v
−= = =
= =
 
 
 1 1 2 2
imp
(11.73 ft/s) (10 s) (17.60 ft/s)n
m v mv
m F m
−⋅ + =
+ =
 
 
2
(980 lb)(17.60 ft/s 11.73 ft/s)
(32.2 ft/s )(10 s)
nF
−= 178.6 lbnF = W 
Note: nF is the net force provided by the sails. The force on the sails is actually greater and includes the force 
needed to overcome the water resistance on the hull.�
 
 
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698 
 
PROBLEM 13.122 
A truck is hauling a 300-kg log out of a ditch using a 
winch attached to the back of the truck. Knowing the 
winch applies a constant force of 2500 N and the 
coefficient of kinetic friction between the ground and 
the log is 0.45, determine the time for the log to reach a 
speed of 0.5m/s. 
 
SOLUTION 
Apply the principle of impulse and momentum to the log. 
 
 1 1 2 2m m→+ Σ =v Imp v 
Components in y-direction: 
 
0 cos 20 0
cos 20
Nt mgt
N mg
+ − ° =
= °
 
Components in x-direction: 
 
2
2
2
0 sin 20
( sin 20 cos 20 )
[ (sin 20 cos 20 )]
k
k
k
Tt mgt Nt mv
T mg mg t mv
T mg t mv
µ
µ
µ
+ − ° − =
− ° − ° =
− ° + ° =
 
Data: 
2
2
2500 N, 300 kg, 0.45,
9.81 m/s , 0.5 m/s
kT m
g v
µ= = =
= =
 
 [2500 (300)(9.81)(sin 20 0.45cos 20 )] (300)(0.5)t− ° + ° = 
 248.95 150t = 0.603 st = W 
 
 
 
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699 
 
PROBLEM 13.123 
A truck is traveling down a road with a 3-percent grade at a 
speed of 55 mi/h when the brakes are applied. Knowing the 
coefficients of friction between the load and the flatbed trailer 
shown are µs = 0.40 and µk = 0.35, determine the shortest 
time in which the rig can be brought to a stop if the load is not 
to shift. 
 
SOLUTION 
Apply the principle impulse-momentum to the crate, knowing that, if the crate does not shift, the velocity of 
the crate matches that of the truck. For impending slip the friction and normal components of the contact force 
between the crate and the flatbed trailer satisfy the following equation: 
 f sF Nµ= 
 1 1 2 2m m→+ Σ =v Imp v 
Components in y-direction: 0 cos 0
cos
Nt mgt
N mg
θ
θ
+ − =
=
 
Components in x-direction: 1 2
1
sin
(sin cos ) 0
s
s
mv mgt Nt mv
mv mgt
θ µ
θ µ θ
+ − =
+ − =
 
 1
( cos sin )s
v
t
g µ θ θ
=
−
 
Data: 1 255 mi/h 80.667 ft/s, 0,v v= = = 
 232.2 ft/s , 0.40, tan 3/100sg µ θ= = = 
 
1.71835
cos sin 0.36983s
θ
µ θ θ
= °
− =
 
 
80.667
(32.2)(0.36983)
t = 6.77 st = W 
 
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700 
PROBLEM 13.124 
Steep safety ramps are built beside mountain highways to enable 
vehicles with defective brakes to stop. A 10-ton truck enters a 15° ramp 
at a high speed v0 = 108 ft/s and travels for 6 s before its speed is 
reduced to 36 ft/s. Assuming constant deceleration, determine (a) the 
magnitude of the braking force, (b) the additional time required for the 
truck to stop. Neglect air resistance and rolling resistance. 
 
SOLUTION 
 
 
 
 
 
 20,000 lbW = 
 2
20,000
621.118 lb s /ft
32.2
m = = ⋅ 
 Momentum in the x direction 
 0 1: ( sin15 )x mv F mg t mv− + ° = 
 621.118(108) ( sin15 )6 (621.118)(36)F mg− + ° = 
 sin15 7453.4F mg+ ° = 
(a) 7453.4 20,000 sin15 2277 lbF = − ° = 
2280 lbF = W�
(b) 0 ( sin15 ) 0mv F mg t− + ° = t = total time 
 621.118(108) 7453.4 0;t− = t = 9.00 s 
 Additional time = 9 – 6 3.00 st = W 
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701 
 
PROBLEM 13.125 
Baggage on the floor of the baggage car of a high-speed train is not prevented from moving other than by 
friction. The train is travelling down a 5 percent grade when it decreases its speed at a constant rate from 
120 mi/h to 60 mi/h in a time interval of 12 s. Determine the smallest allowable value of the coefficient of 
static friction between a trunk and the floor of the baggage car if the trunk is not to slide. 
 
SOLUTION 
 
 1
2
120 mi/h 176 ft/s
60 mi/h 88 ft/s
v
v
= =
= =
 
 1 2 12 st − = 
 1 2 1 2 cosNt Wt θ− −= 
 m 1 sv mµ− 1 2 cosgt mθ− + 1 2 singt mθ− = 2v 
 2 2(176 ft/s) (32.2 ft/s )(12 s)(cos 2.86 ) (32.2 ft/s )(12 s)(sin 2.86 ) 88 ft/ssµ− ° + ° =176 88 (32.2)(12)(sin 2.86 )
(32.2)(12)(cos2.86 )s
µ − + °=
°
 0.278sµ = W�
 
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702 
 
PROBLEM 13.126 
A 2-kg particle is acted upon by the force, expressed in newtons, 2(8 6 ) (4 ) (4 ) .t t t= − + − + +F i j k Knowing 
that the velocity of the particle is (150 m/s) (100 m/s) (250 m/s)= + −v i j k at 0,t = determine (a) the time at 
which the velocity of the particle is parallel to the yz plane, (b) the corresponding velocity of the particle. 
 
SOLUTION 
 0m dt m+ =³v F v (1) 
Where 2
0
2 3 2
[(8 6 ) (4 ) (4 ) ]
1 1
(8 3 ) 4 4
3 2
t
dt t t t dt
t t t t t t
= − + − + +
§ · § ·= − + − + +¨ ¸ ¨ ¸
© ¹ © ¹
³ ³F i j k
i j k
 
Substituting 2 kg,m = 0 150 100 250v = + −i j k into (1): 
 2 3 2
1 1
(2 kg)(150 100 250 ) (8 3 ) 4 4 (2 kg)
3 2
t t t t t t§ · § ·+ − + − + − + + =¨ ¸ ¨ ¸
© ¹ © ¹
i j k i j k v 
 2 3 2
3 1 1
150 4 100 2 250 2
2 6 4
t t t t t t§ · § · § ·= + − + + − + − + +¨ ¸ ¨ ¸ ¨ ¸
© ¹ © ¹ © ¹
v i j k 
(a) v is parallel to yz plane when 0,xv = that is, when 
 2
3
150 4 0 11.422 s
2
t t t+ − = = 11.42t = s W 
(b) 3
2
1
100 2(11.422) (11.422)
6
1
250 2(11.422) (11.422)
4
ª º= + −« »¬ ¼
ª º+ − + +« »¬ ¼
v j
k
 
 (125.5 m/s) (194.5 m/s)= − −v j k W�
 
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703 
 
PROBLEM 13.127 
A truck is traveling down a road with a 4-percent grade at a speed of 60 mi/h when its brakes are applied to 
slow it down to 20 mi/h. An antiskid braking system limits the braking force to a value at which the wheels of 
the truck are just about to slide. Knowing that the coefficient of static friction between the road and the wheels 
is 0.60, determine the shortest time needed for the truck to slow down. 
 
SOLUTION 
 
 
 1
4
tan 2.29
100
θ −= = ° 
 1 1 2 2 m m−+ Σ =v imp v 
 1 2sinmv Wt Ft mvθ+ − = 
 1
2
60 mi/h 88 ft/s cos
20 mi/h 29.33 ft/s coss s
v N W W mg
v F N W
θ
µ µ θ
= = = =
= = = =
 
 ( m )(88 ft/s) ( m+ 2)(32.2 ft/s )( )(sin 2.29 ) (0.60)(t m° − 2)(32.2 ft/s )(cos2.29 )( ) (t m° = )(29.33 ft/s) 
 
88 29.33
32.2[(0.60)cos 2.29 sin 2.29 ]
t
−=
° − °
 3.26 st = W 
 
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704 
 
 
PROBLEM 13.128 
Skid marks on a drag race track indicate that the rear (drive) 
wheels of a car slip for the first 20 m of the 400-m track. 
(a) Knowing that the coefficient of kinetic friction is 0.60, 
determine the shortest possible time for the car to travel the 
initial 20-m portion of the track if it starts from rest with its 
front wheels just off the ground. (b) Determine the minimum 
time for the car to run the whole race if, after skidding for 20 m, 
the wheels roll without sliding for the remainder of the race. 
Assume for the rolling portion of the race that 65 percent of the 
weight is on the rear wheels and that the coefficient of static 
friction is 0.85. Ignore air resistance and rolling resistance. 
 
SOLUTION 
(a) First 20 m 
 Velocity at 20 m. Rear wheels skid to generate the maximum force resulting in maximum velocity and 
 minimum time since all the weight is on the rear wheel: This force is 0.60 .kF N Wµ= = 
 Work and energy. 0 0 20 20T U T−+ = 
 20 0 20 20 20
1
0 ( )(20)
2
T U F T mv−= = = 
 
2
20
2 2
20
20
1
0 (20)
2
(2)(0.60)(20 m)(9.81 m/s )
15.344 m/s
k mg mv
v
v
µ+ =
=
=
 
 Impulse-momentum. 
 
 0 20 20 200 15.344 m/sk mgt mv vµ −+ = = 
 0 20 2
15.344 m/s
(0.60)(9.81 m/s )
t − = 0 20 2.61st − = W 
 
 
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705 
PROBLEM 13.128 (Continued) 
 
(b) For the whole race: 
 The maximum force on the wheels for the first 20 m is 0.60 .kF mg mgµ= = For remaining 360 m, the 
 maximum force, if there is no sliding and 65 percent of the weight is on the rear (drive) wheels, is 
 (0.65) (0.85)(0.65) 0.5525sF mg mg mgµ= = = 
 Velocity at 400 m. 
 Work and energy. 0 0 20 20 400 400T U U T− −+ + = 
 
0 0 20 60 400
2
400 400
0 (0.60 )(20 m), (0.5525 )(380 m)
1
2
T U mg U mg
T mv
− −= = =
=
 
 2400
1
0 12 (0.5525)(380)
2
mg mg mv+ + = 
 400 65.990 m/sv = 
 Impulse–momentum. 
 From 20 m to 400 m 
 
 20
400
0.510
15.344 m/s
65.990 m/s
sF N mg
v
v
µ= =
=
=
 
 20 400 20 400(15.344) 0.5525 (65.990); 9.3442 sm mgt m t− −+ = = 
 0 400 0 20 20 400 2.61 9.34t t t− − −= + = + 0 400 11.95 st − = W�
 
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706 
 
 
PROBLEM 13.129 
The subway train shown is traveling at a speed of 30 mi/h 
when the brakes are fully applied on the wheels of cars B and 
C, causing them to slide on the track, but are not applied on the 
wheels of car A. Knowing that the coefficient of kinetic 
friction is 0.35 between the wheels and the track, determine 
(a) the time required to bring the train to a stop, (b) the force in 
each coupling. 
 
SOLUTION 
Weights of cars: 80,000 lb, 100,000 lbA C BW W W= = = 
Masses of cars: 2 22484 lb s /ft, 3106 lb s /ftA C Bm m m= = ⋅ = ⋅ 
For each car the normal force (upward) is equal in magnitude to the weight of the car. 
 80,000 lb 100,000 lbA C BN N N= = = 
Friction forces: 0 (brakes not applied)
(0.35)(100,000) 35000 lb
(0.35)(80,000) 28,000 lb
A
B
C
F
F
F
=
= =
= =
 
Stopping data: 1 230 mi/h 44 ft/s, 0.v v= = = 
(a) Apply the principle of impulse-momentum to the entire train. 
 
2
1 2
8074 lb s /ft
63,000 lb
A B C
A B C
m m m m
F F F F
mv Ft mv
= + + = ⋅
= + + =
− + =
 
 1 2
( ) (8074)(44)
5.639 s
63,000
m v v
t
F
−= = = 5.64 st = W�
(b) Coupling force FAB: 
 Apply the principle of impulse-momentum to car A alone. 
 
 1
0
(2484)(44) 0 (5.639) 0
A A AB
AB
m v F t F t
F
− + + =
− + + =
 
 19,390 lbABF = 19,390 lb (tension)ABF = W 
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707 
PROBLEM 13.129 (Continued) 
 
Coupling force :BCF 
Apply the principle of impulse-momentum to car C alone. 
 
 1
0
(2484)(44) (28000)(5.639) (5.639) 0
C C BC
BC
m v F t F t
F
− + − =
− + − =
 
 8620 lbBCF = 8620 lb (tension)BCF = W 
 
 
 
 
 
 
 
 
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708 
 
PROBLEM 13.130 
Solve Problem 13.129 assuming that the brakes are applied only on the wheels of car A. 
PROBLEM 13.129 The subway train shown is traveling at a speed of 30 mi/h when the brakes are fully 
applied on the wheels of cars B and C, causing them to slide on the track, but are not applied on the wheels of 
car A. Knowing that the coefficient of kinetic friction is 0.35 between the wheels and the track, determine 
(a) the time required to bring the train to a stop, (b) the force in each coupling. 
 
SOLUTION 
Weights of cars: 80,000 lb, 100,000 lbA C BW W W= = = 
Masses of cars: 2 22484 lb s /ft, 3106 lb s /ftA C Bm m m= = ⋅ = ⋅ 
For each car the normal force (upward) is equal in magnitude to the weight of the car. 
 80,000 lb 100,000 lbA C BN N N= = = 
Friction forces: (0.35)(80,000) 28,000 lb
0
(brakes not applied)
0
A
B
C
F
F
F
= =
= ½°
¾= °¿
 
Stopping data: 1 230 mi/h 44 ft/s, 0.v v= = = 
(a) Apply the principle of impulse-momentum to the entire train. 
 
2
1 2
8074 lb s /ft
28,000 lb
Ft
A B C
A B C
m m m m
F F F F
mv mv
= + + = ⋅
= + + =
− + =
 
 1 2
( ) (8074)(44)
12.688 s
28,000
m v v
t
F
−= = = 12.69 st = W�
(b) Coupling force FAB: 
 Apply the principle of impulse-momentum to car A alone. 
 
 1
0
(2484)(44) (28,000)(12.688) (12.688) 0
A A AB
AB
m v F t F t
F
− + + =
− + + =
 
 19,390 lbABF = − 19,390 lb (compression)ABF = W 
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709 
PROBLEM 13.130 (Continued) 
 
Coupling force :BCF 
Apply the principle of impulse-momentum to car C alone. 
 
 1
0
(2484)(44) (0) (12.688) 0
C C BC
BC
m v F t F t
F
− + − =
− + − =
 
 8620 lbBCF = − 8620 lb (compression)BCF = W 
 
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710 
 
 
PROBLEM 13.131 
A trailer truck with a 2000-kg cab and an 8000-kg trailer is 
traveling on a level road at 90 km/h. The brakes on the trailer 
fail and the antiskid system of the cab provides the largest 
possible force which will not cause the wheels of the cab to 
slide. Knowing that the coefficient of static friction is 0.65, 
determine (a) the shortest time for the rig to come to a stop, 
(b) the force in the coupling during that time. 
 
SOLUTION 
 90 km/h 25 m/sv = = 
(a) The shortest time for the rig to come to a stop will be when the friction force on the wheels is maximum. 
The downward force exerted by the trailer on the cab is assumed to be zero. Since the trailer brakes fail, 
all of the braking force is supplied by the wheels of the cab, which is maximum when the wheels of the 
cab are at impending sliding. 
 
 1 2 1 2
1 2
(2000)
(0.65)(2000)
s C C CFt N t N m g g
Ft gt
µ− −
−
= = =
=
 
 2 1[( ) ] [( ) ]C T C Tm m v Ft m m v+ = − + + 
 2 1 20 (0.65)(2000 kg)(9.81 m/s )( ) 10,000 kg(25 m/s)t −= − = 
 1 2 19.60 st − = W 
(b) For the trailer: 
 
 2 1 2 1[ ] [ ]T Tm v Qt m v−= − + 
 From (a), 1 2 19.60 s
0 (19.60 s) (8000 kg)(25 m/s)
10,204 N
t
Q
Q
− =
= − +
=
 
 10.20 kN (compression)Q = W�
 
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711 
PROBLEM 13.132 
The system shown is at rest when a constant 150-N force 
is applied to collar B. Neglecting the effect of friction, 
determine (a) the time at which the velocity of collar B 
will be 2.5 m/s to the left, (b) the corresponding tension in 
the cable. 
 
SOLUTION 
 
Constraint of cord. When the collar B moves 1 unit to the left, the weight A moves up 2 units. Thus 
 
1
2
2A B B A
v v v v= = 
Masses and weights: 3 kg 29.43 N
8 kg
A A
B
m W
m
= =
=
 
Let T be the tension in the cable. 
Principle of impulse and momentum applied to collar B. 
 
 : 20 150 2 ( )B Bt Tt m v+ − = 
For 2( ) 2.5 m/sBv = 150 2 (8 kg)(2.5 m/s)t Tt− = 
 150 2 20t Tt− = (1) 
 
 
 
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712 
PROBLEM 13.132 (Continued) 
 
Principle of impulse and momentum applied to weight A. 
 
 : 20 ( )A A ATt W t m v+ − = 
 2(2 )A A BTt W t m V+ = 
 29.43 (3 kg)(2)(2.5 m/s)Tt t− = 
 29.43 15Tt t− = (2) 
To eliminate T multiply Eq. (2) by 2 and add to Eq. (1). 
(a) Time: 91.14 50t = 0.549 st = W 
 From Eq. (2), 
15
29.43T
t
= + 
(b) Tension in the cable. 56.8 NT = W 
 
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713 
 
 
PROBLEM 13.133 
An 8-kg cylinder C rests on a 4-kg platform A supported by a cord which 
passes over the pulleys D and E and is attached to a 4-kg block B. Knowing 
that the system is released from rest, determine (a) the velocity of block B 
after 0.8 s, (b) the force exerted by the cylinder on the platform. 
 
SOLUTION 
�
�
�
�
�
�
�
�
(a) Blocks A and C: 
 
 1 1 2 1 2 2[( ) ] ( ) ( ) [( ) ]A C A C A Cm m v T t m m gt m m v− −+ − + + = + 
 0 (12 )(0.8) 12g T v+ − = (1) 
 Block B: 
 
 1 1 2 1 2 2[ ] ( ) ( )B B Bm v T t m gt m v− −+ − = 
 0 ( 4 )(0.8) 4T g v+ − = (2) 
 Adding (1) and (2), (eliminating T) 
 (12 4 )(0.8) (12 4)g g− = + 
 
2(8 kg)(9.81 m/s )(0.8 s)
16 kg
v = 3.92 m/sv = W 
(b) Collar A: 
 
 1( ) 0 0 ( )A C Am v F m g= + + (3) 
 From Eq. (2) with3.92 m/sv = 
 2
4
4
0.8
(4 kg)(3.92 m/s)
(4 kg)(9.81 m/s )
(0.8 s)
58.84 N
v
T g
T
T
= +
= +
=
 
Solving for CF in (3) 
 2
(4 kg)(3.92 m/s)
(4 kg)(9.81 m/s ) 58.84 N
(0.8 s)C
F = − + 
 39.2 NCF = W�
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714 
PROBLEM 13.134 
An estimate of the expected load on over-the-shoulder 
seat belts is to be made before designing prototype 
belts that will be evaluated in automobile crash tests. 
Assuming that an automobile traveling at 45 mi/h is 
brought to a stop in 110 ms, determine (a) the average 
impulsive force exerted by a 200-lb man on the belt, 
(b) the maximum force mF exerted on the belt if the 
force-time diagram has the shape shown. 
 
SOLUTION 
(a) Force on the belt is opposite to the direction shown. 
 1
45 mi/h 66 ft/s,
200 lb
v
W
= =
=
 
 1 2m dt m− =³v F v 
 aveFdt F t= ∆³ 
 
ave2
0.110 s
(200 lb)(66 ft/s)
(0.110 s) 0
(32.2 ft/s )
t
F
∆ =
− = 
 ave
(200)(66)
3727 lb
(32.2)(0.110)
F = = ave 3730 lbF = W 
 
(b) Impulse = area under 1diagram (0.110 s)
2 m
F t F− = 
 From (a), aveimpulse (3727 lb)(0.110 s)F t= ∆ = 
 
1
(0.110) (3727)(0.110)
2 m
F = 
 7450 lbmF = W 
 
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715 
 
 
PROBLEM 13.135 
A 60-g model rocket is fired vertically. The engine 
applies a thrust P which varies in magnitude as shown. 
Neglecting air resistance and the change in mass of the 
rocket, determine (a) the maximum speed of the rocket 
as it goes up, (b) the time for the rocket to reach its 
maximum elevation. 
 
SOLUTION 
Mass: 0.060 kgm = 
Weight: (0.060)(9.81) 0.5886 Nmg = = 
Forces acting on the model rocket: 
Thrust: ( )(given function of )P t t 
Weight: W (constant) 
Support: S (acts until )P W> 
Over 0 0.2 s:t< < 
 
13
65
0.2
0.5886 N
P t t
W
= =
=
 
Before the rocket lifts off, 0.5886 65S W P t= − = − 
S become zero when 1.t t= 
 1 10 0.5886 65 0.009055 s.t t= − = 
Impulse due to 1: ( )S t t> 
 
1
0 0
1
1
 
2
(0.5)(0.5886)(0.009055)
0.00266 N s
t t
Sdt Sdt
mgt
=
=
=
= ⋅
³ ³
 
The maximum speed occurs when 0.
dv
a
dt
= = 
At this time, 0,W P− = which occurs at 2 0.8 s.t = 
 
 
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716 
PROBLEM 13.135 (Continued) 
 
(a) Maximum speed (upward motion): 
 Apply the principle of impulse-momentum to the rocket over 20 .t t≤ ≤ 
0.8
0
area under the given thrust-time plot.
1 1
(0.2)(13) (0.1)(13 5) (0.8 0.3)(5)
2 2
4.7 N s
Pdt =
= + + + −
= ⋅
³
 
0.8
0
(0.5886)(0.8) 0.47088 N SWdt = = ⋅³ 
0.8 0.8 0.8
1 1 2
0 0 0
20 4.7 0.00266 0.47088 0.060 
m v Pdt Sdt Wdt mv
v
+ + − =
+ + − =
³ ³ ³ 
 2 70.5 m/sv = W 
(b) Time t3 to reach maximum height: 3( 0)v = 
 
3 3
1 3 3
0 0
t t
mv Pdt Sdt Wt mv+ + − =³ ³ 
 30 4.7 0.00266 0.5886 0t+ + − = 3 7.99 st = W 
 
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717 
 
 
PROBLEM 13.136 
A simplified model consisting of a single straight line is to be obtained for the 
variation of pressure inside the 10-mm-diameter barrel of a rifle as a 20-g 
bullet is fired. Knowing that it takes 1.6 ms for the bullet to travel the length of 
the barrel and that the velocity of the bullet upon exit is 700 m/s, determine the 
value of p0. 
 
SOLUTION 
 
At 0,t = 0 1 2
1 0
p p c c t
c p
= = −
=
 
At 31.6 10 s,t −= × 
3
1 2
0
2 3
0
0 (1.6 10 s)
1.6 10 s
p
c c
p
c
−
−
=
= − ×
=
×
 
 320 10 kgm −= × 
 
31.6 10 s
2
0
0 A pdt mv
−×
+ =³ 
 
3 2
6 2
(10 10 )
4
78.54 10 m
A
A
π −
−
×=
= ×
 
 
3 31.6 10 s
1 2
0
20 10
0 ( )A c c t dt
g
− −× ×+ − =³ 
 
3 2
6 2 3 32
1
( )(1.6 10 s)
(78.54 10 m ) ( )(1.6 10 s) (20 10 kg)(700 m/s)
2
c
c
−
− − −ª º×× × − = ׫ »
« »¬ ¼
 
 
3 6 3
1 2
6 2 2
3 2 3
0 03
6 2
0
1.6 10 1.280 10 178.25 10
(1.280 10 m s )
(1.6 10 m s) 178.25 10 kg m/s
(1.6 10 s)
222.8 10 N/m
c c
p p
p
− −
−
−
−
× − × = ×
× ⋅× ⋅ − = × ⋅
×
= × 0 223 MPap = W �
 
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718 
 
 
PROBLEM 13.137 
A 125-lb block initially at rest is acted upon by a force P which 
varies as shown. Knowing that the coefficients of friction between 
the block and the horizontal surface are 0.50sµ = and 0.40,kµ = 
determine (a) the time at which the block will start moving, (b) the 
maximum speed reached by the block, (c) the time at which the 
block will stop moving. 
 
SOLUTION 
 
 0 Pdt Fdt mv+ − =³ ³ 
At any time: 
1
v Pdt Fdt
m
ª º= −« »¬ ¼³ ³ (1) 
(a) Block starts moving at t. 
 
 (0.50)(125 lb) 62.5 lbs sP F Wµ= = = = 
 1 1
8 s 8 s
;
100 lb 62.5 lb 100 lbs
t t
F
= = 1 5.00 st = W 
(b) Maximum velocity: At mt t= 
 where 0.4(125) 50 lbk kP F Wµ= = = = 
 Block moves at 5 s.t = 
 Shaded area is maximum net impulse RPdt F dt−³ ³ 
 when 1m mt t v v= = 
 Eq. (1): 
shaded1 1 1 1 1
(12.5 50)(3) (50)(4) (193.75)
 area 2 2m
v
m m m
ª º ª º= = + + =« » « »¬ ¼¬ ¼
 
 125 lb
32.2
1
[193.75] 49.91 ft/smv = = 49.9 ft/sm =v W 
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719 
PROBLEM 13.137 (Continued) 
 
(c) Block stops moving when 0; orPdt Fdt Qdt Fdtª º− = =« »¬ ¼³ ³ ³ ³ 
 Assume 16 s.mt > 
 
 
1
(100)(16) 800 lb s
2
1
(62.5)(5) (50)( 5)
2 m
Pdt
Fdt t
= = ⋅
= + −
³
³
 
 800 [156.25 50( 5)] 0mPdt Fdt t− = − + − =³ ³ 17.875 smt = 
 16 s OKmt > 17.88 smt = W�
 
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720 
 
 
PROBLEM 13.138 
Solve Problem 13.137, assuming that the weight of the block is 175 lb. 
PROBLEM 13.137 A 125-lb block initially at rest is acted upon by a 
force P which varies as shown. Knowing that the coefficients of 
friction between the block and the horizontal surface are 0.50sµ = 
and 0.40,kµ = determine (a) the time at which the block will start 
moving, (b) the maximum speed reached by the block, (c) the time at 
which the block will stop moving. 
 
SOLUTION 
See solution of Problem 13.137. 
 
1
175 lbW v Pdt Fdt
m
ª º= = −« »¬ ¼³ ³ (1) 
(a) Block starts moving: (0.50)(175) 87.5 lbs sP F Wµ= = = = 
 See first figure of Problem 13.137. 
 1 1
8 s 8 s
;
100 lb 87.5 lb 100 ss
t t
F
= = 1 7.00 st = W 
(b) Maximum velocity: 0.4(175) 70 lbk kP F Wµ= = = = 
 
16 8 s
70 lb 100 lb
8
16 70 5.6
100
10.40 s
m
m
m
t
t
t
− =
§ ·− = =¨ ¸
© ¹
=
 
 Eq. (1): 
shaded1
area
1 1 1
(17.5 30)(1.0) (30)(10.4 8)
2 2
1
(59.75)
mv m
m
m
ª º= « »
¬ ¼
ª º= + + −« »¬ ¼
=
 
 
175 lb
32.2
1
[59.75]
10.994 ft/s
mv =
= 10.99 ft/sm =v W�
 
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721 
PROBLEM 13.138 (Continued) 
 
(c) Block stops moving when net impulse ( ) 0P F dtª º− =« »¬ ¼³ 
 Assume , 16 s.st < 
 
 
0
2
(16 )1 1
(100)(8) 100 100 ( 8)
2 2 8
1 1 100
(100)(16) (16 )
2 2 8
st s
s
s
t
Pdt t
t
−ª º= + + −« »¬ ¼
§ ·= − −¨ ¸
© ¹
³
 
 
0
1
(87.5)(7) (70)( 7)
2
st
sFdt t= + −³ 
 2
100
800 (16 ) 306.25 70( 7) 0
16 s s
Pdt Fdt t t− = − − − − − =³ ³ 
 Solving for ,st 13.492 sst = 13.49 sst = W�
 
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722 
 
 
PROBLEM 13.139 
A baseball player catching a ball can soften the impact by 
pulling his hand back. Assuming that a 5-oz ball reaches 
his glove at 90 mi/h and that the player pulls his hand back 
during the impact at an average speed of 30 ft/s over a 
distance of 6 in., bringing the ball to a stop, determine the 
average impulsive force exerted on the player’s hand. 
 
SOLUTION 
 
 
90 mi/h 132 ft/s
5
/g
16
v
m
= =
=
 
 
( ) ( )612 1 s6030av
d
t
v
= = = 
 0 av av
Wv
F t mv F
gt
= + = 
 av
mv
F
t
= 
 
( )
( )
5
16
2 1
60
lb (132 ft/s)
(32.2 ft/s ) s
= 76.9 lbavF = W�
 
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723 
 
PROBLEM 13.140 
A 1.62 ounce golf ball is hit with a golf club and leaves it with a velocity of 100 mi/h. We assume that for 
0 t≤ ≤ t0, where t0 is the duration of the impact, the magnitude F of the force exerted on the ball can be 
expressed as F 0sin ( / ).mF t tπ= Knowing that 0 0.5 ms,t = determine the maximum value mF of the force 
exerted on the ball. 
 
SOLUTION 
 
3
3
1.62 ounces 0.10125 lb 3.1444 10 slug
0.5 ms 0.5 10 s
100 mi/h 146.67 ft/s
W m
t
v
−
−
= = = ×
= = ×
= =
 
The impulse applied to the ball is 
 
0
0 0 0
0 0 0 0 0
0 0
sin cos
2
(cos cos 0)
t
t t
m
m
m m
F tt t
Fdt F dt
t t
F t F t
π π
π
π
π π
= = −
= − − =
³ ³
 
Principle of impulse and momentum. 
 
0
1 2
0
t
m dt m+ =³v F v 
with 1 0,=v 
 0 2
2
0 m
F t
mv
π
+ = 
Solving for Fm, 
3
32
3
0
(3.1444 10 )(146.67)
1.4488 10 lb
2 (2)(0.5 10 )
m
mv
F
t
π π −
−
×= = = ×
×
 1.45 kipmF = W �
 
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724 
 
 
PROBLEM 13.141 
The triple jump is a track-and-field event in which an athlete 
gets a running start and tries to leap as far as he can with a 
hop, step, and jump. Shown in the figure is the initial hop of 
the athlete. Assuming that he approaches the takeoff line from 
the left with a horizontal velocity of 10 m/s, remains in 
contact with the ground for 0.18 s, and takes off at a 50° angle 
with a velocity of 12 m/s, determine the vertical component of 
the average impulsive force exerted by the ground on his foot. 
Give your answer in terms of the weight W of the athlete. 
 
SOLUTION 
 
 1 2( ) 0.18 sm t m t+ − ∆ = ∆ =v P W v 
Vertical components 
 
0 ( )(0.18) (12)(sin 50 )
(12)(sin 50 )
(9.81)(0.18)
v
v
W
P W
g
P W W
+ − = °
°= +
 
 6.21vP W= W 
 
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725 
 
 
PROBLEM 13.142 
The last segment of the triple jump track-and-field event is the 
jump, in which the athlete makes a final leap, landing in a 
sand-filled pit. Assuming that the velocity of a 80-kg athlete 
just before landing is 9 m/s at an angle of 35° with the 
horizontal and that the athlete comes to a complete stop in 
0.22 s after landing, determine the horizontal component of 
the average impulsive force exerted on his feet during landing. 
 
SOLUTION 
 
 
80 kg
0.22 s
m
t
=
∆ =
 
 1 2( )m t m+ − ∆ =v P W v 
Horizontal components 
 
(9)(cos 35 ) (0.22) 0
(80 kg)(9 m/s)(cos 35 )
2.6809 kN
(0.22 s)
H
H
m P
P
° − =
°= = 2.68 kNHP = W 
 
 
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726 
 
 
PROBLEM 13.143 
The design for a new cementless hip implant is to be studied using an 
instrumented implant and a fixed simulated femur. Assuming the punch 
applies an average force of 2 kN over a time of 2 ms to the 200 g implant 
determine (a) the velocity of the implant immediately after impact, (b) the 
average resistance of the implant to penetration if the implant moves 
1 mm before coming to rest. 
 
SOLUTION 
 ave
200 g 0.200 kg
2 kN 2000 N
2 ms 0.002 s
m
F
t
= =
= =
∆ = =
 
(a) Velocity immediately after impact:Use principle of impulse and momentum: 
 1 2 1 2 ave
1 1 2 2
0 ? Imp ( )v v F t
m m
→
→
= = = ∆
+ =v Imp v
 
 ave 20 ( )F t mv+ ∆ = 
 ave2
( ) (2000)(0.002)
0.200
F t
v
m
∆= = 2 20.0 m/sv = W 
(b) Average resistance to penetration: 
 2
3
1 mm 0.001 m
20.0 ft/s
0
x
v
v
∆ = =
=
=
 
 Use principle of work and energy. 
 22 2 3 3 2 ave
1
or ( ) 0
2
T U T mv R x→+ = − ∆ = 
 
2 2
32
ave
(0.200)(20.0)
40 10 N
2( ) (2)(0.001)
mv
R
x
= = = ×
∆
 ave 40.0 kNR = W 
 
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727 
 
 
PROBLEM 13.144 
A 25-g steel-jacketed bullet is fired horizontally with a 
velocity of 600 m/s and ricochets off a steel plate along the 
path CD with a velocity of 400 m/s. Knowing that the 
bullet leaves a 10-mm scratch on the plate and assuming 
that its average speed is 500 m/s while it is in contact with 
the plate, determine the magnitude and direction of the 
average impulsive force exerted by the bullet on the plate. 
 
SOLUTION 
Impulse and momentum. 
Bullet alone: 
 
 1 2m t m+ ∆ =v F v 
t direction: 1 2cos1 5 cos 20tmv F t mv° − ∆ = ° 
 
6
6 3 2
(0.025 kg)[600 m/s cos 15 400 m/s cos 20 ] 5.092 kg m/s
0.010 m
20 10 s
500 m/s
(5.092 kg m/s)/(20 10 s) 254.6 10 kg m/s 254.6 kN
t
BC
AV
t
t
S
t
v
−
−
∆ = ° − ° = ⋅
∆ = = = ×
= ⋅ × = × ⋅ =
F
F
 
n direction: 1 2 sin1 5 sin 20nmv F t mv− ° + ∆ = ° 
 (0.025 kg)[600 m/s sin 15 400 m/s sin 20 ] 7.3025 kg m/sn t∆ = ° + ° = ⋅F 
 6 3 2(43025 kg m/s)/(20 10 ) 365.1 10 kg m/s 365.1 kNn
−= ⋅ × = × ⋅ =F 
 
Force on bullet: 2 2 2 2365.1 254.6 445 kNn tF F F= + = + = 
 
365.1
tan 55.1
254.6
15 40.1
n
t
F
F
θ θ
θ
= = = °
− ° = °
 
 445 kN=F 40.1° 
Force on plate: ′ = −F F 445 kN′ =F 40.1° W 
 
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728 
 
 
PROBLEM 13.145 
A 25-ton railroad car moving at 2.5 mi/h is 
to be coupled to a 50 ton car which is at 
rest with locked wheels ( 0.30).kµ = 
Determine (a) the velocity of both cars 
after the coupling is completed, (b) the 
time it takes for both cars to come to rest. 
 
SOLUTION 
 Weight and mass: (Label cars A and B.) 
 
2
2
Car : 50 tons 100,000 lb, 3106 lb s /ft
Car : 25 tons 50,000 lb, 1553 lb s /ft
A A
B B
A W m
B W m
= = = ⋅
= = = ⋅ 
 Initital velocities: 0A =v 
 2.5 mi/h 3.6667 ft/s 3.6667 ft/sB Bv = = =v 
(a) The momentum of the system consisting of the two cars is conserved immediately before and after 
 coupling. 
 
 Let ′v be the common velocity of that cars immediately after coupling. Apply conservation of 
momentum. 
 : B B A Bm v m v m v′ ′= + 
 
(3106)(3.6667)
2.444 ft/s
4569
B B
A B
m v
v
m m
′ = = =
+
 1.667 mi/h′ =v W 
(b) After coupling: The friction force acts only on car A. 
 + 0 : 0A A A A AF N W N WΣ = − = = 
 
(sliding)
0 (Car is rolling.)
A k A k A
B
F N W
F B
µ µ= =
=
 
 Apply impuslse-momentum to the coupled cars. 
 
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729 
PROBLEM 13.145 (Continued) 
: ( ) 0A B Am m v F t′− + + = 
1
1( )
(1553)(3.6667)
0.1898
(0.30)(100,000)
A B B B
A k A
m m v m v
t
F W
t
µ
+= =
= =
 
 0.190 st = W�
 
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730 
 
 
PROBLEM 13.146 
At an intersection car B was traveling south and car A was 
traveling 30° north of east when they slammed into each other. 
Upon investigation it was found that after the crash the two cars 
got stuck and skidded off at an angle of 10° north of east. Each 
driver claimed that he was going at the speed limit of 50 km/h and 
that he tried to slow down but couldn’t avoid the crash because the 
other driver was going a lot faster. Knowing that the masses of 
cars A and B were 1500 kg and 1200 kg, respectively, determine 
(a) which car was going faster, (b) the speed of the faster of the 
two cars if the slower car was traveling at the speed limit. 
 
SOLUTION 
(a) Total momentum of the two cars is conserved. 
 
 , :mv xΣ cos 30 ( ) cos 10A A A Bm v m m v° = + ° (1) 
 , :mv yΣ sin 30 ( ) sin 10A A B B A Bm v m v m m v° − = + ° (2) 
 Dividing (1) into (2), 
 
sin 30 sin 10
cos 30 cos 30 cos 10
B B
A A
m v
m v
° °− =
° ° °
 
 
(tan 30 tan 10 )( cos 30 )
(1500)
(0.4010) cos 30
(1200)
0.434 2.30
B A
A B
B
A
B
A B
A
v m
v m
v
v
v
v v
v
° − ° °=
= °
= =
 
 Thus, A was going faster. W 
(b) Since Bv was the slower car. 
 50 km/hBv = 
 (2.30)(50)Av = 115.2 km/hAv = W 
 
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731 
 
 
PROBLEM 13.147 
The 650-kg hammer of a drop-hammer pile driver falls from a height 
of 1.2 m onto the top of a 140-kg pile, driving it 110 mm into the 
ground. Assuming perfectly plastic impact ( 0),e = determine the 
average resistance of the ground to penetration. 
 
SOLUTION 
Velocity of the hammer at impact: 
Conservation of energy. 
 1 0T = 
 2
1
 (1.2 m)
(650 kg)(9.81 m/s )(1.2 m)
7652 J
H
H
V mg
V
V
=
=
=
 
 
2
2 2 2
2
1
2
650
325 
2
0
H H
T m
V v v
V
=
= =
=
 
 1 1 2 2
20 7652 325
T V T V
v
+ = +
+ =
 
 
2 2 223.54 m /s
4.852 m/s
v
v
=
=
 
Velocity of pile after impact: 
Since the impact is plastic ( 0),e = the velocity of the pile and hammer are the same after impact. 
Conservation of momentum: 
 
The ground reaction and the weights are non-impulsive. 
 
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732 
PROBLEM 13.147 (Continued) 
 
Thus, ( )
(650)
(4.852 m/s) 3.992 m/s
( ) (650 140)
H H H p
H H
H p
m v m m v
m v
v
m m
′= +
′ = = =
+ +
 
Work and energy: 0.110 md = 
 2 2 3 3T U T−+ = 
 
2
2
3
2
2
3
21
( )( )
2
0
1
(650 140)(3.992)
2
6.295 10 J
H HT m m v
T
T
T
′= +
=
= +
= ×
 
 
2 3
2 3
( )
(650 140)(9.81)(0.110) (0.110)
852.49 (0.110)
H p AV
AV
AV
U m m gd F d
F
U F
−
−
= + −
= + −
= −
 
 2 2 3 3T U T−+ = 
 36.295 10 852.49 (0.110) 0AVF× + − = 
 3(7147.5)/(0.110) 64.98 10 NAVF = = × 65.0 kNAVF = W 
 
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733 
 
 
PROBLEM 13.148 
A small rivet connecting two pieces of sheet metal is being clinched 
by hammering. Determine the impulse exerted on the rivet and the 
energy absorbed by the rivet under each blow, knowing that the head 
of the hammer has a weight of 1.5 lbs and that it strikes the rivet with 
a velocity of 20 ft/s. Assume that the hammer does not rebound and 
that the anvil is supported by springs and (a) has an infinite mass 
(rigid support), (b) has a weight of 9 lb. 
 
SOLUTION 
Weight and mass: 
 
2
2
Hammer: 1.5 lb 0.04658 lb s /ft
Anvil: Part : 
Part : 9 lb 0.2795 lb s /ft
H H
A A
A A
W m
a W m
b W m
= = ⋅
= ∞ = ∞
= = ⋅
 
Kinetic energy before impact: 
 2 21
1 1
(0.04658)(20) 9.316 ft lb
2 2H H
T m v= = = ⋅ 
Let v2 be the velocity common to the hammer and anvil immediately after impact. Apply the principle of 
conservation of momentum to the hammer and anvil over the duration of the impact. 
 
 
 
 
 
 
 
 
 : 1 2mv mvΣ = Σ 
 2( )H H H Am v m m v= + 
 2
H H
H A
m v
v
m m
=
+
 (1) 
Kinetic energy after impact: 
 
2 2
2
2
1 1
( )
2 2
H H
A H A
H A
m v
T m m v
m m
= + =
+
 
 2 1
H
H A
m
T T
m m
=
+
 (2) 
 
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734 
PROBLEM 13.148 (Continued) 
 
Impulse exerted on the hammer: 
 
 
 
 
 
 
 : 2( )H H Hm v F t m v− ∆ = 
 2( )H HF t m v v∆ = − (3) 
(a) :AW = ∞ 
 2
2
By Eq. (1), 0
By Eq. (2), 0
v
T
=
=
 
 Energy absorbed: 1 2 9.32 ft lbT T− = ⋅ W 
 By Eq. (3), ( ) (0.04658)(20 0) 0.932 lb sF t∆ = − = ⋅ 
The impulse exerted on the rivet the same magnitude but opposite to direction. 
 0.932 lb sF t∆ = ⋅ W 
(b) 9 lb:AW = 
 By Eq. (1), 2
(0.04658)(20)
2.857 ft/s
0.32608
v = = 
 By Eq. (2), 2
(0.04658)(9.316)
1.331 ft lb
0.32608
T = = ⋅ 
 Energy absorbed: 1 2 7.99 ft lbT T− = ⋅ W 
 By Eq. (3), ( ) (0.04658)(20 2.857)F t∆ = − ( ) 0.799 lb sF t∆ = ⋅ W 
 
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735 
 
PROBLEM 13.149 
Bullet B weighs 0.5 oz and blocks A and C both weigh 
3 lb. The coefficient of friction between the blocks 
and the plane is 0.25.kµ = Initially the bullet is 
moving at v0 and blocks A and C are at rest (Figure 1). 
After the bullet passes through A it becomes 
embedded in block C and all three objects come to 
stop in the positions shown (Figure 2). Determine the 
initial speed of the bullet v0. 
 
SOLUTION 
Masses: 
Bullet: 6 2
0.5
970.5 10 lb s /ft
(16)(32.2)B
m −= = × ⋅ 
Blocks A and C: 3 2
3
93.168 10 lb s /ft
32.2A C
m m −= = = × ⋅ 
Block C + bullet: 3 294.138 10 lb s /ftC Bm m
−+ = × ⋅ 
Normal forces for sliding blocks from 0N mg− = 
Block A: 3.00 lb.A AN m g= = 
Block C + bullet: ( ) 3.03125 lb.C C BN m m g= + = 
Let v0 be the initial speed of the bullet; 
 v1 be the speed of the bullet after it passes through block A; 
 vA be the speed of block A immediately after the bullet passes through it; 
 vC be the speed block C immediately after the bullet becomes embedded in it. 
Four separate processes and their governing equations are described below. 
1. The bullet hits block A and passes through it. Use the principle of conservation of momentum. 
 0( ) 0Av = 
 
0 0 1
0 1
( )B A A B A A
A A
B
m v m v m v m v
m v
v v
m
+ = +
= + (1) 
2. The bullet hits block C and becomes embedded in it. Use the principle of conservation of momentum. 
 0( ) 0Cv = 
 
1 0
1
( ) ( )
( )
B C C B C C
B C C
B
m v m v m m v
m m v
v
m
+ = +
+= (2) 
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736 
PROBLEM 13.149 (Continued) 
 
3. Block A slides on the plane. Use principle of work and energy. 
 
1 1 2 2
2 21 0 or
2
k A A
A A k A A A
A
T U T
N d
m v N d v
m
µµ
→+ =
− = = (3) 
4. Block C with embedded bullet slides on the plane. Use principle of work and energy. 
 4 in. 0.33333 ftCd = = 
 
1 1 2 2
2 21 ( ) 0 or
2
k C C
C B C k C C C
C B
T U T
N d
m m v N d v
m m
µµ
→+ =
+ − = =
+
 (4) 
Applying the numerical data: 
From Eq. (4), 
3
(2)(0.25)(3.03125)(0.33333)
94.138 10
2.3166 ft/s
Cv −= ×
=
 
From Eq. (3), 
3
(2)(0.25)(3.00)(0.5)
93.168 10
2.8372 ft/s
Av −= ×
=
 
From Eq. (2), 
3
1 6
(94.138 10 )(2.3166)
970.5 10
224.71 ft/s
v
−
−
×=
×
=
 
From Eq. (1), 
3
0 6
(93.138 10 )(2.8372)
224.71
970.5 10
v
−
−
×= +
×
 0 497 ft/sv = W 
 
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737 
 
PROBLEM 13.150 
A 180-lb man and a 120-lb woman stand at opposite ends of a 
300-lb boat, ready to dive, each with a 16-ft/s velocity relative 
to the boat. Determine the velocity of the boat after they have 
both dived, if (a) the woman dives first, (b) the man dives first. 
 
SOLUTION 
(a) Woman dives first: 
 Conservation of momentum: 
 1 1
120 300 180
(16 ) 0v v
g g
+− − + = 
 1
(120)(16)
3.20 ft/s
600
v = = 
 Man dives next. Conservation of momentum: 
 
 
 
 1 2 2
300 180 300 180
(16 )v v v
g g g
+ = − + − 
 12
480 (180)(16)
2.80 ft/s
480
v
v
−= = 2 2.80 ft/s=v W 
(b) Man dives first: 
 Conservation of momentum: 
 1 1
180 300 120
(16 ) 0v v
g g
+′ ′− − = 
 1
(180)(16)
4.80 ft/s
600
v′ = = 
 Woman dives next. Conservation of momentum: 
 1 2 2
300 120 300 120
(16 )v v v
g g g
+ ′ ′ ′− = + − 
 12
420 (120)(16)
0.229 ft/s
420
v
v
′− +′ = = − 
 2 0.229 ft/s′ =v W�
 
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738 
 
PROBLEM 13.151 
A 75-g ball is projected from a height of 1.6 m with a horizontal velocity 
of 2 m/s and bounces from a 400-g smooth plate supported by springs. 
Knowing that the height of the rebound is 0.6 m, determine (a) the 
velocity of the plate immediately after the impact, (b) the energy lost due 
to the impact. 
 
 SOLUTION 
Just before impact 
 
2 (1.6) 5.603 m/syv g= = 
Just after impact 
 
2 (0.6) 3.431 m/syv g= = 
(a) Conservation of momentum: ( y+ ) 
 ball ball plate plate0y ym v m v m v′ ′+ = − + 
 plate(0.075)(5.603) 0 0.075(3.431) 0.4v′+ = − + 
 plate 1.694 m/sv′ = W�
(b) Energy loss 
 Initial energy 21
1
( ) (0.075)(2) 0.075 (1.6)
2
T V g+ = + 
 Final energy 2 22
1 1
( ) (0.075)(2) 0.075 (0.6) (0.4)(1.694)
2 2
T V g+ = + + 
 Energy lost (1.3272 1.1653) J 0.1619J= − = W 
 
 
 
 
 
 
 
 
 
 
 
 
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739 
 
 
PROBLEM 13.152 
A 2-kg sphere A is connected to a fixed Point O by an inextensible cord of length 
1.2 m. The sphere is resting on a frictionless horizontal surface at a distance of 1.5 ft 
from O when it is given a velocity 0v in a direction perpendicular to line OA. It 
moves freely until it reaches position ,A′ when the cord becomes taut. Determine 
the maximum allowable velocity 0v if the impulse of the force exerted on the cord 
is not to exceed 3 N s.⋅ 
 
SOLUTION 
For the sphere at A′ immediately before and after the cord becomes taut 
 
0 Amv F t mv ′+ ∆ = 
 0 sin 0 3 N smv F t F tθ − ∆ = ∆ = ⋅ 
 2 kgm = 
02(sin 65.38 ) 3v° = 
 0 1.650 m/sv = W 
 
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740 
 
 
PROBLEM 13.153 
A 1-oz bullet is traveling with a velocity of 1400 ft/s when it impacts and 
becomes embedded in a 5-lb wooden block. The block can move vertically 
without friction. Determine (a) the velocity of the bullet and block immediately 
after the impact, (b) the horizontal and vertical components of the impulse 
exerted by the block on the bullet. 
 
SOLUTION 
Weight and mass. 2
2
1
Bullet: 1 oz lb 0.001941 lb s /ft.
16
Block: 5 lb 0.15528 lb s /ft.
w m
W M
= = = ⋅
= = ⋅
 
(a) Use the principle of impulse and momentum applied to the bullet and the block together. 
 
1 1 2 2m m→Σ + Σ =v Imp v 
 Components : 0 cos30 0 ( )mv m M v′° + = + 
 
0 cos30 (0.001941)(1400)cos 30°
0.157221
14.968 ft/s
mv
v
m M
v
°′ = =
+
′ =
 
 14.97 ft/s′ =v W 
 
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741 
PROBLEM 13.153 (Continued) 
 
(b) Use the principle of impulse and momentum applied to the bullet alone. 
 
 
 x-components: 0 sin 30 0xmv R t− ° + ∆ = 
 0 sin 30 (0.001941)(1400)sin 30xR t mv∆ = ° = ° 
 1.3587 lb s= ⋅ 1.359 lb sxR t∆ = ⋅ W 
 y-components: 0 cos30 ymv R t mv′− ° + ∆ = − 
 0( cos30 )yR t m v v′∆ = ° − 
 (0.001941)(1400 cos 30° 14.968)= − 2.32 lb syR t∆ = ⋅ W 
 
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742 
 
 
PROBLEM 13.154 
In order to test the resistance of a chain to impact, the chain is suspended 
from a 240-lb rigid beam supported by two columns. A rod attached to the 
last link is then hit by a 60-lb block dropped from a 5-ft height. Determine the 
initial impulse exerted on the chain and the energy absorbed by the chain, 
assuming that the block does not rebound from the rod and that the columns 
supporting the beam are (a) perfectly rigid, (b) equivalent to two perfectly 
elastic springs. 
 
SOLUTION 
Velocity of the block just before impact: 
 
1 1
2
2 2
1 1 2 2
2
0 (60 lb)(5 ft) 300 lb ft
1
0
2
1 60
0 300
2
(600)(32.2)
60
17.94 ft/s
T V Wh
T mv V
T V T V
v
g
v
= = = = ⋅
= =
+ = +
§ ·+ = ¨ ¸
© ¹
=
=
 
(a) Rigid columns: 
 
 
60
0 (17.94)mv F t F t
g
§ ·− + ∆ = = ∆¨ ¸
© ¹
 
 33.43 lb sF t∆ = ⋅ on the block. 33.4 lb sF t∆ = ⋅ W 
 All of the kinetic energy of the block is absorbed by the chain. 
 
21 60 (17.94)
2
300 ft lb
T
g
§ ·
= ¨ ¸
© ¹
= ⋅ 300 ft lbE = ⋅ W 
 
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743 
PROBLEM 13.154 (Continued) 
 
(b) Elastic columns: 
 Momentum of system of block and beam is conserved. 
 
 
( )
60
(17.94 ft/s)
( ) 300
mv M m v
m
v v
m M
′= +
′ = =
+
 3.59 ft/sv′ = 
 Referring to figure in part (a), 
 
( )
60
(17.94 3.59)
mv F t mv
F t m v v
g
′− + ∆ = −
′∆ = −
§ ·= −¨ ¸
© ¹
 26.7 lb sF t∆ = ⋅ W 
 
2 2 2
2 2 2
1 1 1
2 2 2
60 240
[(17.94) (3.59) ] (3.59)
2 2
E mv mv Mv
g g
′ ′= − −
= − − 240 ft lbE = ⋅ W�
 
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744 
 
 
PROBLEM 13.CQ6 
A 5 kg ball A strikes a 1 kg ball B that is initially at rest. Is it possible 
that after the impact A is not moving and B has a speed of 5v? 
(a) Yes 
(b) No 
Explain your answer. 
 
SOLUTION 
Answer: (b) No. 
Conservation of momentum is satisfied, but the coefficient of restitution equation is not. The coefficient of 
restitution must be less than 1. 
 
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745 
 
 
PROBLEM 13.F6 
A sphere with a speedv0 rebounds after striking a frictionless inclined plane as 
shown. Draw impulse-momentum diagrams that could be used to find the 
velocity of the sphere after the impact. 
 
 
 
SOLUTION 
Answer: 
 
 
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746 
 
 
PROBLEM 13.F7 
An 80-Mg railroad engine A coasting at 6.5 km/h strikes a 20-Mg 
flatcar C carrying a 30-Mg load B which can slide along the floor of 
the car (µk = 0.25). The flatcar was at rest with its brakes released. 
Instead of A and C coupling as expected, it is observed that A 
rebounds with a speed of 2 km/h after the impact. Draw impulse-
momentum diagrams that could be used to determine (a) the 
coefficient of restitution and the speed of the flatcar immediately 
after impact, and (b) the time it takes the load to slide to a stop 
relative to the car. 
 
 
SOLUTION 
Answer: 
(a) Look at A and C (the friction force between B and C is not impulsive) to find the velocity after impact. 
 
 
(b) Consider just B and C to find their final velocity. 
 
 Consider just B to find the time. 
 
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747 
 
 
PROBLEM 13.F8 
Two frictionless balls strike each other as shown. The coefficient of restitution 
between the balls is e. Draw the impulse-momentum diagrams that could be 
used to find the velocities of A and B after the impact. 
 
 
SOLUTION 
Answer: 
 
 
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748 
 
 
PROBLEM 13.F9 
A 10-kg ball A moving horizontally at 12 m/s strikes a 10-kg block B. The 
coefficient of restitution of the impact is 0.4 and the coefficient of kinetic 
friction between the block and the inclined surface is 0.5. Draw impulse-
momentum diagrams that could be used to determine the speeds of A and B 
after the impact. 
 
SOLUTION 
Answer: 
 
 
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749 
 
 
PROBLEM 13.F10 
Block A of mass mA strikes ball B of mass mB with a speed of vA as shown. 
Draw impulse-momentum diagrams that could be used to determine the 
speeds of A and B after the impact and the impulse during the impact. 
 
 
 
 
 
SOLUTION 
Answer: 
 
 
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750 
 
 
PROBLEM 13.155 
The coefficient of restitution between the two collars is known to be 
0.70. Determine (a) their velocities after impact, (b) the energy loss 
during impact. 
 
SOLUTION 
Impulse-momentum principle (collars A and B): 
 1 1 2 2m m→Σ + Σ = Σv Imp v 
 
 
Horizontal components : A A B B A A B Bm v m v m v m v′ ′+ = + 
Using data, (5)(1) (3)( 1.5) 5 3A Bv v′ ′+ − = + 
or 5 3 0.5A Bv v′ ′+ = (1) 
Apply coefficient of restitution. 
 
( )
0.70[1 ( 0.5)]
B A A B
B A
v v e v v
v v
′ ′− = −
′ ′− = − −
 
 1.75B Av v′ ′− = (2) 
(a) Solving Eqs. (1) and (2) simultaneously for the velocities, 
 0.59375 m/sAv′ = − 0.594 m/sA =v W�
 1.15625 m/sBv′ = 1.156 m/sB =v W�
 Kinetic energies: 2 2 2 21
2 2 2 2
2
1 1 1 1
(5)(1) (3)( 1.5) 5.875 J
2 2 2 2
1 1 1 1
( ) ( ) (5)( 0.59375) (3)(1.15625) 2.8867 J
2 2 2 2
A A B B
A A B B
T m v m v
T m v m v
= + = + − =
′ ′= + = − + =
 
(b) Energy loss: 1 2 2.99 JT T− = W 
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751 
 
 
PROBLEM 13.156 
Collars A and B, of the same mass m, are moving toward each other 
with identical speeds as shown. Knowing that the coefficient of 
restitution between the collars is e, determine the energy lost in the 
impact as a function of m, e and v. 
 
SOLUTION 
Impulse-momentum principle (collars A and B): 
 1 1 2 2m m→Σ + Σ = Σv Imp v 
 
 
Horizontal components : A A B B A A B Bm v m v m v m v′ ′+ = + 
Using data, ( ) A Bmv m v mv mv′ ′+ − = + 
or 0A Bv v′ ′+ = (1) 
Apply coefficient of restitution. 
 
( )
[ ( )]
B A A B
B A
v v e v v
v v e v v
′ ′− = −
′ ′− = − −
 
 2B Av v ev′ ′− = (2) 
Subtracting Eq. (1) from Eq. (2), 2 2Av ev− = 
 Av ev= − A ev=v 
Adding Eqs. (1) and (2), 2 2Bv ev= 
 Bv ev= B ev=v 
Kinetic energies: 2 2 2 2 21
2 2 2 2 2 2
2
1 1 1 1
( )
2 2 2 2
1 1 1 1
( ) ( ) ( ) ( )
2 2 2 2
A A B B
A A B B
T m v m v mv m v mv
T m v m v m ev m ev e mv
= + = + − =
′ ′= + = + =
 
Energy loss: 2 21 2 (1 ) T T e mv− = − W 
 
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752 
 
PROBLEM 13.157 
One of the requirements for tennis balls to be used in official competition is that, when dropped onto a rigid 
surface from a height of 100 in., the height of the first bounce of the ball must be in the range 53 in. 58 in.h≤ ≤ 
Determine the range of the coefficient of restitution of the tennis balls satisfying this requirement. 
 
SOLUTION 
Uniform accelerated motion: 
 
2
2
v gh
v gh
=
′ ′=
 
Coefficient of restitution: 
v
e
v
h
e
h
′
=
′
=
 
Height of drop 100 in.h = 
Height of bounce 53 in. 58 in.h′≤ ≤ 
Thus, 
53 58
100 100
e≤ ≤ 0.728 0.762e≤ ≤ W 
 
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753 
 
 
PROBLEM 13.158 
Two disks sliding on a frictionless horizontal plane with opposite velocities of 
the same magnitude 0v hit each other squarely. Disk A is known to have a 
weight of 6-lb and is observed to have zero velocity after impact. Determine 
(a) the weight of disk B, knowing that the coefficient of restitution between the 
two disks is 0.5, (b) the range of possible values of the weight of disk B if the 
coefficient of restitution between the two disks is unknown. 
 
SOLUTION 
Total momentum conserved: 
 A A B B A A Bm v m v m v m v′ ′+ = + 
 0 0( ) ( ) 0A B Bm v m v m v′+ − = + 
 01
A
B
m
v v
m
§ ·′ = −¨ ¸
© ¹
 (1) 
Relative velocities: 
 0
( )
2
B A A Bv v e v v
v ev
′ ′− = −
′ = (2) 
Subtracting Eq. (2) from Eq. (1) and dividing by 0 ,v 
 1 2 0 1 2
1 2
A A A
B
B B
m m m
e e m
m m e
− − = = + =
+
 
Since weight is proportional to mass, 
1 2
A
B
W
W
e
=
+
 (3) 
(a) With 6 lbAW = and 0.5,e = 
 
6
1 (2)(0.5)B
W =
+
 3.00 lbBW = W�
(b) With 6 lbAW = and 1,e = 
 
6
2 lb
1 (2)(1)B
W = =
+
 
 With 6 lbAW = and 0,e = 
 
6
6 lb
1 (2)(0)B
W = =
+
 
� Range: 2.00 lb 6.00 lbBW≤ ≤ W�
 
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754 
 
 
PROBLEM 13.159 
To apply shock loading to an artillery shell, a 20-kg pendulum A is released 
from a known height and strikes impactor B at a known velocity v0. 
Impactor B then strikes the 1-kg artillery shell C. Knowing the coefficient of 
restitution between all objects is e, determine the mass of B to maximize the 
impulse applied to the artillery shell C. 
 
SOLUTION 
First impact: A impacts B. 20 kg, ?A Bm m= = 
Impulse-momentum: 1 2 2m m→Σ + Σ = Σv Imp v 
Components directed left: 0A A A B Bm v m v m v′ ′= + 
 020 20 A B Bv v m v′ ′= + (1) 
Coefficient of restitution: ( )B A A Bv v e v v′ ′− = − 
 0B Av v ev′ ′− = 
 0A Bv v ev′ ′= − (2) 
Substituting Eq. (2) into Eq. (1) yields 
 0 020 20( )B B Bv v ev m v′ ′= − + 
 020 (1 ) ( )B Bv e m v′+ = + 
 
020 (1 )
20B B
v e
v
m
+′ =
+ (3)
 
Second impact: B impacts C. ?, 1 kgB Cm m= = 
Impulse-momentum: 2 2 3 3m m→Σ + Σ = Σv Imp v 
 
 
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755 
PROBLEM 13.159 (Continued) 
 
Components directed left: 
 B B B B C Cm v m v m v′ ′′ ′′= + 
 B B B B Cm v m v v′ ′′ ′′= + (4) 
Coefficient of restitution: ( )C B B Cv v e v v′′ ′′ ′ ′− = − 
 C B Bv v ev′′ ′′ ′− = 
 B C Cv v ev′′ ′′ ′− = (5) 
Substituting Eq. (4) into Eq. (5) yields 
 ( )B B B C B C Cm v m v ev m v′ ′′ ′ ′′= − + 
 (1 ) (1 )B B B Cm v e m v′ ′′+ = + 
 
(1 )
1
B B
C
B
m v e
v
m
′ +′′ =
+
 (6) 
Substituting Eq. (3) for Bv′ in Eq. (6) yields 
 
2
020 (1 )
(20 )(1 )
B
C
B B
m v e
v
m m
+′′ =
+ +
 
The impulse applied to the shell C is 
 
2
0(1)(20) (1 )
(20 )(1 )
B
C C
B B
m v e
m v
m m
+′′ =
+ +
 
To maximize this impulse choose mB such that 
 
(20 )(1 )
B
B B
m
Z
m m
=
+ +
 
is maximum. Set / BdZ dm equal to zero. 
 
2 2
2
2
(20 )(1 ) [(20 ) (1 )]
0
(20 ) (1 )
20 21 (21 2 ) 0
20 0
B B B B B
B B B
B B B B
B
m m m m mdZ
dm m m
m m m m
m
+ + − + + += =
+ +
+ + − + =
− =
 
 4.47 kgBm = W 
 
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756 
 
PROBLEM 13.160 
Two identical cars A and B are at rest on a loading dock with brakes released. Car C, of a slightly different 
style but of the same weight, has been pushed by dockworkers and hits car B with a velocity of 1.5 m/s. 
Knowing that the coefficient of restitution is 0.8 between B and C and 0.5 between A and B, determine the 
velocity of each car after all collisions have taken place. 
 
 
SOLUTION 
 A B Cm m m m= = = 
Collision between B and C: 
The total momentum is conserved: 
 
 B C B Cmv mv mv mv′ ′+ = + 
 0 1.5B Cv v′ ′+ = + (1) 
Relative velocities: 
 
( )( ) ( )
( 1.5)(0.8) ( )
1.2
B C BC C B
C B
C B
v v e v v
v v
v v
′ ′− = −
′ ′− = −
′ ′− = − (2) 
Solving (1) and (2) simultaneously, 
 
1.35 m/s
0.15 m/s
B
C
v
v
′ =
′ = 0.150 m/sC′ =v W 
Since ,B Cv v′ ′> car B collides with car A. 
Collision between A and B: 
 
 0 1.35
A B A B
A B
mv mv mv mv
v v
′ ′′ ′+ = +
′ ′′+ = + (3) 
 
 
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757 
PROBLEM 13.160 (Continued) 
 
Relative velocities: 
 
( ) ( )
(0 1.35)(0.5)
0.675
A B AB B A
B A
A B
v v e v v
v v
v v
′ ′′ ′− = −
′′ ′− = −
′ ′′− = (4) 
Solving (3) and (4) simultaneously, 
 2 1.35 0.675Av′ = + 
 1.013 m/sA′ =v W 
� 0.338 m/sB′′ =v W�
Since ,C B Av v v′ ′′ ′< < there are no further collisions.�
 
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758 
 
 
PROBLEM 13.161 
Three steel spheres of equal weight are suspended from the ceiling by cords of 
equal length which are spaced at a distance slightly greater than the diameter of 
the spheres. After being pulled back and released, sphere A hits sphere B, which 
then hits sphere C. Denoting by e the coefficient of restitution between the 
spheres and by 0v the velocity of A just before it hits B, determine (a) the 
velocities of A and B immediately after the first collision, (b) the velocities of B 
and C immediately after the second collision. (c) Assuming now that n spheres 
are suspended from the ceiling and that the first sphere is pulled back and 
released as described above, determine the velocity of the last sphere after it is hit 
for the first time. (d ) Use the result of Part c to obtain the velocity of the last 
sphere when n = 5 and e = 0.9. 
 
SOLUTION 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
(a) First collision (between A and B): 
 The total momentum is conserved: 
 0
A B A B
A B
mv mv mv mv
v v v
′ ′+ = +
′ ′= + (1) 
 Relative velocities: 
 ( ) ( )A B B Av v e v v′ ′− = − 
 0 B Av e v v′ ′= − (2) 
 Solving Equations (1) and (2) simultaneously,0
(1 )
2A
v e
v
−′ = W 
 0
(1 )
2B
v e
v
+′ = W 
(b) Second collision (between B and C): 
 The total momentum is conserved. 
 B C B Cmv mv mv mv′ ′′ ′+ = + 
 Using the result from (a) for Bv′ 
 0
(1 )
0
2 B C
v e
v v
+ ′′ ′+ = + (3) 
 Relative velocities: 
 ( 0)B C Bv e v v′ ′ ′′− = − 
 
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759 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 PROBLEM 13.161 (Continued) 
 
 Substituting again for Bv′ from (a) 
 0
(1 )
( )
2 C B
e
v e v v
+ ′ ′′= − (4) 
 Solving equations (3) and (4) simultaneously, 
 0 0
(1 )1 ( )
(1 )
2 2 2C
v e e
v v e
+ª º′ = + +« »¬ ¼
 
 
2
0 (1 )
4C
v e
v
+′ = W 
 
2
0 (1 )
4B
v e
v
−′′ = W 
(c) For n spheres 
 n balls 
 ( 1)th collision,n − 
 we note from the answer to part (b) with 3n = 
 
2
0
3
(1 )
4n C
v e
v v v
+′ ′ ′= = = 
 or 
(3 1)
0
3 (3 1)
(1 )
2
v e
v
−
−
+′ = 
 Thus, for n balls 
 
( 1)
0
( 1)
(1 )
2
n
n n
v e
v
−
−
+′ = W 
(d ) For 5, 0.90,n e= = 
 from the answer to part (c) with 5n = 
 
(5 1)
0
(5 1)
4
0
4
(1 0.9)
2
(1.9)
(2)
B
v
v
v
−
−
+′ =
=
 
 00.815Bv v′ = W 
 
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760 
 
PROBLEM 13.162 
At an amusement park there are 200-kg bumper cars A, B, and C that have riders with masses of 40 kg, 60 kg, 
and 35 kg respectively. Car A is moving to the right with a velocity 2 m/sA =v and car C has a velocity 
vB 1.5 m/s= to the left, but car B is initially at rest. The coefficient of restitution between each car is 0.8. 
Determine the final velocity of each car, after all impacts, assuming (a) cars A and C hit car B at the same 
time, (b) car A hits car B before car C does. 
 
 
SOLUTION 
Assume that each car with its rider may be treated as a particle. The masses are: 
 
200 40 240 kg,
200 60 260 kg,
200 35 235 kg.
A
B
C
m
m
m
= + =
= + =
= + =
 
Assume velocities are positive to the right. The initial velocities are: 
 2 m/s 0 1.5 m/sA B Cv v v= = = − 
Let , , andA B Cv v v′ ′ ′ be the final velocities. 
(a) Cars A and C hit B at the same time. Conservation of momentum for all three cars. 
 (240)(2) 0 (235)( 1.5) 240 260 235
A A B B C C A A B B C C
A B C
m v m v m v m v m v m v
v v v
′ ′ ′+ + = + +
′ ′ ′+ + − = + + (1) 
 Coefficient of restition for cars A and B. 
 ( ) (0.8)(2 0) 1.6B A A Bv v e v v′ ′− = − = − = (2) 
 Coefficient of restitution for cars B and C. 
 ( ) (0.8)[0 ( 1.5)] 1.2C B B Cv v e v v′ ′− = − = − − = (3) 
 Solving Eqs. (1), (2), and (3) simultaneously, 
 1.288 m/s 0.312 m/s 1.512 m/sA B Cv v v′ ′ ′= − = = 
 1.288 m/sA′ =v W 
 0.312 m/sB′ =v W 
 1.512 m/sC′ =v W 
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761 
PROBLEM 13.162 (Continued) 
 
(b) Car A hits car B before C does. 
 First impact. Car A hits car B. Let andA Bv v′ ′ be the velocities after this impact. Conservation of 
 momentum for cars A and B. 
 (240)(2) 0 240 260
A A B B A A B B
A B
m v m v m v m v
v v
′ ′+ = +
′ ′+ = + (4) 
 Coefficient of restitution for cars A and B. 
 ( ) (0.8)(2 0) 1.6B A A Bv v e v v′ ′− = − = − = (5) 
 Solving Eqs. (4) and (5) simultaneously, 
 0.128 m/s, 1.728 m/sA Bv v′ ′= = 
 0.128 m/sA′ =v 
 1.728 m/sB′ =v 
 Second impact. Cars B and C hit. Let andB Cv v′′ ′′ be the velocities after this impact. Conservation of 
momentum for cars B and C. 
 B B C C B B C Cm v m v m v m v′ ′′ ′′+ = + 
 (260)(1.728) (235)( 1.5) 260 235B Cv v′′ ′′+ − = + (6) 
 Coefficient of restitution for cars B and C. 
 ( ) (0.8)[1.728 ( 1.5)] 2.5824C B B Cv v e v v′′ ′′ ′− = − = − − = (7) 
 Solving Eqs. (6) and (7) simultaneously, 
 1.03047 m/s 1.55193 m/sB Cv v′′ ′′= − = 
 1.03047 m/sB′′ =v 
 1.55193 m/sC′′ =v 
Third impact. Cars A and B hit again. Let andA Bv v′′′ ′′′ be the velocities after this impact. Conservation of 
momentum for cars A and B. 
 (240)(0.128) (260)( 1.03047) 240 260
A A B B A A B B
A B
m v m v m v m v
v v
′ ′′ ′′′ ′′′+ = +
′′′ ′′′+ − = + (8) 
 Coefficient of restitution for cars A and B. 
 ( ) (0.8)[0.128 ( 1.03047)] 0.926776B A A Bv v e v v′′′ ′′′ ′ ′′− = − = − − = (9) 
 Solving Eqs. (8) and (9) simultaneously, 
 
0.95633 m/s
0.02955 m/s
A
B
v
v
′′′ = −
′′′ = −
 
 0.95633 m/sAA′′′ =v 
 0.02955 m/sB′′′ =v 
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762 
PROBLEM 13.162 (Continued) 
 
 There are no more impacts. The final velocities are: 
 0.956 m/sAA′′′ =v W 
 0.0296 m/sB′′′ =v W 
 1.552 m/sC′′ =v W 
We may check our results by considering conservation of momentum of all three cars over all three 
impacts. 
 
(240)(2) 0 (235)( 1.5)
127.5 kg m/s
A A B B C Cm v m v m v+ + = + + −
= ⋅
 
 
(240)( 0.95633) (260)( 0.02955) (235)(1.55193)
127.50 kg m/s.
A A B B C Cm v m v m v′′′ ′′′ ′′+ + = − + − +
= ⋅
 
 
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763 
 
PROBLEM 13.163 
At an amusement park there are 200-kg bumper cars A, B, and C that have riders with masses of 40 kg, 60 kg, 
and 35 kg respectively. Car A is moving to the right with a velocity 2 m/sA =v when it hits stationary car B. 
The coefficient of restitution between each car is 0.8. Determine the velocity of car C so that after car B 
collides with car C the velocity of car B is zero. 
 
 
SOLUTION 
Assume that each car with its rider may be treated as a particle. The masses are: 
 
200 40 240 kg
200 60 260 kg
200 35 235 kg
A
B
C
m
m
m
= + =
= + =
= + =
 
Assume velocities are positive to the right. The initial velocities are: 
 2 m/s, 0, ?A B Cv v v= = = 
First impact. Car A hits car B. Let andA Bv v′ ′ be the velocities after this impact. Conservation of momentum 
for cars A and B. 
 (240)(2) 0 240 260
A A B B A A B B
A B
m v m v m v m v
v v
′ ′+ = +
′ ′+ = + (1) 
Coefficient of restitution for cars A and B. 
 ( ) (0.8)(2 0) 1.6B A A Bv v e v v′ ′− = − = − = (2) 
Solving Eqs. (1) and (2) simultaneously, 
 
0.128 m/s
1.728 m/s
A
B
v
v
′ =
′ =
 
 0.128 m/sA′ =v1.728 m/sB′ =v 
Second impact. Cars B and C hit. Let andB Cv v′′ ′′ be the velocities after this impact. 0.Bv′′ = Coefficient of 
restitution for cars B and C. 
 
( ) (0.8)(1.728 )
1.3824 0.8
C B B C C
C C
v v e v v v
v v
′′ ′′ ′− = − = −
′′ = −
 
 
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764 
PROBLEM 13.163 (Continued) 
 
Conservation of momentum for cars B and C. 
 B B C C B B C Cm v m v m v m v′ ′′ ′′+ = + 
 (260)(1.728) 235 (260)(0) (235)(1.3824 0.8 )C Cv v+ = + − 
 (235)(1.8) (235)(1.3824) (260)(1.728)Cv = − 
 0.294 m/sCv = − 0.294 m/sC =v W 
Note: There will be another impact between cars A and B.�
 
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765 
 
 
PROBLEM 13.164 
Two identical billiard balls can move freely on a horizontal table. Ball 
A has a velocity v0 as shown and hits ball B, which is at rest, at 
a Point C defined by θ = 45°. Knowing that the coefficient of restitution 
between the two balls is e = 0.8 and assuming no friction, determine the 
velocity of each ball after impact. 
 
SOLUTION 
 
 
 
 
 
 
 
 Ball A: t-dir 0 0sin sinAt Atmv mv v vθ θ′ ′= � = 
 Ball B: t-dir 
 0 0B Bt Btm v v′ ′= � = 
 Balls A + B: n-dir 
 0 cos 0 An Bnmv m v m vθ ′ ′+ = + (1) 
 Coefficient of restitution 
 ( )Bn An An Bnv v e v v− = −′ ′ 
 0( cos 0)Bn Anv v e v θ′ ′− = − (2) 
 Solve (1) and (2) 
0 0
1 1
cos ; cos
2 2An Bn
e e
v v v vθ θ− +§ · § ·′ ′= =¨ ¸ ¨ ¸
© ¹ © ¹
 
 With numbers 
 0.8; 45e θ= = ° 
 0 0sin 45 0.707Atv v v= °=′ 
 0 0
1 0.8
cos45 0.0707
2An
v v v
−§ ·′ = ° =¨ ¸
© ¹
 
 0Btv′ = 
 0 0
1 0.8
cos 45 0.6364
2Bn
v v v
+§ ·′ = °=¨ ¸
© ¹
 
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766 
PROBLEM 13.164 (Continued) 
(A) 
 
 
1
22 2
0 0 0[(0.707 ) (0.0707 ) ] 0.711Av" v v v= + = 
� 00.711v= �
 1
0.0707
tan 5.7106
0.707
β − § ·= = °¨ ¸
© ¹
 
 So 45 5.7106 39.3θ = − = ° 
 
 
(B) 
 
 
 00.711Av v′ = 39.3° W 
 00.636Bv v′ = 45° W 
 
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767 
 
PROBLEM 13.165 
The coefficient of restitution is 0.9 between the two 2.37-in. diameter 
billiard balls A and B. Ball A is moving in the direction shown with 
a velocity of 3 ft/s when it strikes ball B, which is at rest. Knowing 
that after impact B is moving in the x direction, determine (a) the 
angle ,θ (b) the velocity of B after impact. 
 
SOLUTION 
(a) Since Bv′ is in the x-direction and (assuming no friction), the common tangent between A and B at 
 impact must be parallel to the y-axis, 
 
 
1
10
tan
6
10
tan
6 2.37
70.04
D
θ
θ −
=
−
=
−
= °
 
 70.0θ = ° W 
(b) Conservation of momentum in x(n) direction: 
 
cos ( ) ( )
(3)(cos70.04) 0 ( )
1.0241 ( ) ( )
A B n A n B
A n B
A n B
mv m v m v mv
v v
v v
θ ′ ′+ = +
′ ′+ = +
′ ′= + (1) 
 Relative velocities in the n direction: 
 0.9e = ( cos ( ) ) ( )A B n B A nv v e v vθ ′ ′− = − 
 (1.0241 0)(0.9) ( )B A nv v′ ′− = − (2) 
 (1) (2)+ 2 1.0241(1.9)Bv′ = 0.972 ft/sB′ =v W 
 
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768 
 
 
PROBLEM 13.166 
A 600-g ball A is moving with a velocity of magnitude 6 m/s when it is 
hit as shown by a 1-kg ball B which has a velocity of magnitude 4 m/s. 
Knowing that the coefficient of restitution is 0.8 and assuming no 
friction, determine the velocity of each ball after impact. 
 
SOLUTION 
 
 Before After 
 6 m/s
( ) (6)(cos 40 ) 4.596 m/s
( ) 6(sin 40°) 3.857 m/s
( ) 4 m/s
( ) 0
A
A n
A t
B B n
B t
v
v
v
v v
v
=
= ° =
= − = −
= = −
=
 
t-direction: 
Total momentum conserved: 
 
( ) ( ) ( ) ( )
(0.6 kg)( 3.857 m/s) 0 (0.6 kg)( ) (1 kg)( )
2.314 m/s 0.6 ( ) ( )
A A t B B t A B t B B t
A t B
A t B t
m v m v m v m v
v v t
v v
′ ′+ = +
′ ′− + = +
′ ′− = + (1) 
Ball A alone: 
Momentum conserved: 
 
( ) ( ) 3.857 ( )
( ) 3.857 m/s
A A t A A t A t
A t
m v m v v
v
′ ′= − =
′ = − (2) 
Replacing ( )A tv′ in (2) in Eq. (1) 
 2.314 (0.6)( 3.857) ( )
2.314 2.314 ( )
( ) 0
B t
B t
B t
v
v
v
′− = − +
′− = − +
′ =
 
 
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769 
PROBLEM 13.166 (Continued) 
 
n-direction: 
Relative velocities: 
 
[( ) ( ) ] ( ) ( )
[(4.596) ( 4)](0.8) ( ) ( )
6.877 ( ) ( )
A n B n B n A n
B n A n
B n A n
v v e v v
v v
v v
′ ′− = −
′ ′− − = −
′ ′= − (3) 
Total momentum conserved: 
 
( ) ( ) ( ) ( )
(0.6 kg)(4.596 m/s) (1 kg)( 4 m/s) (1 kg)( ) (0.6 kg)( )
1.2424 ( ) 0.6( )
A A n B B n A A n B B n
B n A n
B n A n
m v m v m v m v
v v
v v
′ ′+ = +
′ ′+ − = +
′ ′− = + (4) 
Solving Eqs. (4) and (3) simultaneously, 
 
( ) 5.075 m/s
( ) 1.802 m/s
A n
B n
v
v
′ =
′ =
 
Velocity of A: 
 
2 2
|( ) |
tan
|( ) |
3.857
5.075
37.2 40 77.2
(3.857) (5.075)
6.37 m/s
A t
A n
A
v
v
v
β
β β
=
=
= ° + ° = °
′ = +
=
 
 6.37 m/sA′ =v 77.2° W 
Velocity of B: 1.802 m/sB′ =v 40° W 
 
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770 
 
 
PROBLEM 13.167 
Two identical hockey pucks are moving on a hockey rink at the same speed of 3 m/s 
and in perpendicular directions when they strike each other as shown. Assuming a 
coefficient of restitution e = 0.9, determine the magnitude and direction of thevelocity of each puck after impact. 
 
SOLUTION 
Use principle of impulse-momentum: 1 1 2 2m m→Σ + Σ = Σv Imp v 
 
t-direction for puck A: 
 
sin 20 0 ( )
( ) sin 20 3sin 20 1.0261 m/s
A A t
A t A
mv m v
v v
′− ° + =
′ = ° = ° =
 
t-direction for puck B: 
 
cos 20 0 ( )
( ) cos 20 3cos 20 2.8191 m/s
B B t
B t B
mv m v
v v
′− ° + =
′ = ° = − ° = −
 
n-direction for both pucks: 
 cos 20 sin 20 ( ) ( )A B A n B nmv mv m v m v′ ′° − ° = + 
 ( ) ( ) cos 20 sin 20A n B n A Bv v v v′ ′+ = ° − ° 
 3cos 20 3sin 20= ° − ° (1) 
Coefficient of restitution: 0.9e = 
 ( ) ( ) [( ) ( ) ]B n A n A n B nv v e v v′ ′− = − 
 0.9[3cos 20 ( 3)sin 20 ]= ° − − ° (2) 
Solving Eqs. (1) and (2) simultaneously, 
 ( ) 0.8338 m/sA nv′ = − ( ) 2.6268 m/sB nv′ = 
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771 
PROBLEM 13.167 (Continued) 
 
Summary: ( ) 0.8338 m/sA n′ =v 20° 
 ( ) 1.0261 m/sA t′ =v 70° 
 ( ) 2.6268 m/sB n′ =v 20° 
 ( ) 2.8191 m/sB t′ =v 70° 
 2 2(0.8338) (1.0261) 1.322 m/sAv = + = 
 
1.0261
tan
0.8338
α = 50.9α = ° 20 70.9α + ° = ° 
 1.322 m/sA′ =v 70.9° W 
 2 2(2.6268) (2.8191) 3.85 m/sBv′ = + = 
 
2.8191
tan
2.6268
β = 47.0 20 27.0β β= ° − ° = ° 
 3.85 m/sB′ =v 27.0° W 
 
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772 
 
 
PROBLEM 13.168 
Two identical pool balls of 57.15-mm diameter, may move freely on a 
pool table. Ball B is at rest and ball A has an initial velocity 0 .v=v i 
(a) Knowing that 50 mmb = and 0.7,e = determine the velocity of 
each ball after impact. (b) Show that if 1,e = the final velocities of the 
balls from a right angle for all values of b. 
 
SOLUTION 
Geometry at instant of impact: 
 
50
sin
57.15
61.032
b
d
θ
θ
= =
= °
 
Directions n and t are shown in the figure. 
Principle of impulse and momentum: 
 
Ball B: 
 
 
 
Ball A: 
 
 
Ball A, t-direction: 0 0sin 0 ( ) ( ) sinA t A tmv m v v vθ θ+ = = (1) 
Ball B, t-direction: 0 0 ( ) ( ) 0B t B tm v v+ = = (2) 
Balls A and B, n-direction: 0 cos 0 ( ) ( )A n B nmv m v m vθ + + + 
 0( ) ( ) cosA n B nv v v θ+ = (3) 
Coefficient of restitution: 0( ) ( ) [ cos ]B n A nv v e v θ− = (4) 
(a) 0.7.e = From Eqs. (1) and (2), 
 0( ) 0.87489A tv v= (1)′ 
 ( ) 0B tv = (2)′ 
 
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773 
PROBLEM 13.168 (Continued) 
 
 From Eqs. (3) and (4), 
 0( ) ( ) 0.48432A n B nv v v+ = (3)′ 
 0( ) ( ) (0.7)(0.48432 )B n A nv v v− = (4)′ 
 Solving Eqs. (5) and (6) simultaneously, 
 
0 0
2 2
2 2
0 0
0
0
0
( ) 0.072648 ( ) 0.41167
( ) ( )
(0.072648 ) (0.87489
0.87790
( ) 0.072648
tan 0.083037
( ) 0.87489
4.7468
90
90 61.032 4.7468
24.221
A n B n
A A n A t
A n
A t
v v v v
v v v
v v
v
v v
v v
β
β
ϕ θ β
= =
= +
= +
=
= = =
= °
= ° − −
= ° − ° − °
= °
 
 00.878A v=v 24.2° W 
 00.412B v=v 61.0° W 
(b) 1.e = Eqs. (3) and (4) become 
 0( ) ( ) cosA n B nv v v θ+ = (3)′′ 
 0( ) ( ) cosB n A nv v v θ− = (4)′′ 
 Solving Eqs. (3)′′ and (4)′′ simultaneously, 
 0( ) 0, ( ) cosA n B tv v v θ= = 
 But 0( ) sin , and ( ) 0A t B tv v vθ= = 
 vA is in the t-direction and vB is in the n-direction; therefore, the velocity vectors form a right angle. 
 
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774 
 
 
PROBLEM 13.169 
A boy located at Point A halfway between the center O of a 
semicircular wall and the wall itself throws a ball at the wall in 
a direction forming an angle of 45° with OA. Knowing that 
after hitting the wall the ball rebounds in a direction parallel to 
OA, determine the coefficient of restitution between the ball and 
the wall. 
 
SOLUTION 
Law of sines: 
 
2
sin sin135
20.705
45 20.705
24.295
R R
θ
θ
α
°=
= °
= ° − °
= °
 
Conservation of momentum for ball in t-direction: 
 sin sinv vθ α′− = − 
Coefficient of restitution in n: (cos ) cosv e vθ α′= 
Dividing, 
tan
tan
e
θ α= 
 
tan 20.705
tan 24.295
e
°=
°
 0.837e = W 
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775 
 
 
PROBLEM 13.170 
The Mars Pathfinder spacecraft used large airbags to cushion 
its impact with the planet’s surface when landing. Assuming 
the spacecraft had an impact velocity of 18 m/s at an angle 
of 45° with respect to the horizontal, the coefficient of 
restitution is 0.85 and neglecting friction, determine (a) the 
height of the first bounce, (b) the length of the first bounce. 
(Acceleration of gravity on the Mars = 3.73 m/s2.) 
 
SOLUTION 
Use impulse-momentum principle. 
 1 1 2 2m m→Σ + Σ = Σv Imp v 
 
The horizontal direction (x to the right) is the tangential direction and the vertical direction ( y upward) is the 
normal direction. 
Horizontal components: 0 sin 45 0 xmv mv° = = 
 0 sin 45 .xv v= ° 12.728 m/sx =v 
Vertical components, using coefficient of restitution 0.85e = 
 00 [0 ( cos45 )]yv e v− = − − ° 
 (0.85)(18cos 45 )yv = ° 10.819 m/sy =v 
The motion during the first bounce is projectile motion. 
Vertical motion: 20
0
1
( )
2
( )
y
y y
y v t g t
v v gt
= −
= −
 
Horizontal motion: xx v t= 
 
 
 
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776 
PROBLEM 13.170 (Continued) 
 
(a) Height of first bounce: 
 
0
0
2
2
0: 0 ( )
( ) 10.819 m/s
2.901 s
3.73 m/s
1
(10.819)(2.901) (3.73)(2.901)
2
y y
y
v v gt
v
t
g
y
= = =
= = =
= −
 
 15.69 my = W 
(b) Length of first bounce: 
 
210: 10.819 (3.73) 0
2
5.801 s
y t t
t
= − =
=
 
 (12.728)(5.801)x = 73.8 mx = W 
 
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777 
 
PROBLEM 13.171 
A girl throws a ball at an inclined wall from a height 
of 3 ft, hitting the wall at A with a horizontal 
velocity 0v of magnitude 25 ft/s. Knowing that the 
coefficient of restitution between the ball and the 
wall is 0.9 and neglecting friction, determine the 
distance d from the foot of the wall to the Point B 
where the ball will hit the ground after bouncing 
off the wall. 
 
SOLUTION 
 
Momentum in t direction is conserved 
sin 30 tmv mv′° = 
 (25)(sin 30 ) tv′° = 
 12.5 ft/stv′ = 
Coefficient of restitution in n-direction 
 ( cos30 ) nv e v′° = 
(25)(cos30 )(0.9) 19.49 ft/sn nv v′ ′° = = 
 
Write v′ in terms of x and y components 
0( ) (cos30 ) (sin 30 ) 19.49(cos30 ) 12.5(sin 30 )x n tv v v′ ′ ′= ° − ° = ° − ° 
 10.63 ft/s= 
0( ) (sin 30 ) (cos30 ) 19.49(sin 30 ) 12.5(cos30 )y n tv v v′ ′ ′= ° + ° = ° + ° 
 20.57 ft/s= 
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778 
PROBLEM 13.171 (Continued) 
Projectile motion 
2
2 2
0 0
1
( ) 3 ft (20.57 ft/s) (32.2 ft/s )
2 2y
t
y y v t gt t′= + − = + − 
At B, 20 3 20.57 16.1 ; 1.4098 sB B By t t t= = + − = 
 0 0( ) 0 10.63(1.4098); 14.986 ftB x B Bx x v t x′= + = + = 
 3cot 60 (14.986 ft) (3 ft)cot 60 13.254 ftBd x= − ° = − ° = 
 13.25 ftd = W 
 
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779 
 
 
PROBLEM 13.172 
A sphere rebounds as shown after striking an inclined plane 
with a vertical velocity v0 of magnitude 0 5 m/s.v = Knowing 
that 30α = ° and 0.8e = between the sphere and the plane, 
determine the height h reached by the sphere. 
 
SOLUTION 
Rebound at A 
Conservation of momentum in the t-direction: 
 0
sin 30 ( )
( ) (5 m/s)(sin 30 ) 2.5 m/s
A t
A t
mv m v
v
′° =
′ = ° =
 
Relative velocities in the n-direction: 
 0( cos30 0) 0 ( )
( ) (0.8)(5 m/s)(cos30 ) 3.4641 m/s
A n
A n
v e v
v
′− ° − = −
′ = ° =
 
Projectile motion between A and B: 
After rebound 
 0
0
0
0
( ) ( ) cos30 ( ) sin 30
( ) (2.5)(cos30 ) (3.4641)sin 30 3.8971 m/s
( ) ( ) sin 30 ( ) cos30
( ) (2.5)(sin 30 ) (3.4641)cos30 1.750 m/s
x A t A n
x
y A t A n
y
v v v
v
v v v
v
′ ′= ° + °
= ° + =
′ ′= − ° + °
= − ° + ° =
 
x-direction: 0 0( ) ( )
3.8971 3.8971 m/s
x x x
x B
x v t v v
x t v v
= =
= = =
 
y-direction: 20
0
1
( )
2
( )
y
y y
y v t gt
v v gt
= −
= −
 
At A: 0
0 2
0 ( )
1.75 m/s
( ) /
9.81 m/s
0.17839 s
y y AB
AB y
A B
v v gt
t v g
t −
= = −
= =
=
 
At B: 
 
2
0
2
( )
2
9.81
(1.75)(0.17839) (0.17839)
2
A B
y A B
gt
y h v t
h
−
−= = −
= − 0.156 mh = W 
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780 
 
PROBLEM 13.173 
A sphere rebounds as shown after striking an inclined 
plane with a vertical velocity v0 of magnitude 0 6 m/s.v = 
Determine the value of α that will maximize the horizontal 
distance the ball travels before reaching its maximum 
height h assuming the coefficient of restitution between 
the ball and the ground is (a) 1,e = (b) 0.8.e = 
 
SOLUTION 
Directions x, y, n, and t are shown in the sketch. 
Analysis of the impact: Use the principle of impulse and momentum for 
components in the t-direction. 
 0 1sin 0 ( )tmv m vα ′+ = 
 1 0( ) sintv v α= (1) 
Coefficient of restitution: 1 0( ) ( )n ne= −v v 
 1 0( ) cosnv ev α= (2) 
x and y components of velocity immediately after impact: 
 1 1 1 0( ) ( ) sin ( ) cos (1 )sin cosx n tv v v v eα α α α= + = + 
 0
1
(1 )sin 2
2
v e α= + (3) 
 
2 2
1 1 1 0
0
0
( ) ( ) cos ( ) sin ( cos sin )
1
[ (1 cos 2 ) (1 cos 2 )]
2
1
[(1 )cos 2 (1 )]
2
y n tv v v v e
v e
v e e
α α α α
α α
α
= − = −
= + − −
= + − − (4) 
Projectile motion: Use coordinates x and y with the origin at the point of impact. 
 0
0
0
0
x
y
=
=
 
Vertical motion: 1
0
( )
1
[(1 )cos 2 (1 )]
2
y y
y
v v gt
v v e e gtα
= −
= + − − −
 
0yv = at the position of maximum height where 
 1 02
( )
[(1 )cos (1 )]
2
yv vt e e
g g
α= = + − − (5) 
 
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781 
PROBLEM 13.173 (Continued) 
 
Horizontal motion: 1 0
1
1
( ) (1 )sin 2
2
( )
x x
x
v v v e
x v t
α= = +
=
 
At the point of maximum height, 
 
2
0
2 1 2( ) (1 )sin 2 [(1 )cos 2 (1 )]4x
v
x v t e e e
g
α α= = + + − − 
Let 2θ α= and 22 04 / (1 ).Z gx v e= + To determine the value of θ that maximizes 2x (or Z ), differentiate 
Z with respect to θ and set the derivative equal to zero. 
 2
2 2
sin [(1 )cos (1 )]
cos [(1 )cos (1 )] (1 )sin
(1 )cos (1 )cos (1 )(1 cos ) 0
Z e e
dZ
e e e
d
e e e
θ θ
θ θ θ
θ
θ θ θ
= + − −
= + − − − +
= + − − − + − =
 
 22(1 )cos (1 )cos (1 ) 0e e eθ θ+ − − − + = 
This is a quadratic equation for cos .θ 
(a) 1e = 24cos 2 0θ − = 
 
2 1cos
2
2
cos
2
θ
θ
=
= ±
 
 45θ = ± ° and 135± ° 
 22.5α = ° and 67.5° 
 Reject the negative values of θ which make x2 negative. 
 Reject 67.5α = ° since it makes a smaller maximum height. 
 22.5α = ° W 
(b) 0.8e = 23.6cos 0.2cos 1.8 0θ θ− − = 
 
cos 0.73543 and 0.67987
42.656 and 132.833
21.328 and 66.417
θ
θ
α
= −
= ± ° ± °
= ± ° ± ° 21.3α = ° W 
 
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782 
 
PROBLEM 13.174 
Two cars of the same mass run head-on 
into each other at C. After the collision, the 
cars skid with their brakes locked and 
come to a stop in the positions shown in 
the lower part of the figure. Knowing that 
the speed of car A just before impact was 
5 mi/h and that the coefficient of kinetic 
friction between the pavement and the tires 
of both cars is 0.30, determine (a) the 
speed of car B just before impact, (b) the 
effective coefficient of restitutionbetween 
the two cars. 
 
SOLUTION 
(a) At C: 
 
 Conservation of total momentum: 
 5 mi/h 7.333 ft/s
A Bm m m= =
= 
 A A B B A A B Bm v m v m v m v′ ′+ = + 
 7.333 B A Bv v v′ ′− + = + (1) 
 Work and energy. 
 Care A (after impact): 
 
2
1
2
1 2
1 2
1 1 2 2
2
2 2
2
1
( )
2
0
(12)
(12 ft)
1
( ) (12) 0
2
( ) (2)(12 ft)(0.3)(32.2 ft/s )
231.84 ft/s
15.226 ft/s
A A
f
k A
A A k A
A
A
T m v
T
U F
U m g
T U T
m v m m g
v
v
µ
−
−
−
′=
=
=
=
+ =
′ − =
′ =
=
′ =
 
 
 
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783 
PROBLEM 13.174 (Continued) 
 
 Car B (after impact): 
 
2
1
2
1 2
1 1 2 2
1
( )
2
0
(3)
B B
k B
T m v
T
U m g
T U T
µ−
−
′=
=
=
+ =
 
 2
1
( ) (3)
2 B B k B
m v m gµ′ − 
 
2 2
2 2
(2)(3 ft)(0.3)(32.2 ft/s )
( ) 57.96 ft/s
7.613 ft/s
B
B
B
v
v
v
′ =
′ =
′ =
 
 From (1) 7.333
7.333 15.226 7.613
B A Bv v v′ ′= + +
= + +
 
 30.2 ft/s 20.6 mi/hBv = = W�
(b) Relative velocities: 
 ( )A B B Av v e v v′ ′− − = − 
 
( 7.333 30.2) 7.613 15.226
(7.613)
0.2028
(37.53)
e
e
− − = −
−= =
−
 0.203e = W 
 
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784 
 
 
PROBLEM 13.175 
A 1-kg block B is moving with a velocity v0 of magnitude 0 2 m/sv = 
as it hits the 0.5-kg sphere A, which is at rest and hanging from a cord 
attached at O. Knowing that 0.6kµ = between the block and the 
horizontal surface and 0.8e = between the block and the sphere, 
determine after impact (a) the maximum height h reached by the 
sphere, (b) the distance x traveled by the block. 
 
SOLUTION 
Velocities just after impact 
 
Total momentum in the horizontal direction is conserved: 
 
0 (1 kg)(2 m/s) (0.5 kg)( ) (1 kg)( )
A A B B A A B B
A B
m v m v m v m v
v v
′ ′+ = +
′ ′+ = +
 
 4 2A Bv v′ ′= + (1) 
Relative velocities: 
 
( ) ( )
(0 2)(0.8)
1.6
A B B A
B A
B A
v v e v v
v v
v v
′ ′− = −
′ ′− = −
′ ′− = − (2) 
Solving Eqs. (1) and (2) simultaneously: 
 
0.8 m/s
2.4 m/s
B
A
v
v
′ =
′ =
 
(a) Conservation of energy: 
 
2
1 1 1
2
1
1
0
2
1
(2.4 m/s) 2.88
2
A
A A
T m v V
T m m
= =
= =
 
 2
2
0
A
T
V m gh
=
=
 
 1 1 2 2 2.88T V T V m+ = + 0 0A m+ = + (9.81)A h 
 0.294 mh = W 
 
 
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785 
PROBLEM 13.175 (Continued) 
 
(b) Work and energy: 
 
 
2 2
1 1 2
1 2
1 2
1 1 2 2
1 1
(0.8 m/s) 0.32 0
2 2
(0.6)( )(9.81)
5.886
: 0.32 5.886 0
B B B
f k x B B
B
B B
T m v m m T
U F x Nx m gx m x
U m x
T U T m m x
µ µ−
−
−
= = = =
= − = − = − = −
= −
+ = − =
 
 0.0544 mx = 54.4 mmx = W 
 
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786 
 
PROBLEM 13.176 
A 0.25-lb ball thrown with a horizontal velocity v0 strikes a 1.5-lb plate attached to a vertical wall at a height 
of 36 in. above the ground. It is observed that after rebounding, the ball hits the ground at a distance of 24 in. 
from the wall when the plate is rigidly attached to the wall (Figure 1) and at a distance of 10 in. when a foam-
rubber mat is placed between the plate and the wall (Figure 2). Determine (a) the coefficient of restitution 
e between the ball and the plate, (b) the initial velocity v0 of the ball. 
 
 
SOLUTION 
 
 
 
 
 
 
 
 
(a) Figure (1), ball alone relative velocities 
0 1( )Bv e v′= 
 Projectile motion 
time for the ball to hit groundt = 
 02 ft v et= (1) 
 Figure (2), ball and plate relative velocities 
2( ) ( )B A P Bv v e v v′ ′− = + 
0, 0B Pv v v= = 
 0 2( )P Bv e v v′ ′= + (2) 
 Conservation of momentum 
B B P P B B P Pm v m v m v m v′ ′+ = + 
0 2
0.25 0.25 1.5
0 ( )B Pv v vg g g
′ ′+ = − + 
 
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787 
 PROBLEM 13.176 (Continued) 
 
 0 2 0 20.25 0.25( ) 1.5 ( ) 6B p B pv v v v v v′ ′ ′ ′= − + � = − − (3) 
 Solving (2) and (3) for 2( ) ,Bv′ 2 0
(6 1)
( )
7B
e
v v
−′ = 
 Projectile motion 
 0
(6 1)
0.8333
7
e
v t
−= (4) 
 Dividing Equation (4) by Equation (1) 
0.8333 6 1
; 2.91655 6 1
2 7
e
e e
e
−= = − 
 0.324e = W�
(b) From Figure (1) 
 Projectile motion, 2 2
1 1
; 3 (32.2)
2 2
h gt t= = 
 26 32.2t= (5) 
 From Equation (1), 
0
0 0
2 6.1728
2
0.324
v et t
v v
= � = = 
 Using Equation (5) 
2
2
0
0
6.1728
6 32.2 6 1226.947v
v
§ ·
= � =¨ ¸
© ¹
 
2
0 204.49v = 
 0 14.30 ft/sv = W 
 
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788 
 
 
PROBLEM 13.177 
After having been pushed by an airline employee, an empty 40-kg 
luggage carrier A hits with a velocity of 5 m/s an identical 
carrier B containing a 15-kg suitcase equipped with rollers. The 
impact causes the suitcase to roll into the left wall of carrier B. 
Knowing that the coefficient of restitution between the two 
carriers is 0.80 and that the coefficient of restitution between the 
suitcase and the wall of carrier B is 0.30, determine (a) the 
velocity of carrier B after the suitcase hits its wall for the first 
time, (b) the total energy lost in that impact. 
 
SOLUTION 
(a) Impact between A and B: 
 Total momentum conserved: 
 
 40 kgA A B B A A B B A Bm v m v m v m v m m′ ′+ = + = = 
 5 m/s 0 A Bv v′ ′+ = + (1) 
 Relative velocities: 
 ( )A B AB B Av v e v v′ ′− = − 
 (5 0)(0.80) B Av v′ ′− = − (2) 
 Adding Eqs. (1) and (2) 
 
(5 m/s)(1 0.80) 2
4.5 m/s
B
B
v
v
′+ =
′ = 
 Impact between B and C (after A hits B) 
 Total momentum conserved: 
 B B C C B B C Cm v m v m v m v′ ′ ′′ ′′+ = + 
 (40 kg)(4.5 m/s) 0 (40 kg) (15 kg)B Cv v′′ ′′+ = + 
 4.5 0.375B Cv v′′ ′′= + (3) 
 Relative velocities: 
 
( )
(4.5 0)(0.30)B C BC C B
C B
v v e v v
v v
′ ′ ′′ ′′− = −
′′ ′′− = − (4) 
 
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789 
PROBLEM 13.177 (Continued) 
 
 Adding Eqs. (4) and (3) 
 
(4.5)(1 0.3) (1.375)
4.2545 m/s
C
C
v
v
′′+ =
′′ =
 
 4.5 0.375(4.2545) 2.90 m/sB Bv v′′ ′′= − = 2.90 m/sBv′ = W 
(b) 
2 2
2 2
2 2
( ) ( )
1 40
( ) kg (4.5 m/s) 405 J
2 2
1 40
0 ( ) kg (2.90) 168.72 J
2 2
1 15
( ) kg (4.2545 m/s) 135.76 J
2 2
L B C B C
B B B
C B B B
C C C
T T T T T
T m v
T T m v
T m v
′ ′ ′′ ′′∆ = + − +
§ ·′ ′= = =¨ ¸
© ¹
§ ·′ ′′ ′′= = = =¨ ¸
© ¹
§ ·′′ ′′= = =¨ ¸
© ¹
 
 (405 0) (168.72 135.76) 100.5 JLT∆ = + − + = 100.5 JLT∆ = W 
 
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790 
 
 
PROBLEM 13.178 
Blocks A and B each weigh 0.8 lb and 
block C weighs 2.4 lb. The coefficient of 
friction between the blocks and the plane 
is 0.30.kµ = Initially block A is moving 
at a speed 0 15 ft/sv = and blocks B and C 
are at rest (Fig. 1). After A strikes B and B 
strikes C, all three blocks come to a stop 
in the positions shown (Fig. 2). Determine 
(a) the coefficients of restitution between 
A and B and between B and C, (b) the 
displacement x of block C. 
 
SOLUTION 
(a) Work and energy 
 Velocity of A just before impact with B: 
 ( )2 21 0 2 2
1 1
2 2
A A
A
W W
T v T v
g g
= = 
 1 2
1 1 2 2
(1 ft)k AU W
T U T
µ−
−
= −
+ =
 
 ( )2 20 2
1 1
(1)
2 2
A A
k A A
W W
v W v
g g
µ− = 
 
2 2 2 2
2 0
2 2
2 2
( ) 2 (15 ft/s) 2(0.3)(32.2 ft/s )(1 ft)
( ) 205.68 ft/s , ( ) 14.342 ft/s
A k
A A
v v g
v v
µ= − = −
= =
 
 Velocity of A after impact with B: 2( )Av′ 
 
2
2 2 3
2 3
2
2 2 3 3 2
1
( ) 0
2
(3/12)
1
, ( ) ( )( /4) 0
2
A
A
k A
A
A k A
W
T v T
g
U W
W
T U T v W
g
µ
µ
−
−
′ ′= =
= −
′ ′+ = − =
 
 
2 2 2 2
2
2
1
( ) 2(0.3)(32.2 ft/s ) ft 4.83 ft /s
4
( ) 2.198 ft/s
A
A
v
v
§ ·′ = =¨ ¸
© ¹
′ =
 
 
 
 
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791 
PROBLEM 13.178 (Continued) 
 
 Conservation of momentum as A hits B: 
 2
2
( ) 14.342 ft/s
( ) 2.198 ft/s
A
A
v
v
=
′ =
 
 2 2( ) ( )
14.342 0 2.198 12.144 ft/s
A A B B B A B B A B
B B
m v m v m v m v m m
v v
′ ′+ = + =
′ ′+ = + =
 
 Relative velocities A and B: 
 2 2[( ) ] ( )A B AB B Av v e v v′ ′− = − 
 (14.342 0) 12.144 2.198ABe− = − 0.694ABe = W 
 Work and energy. 
 Velocity of B just before impact with C: 
 
2 2
2 2
2 2
4 4 4
2 4
22
4
2 2 4 4
1
( ) (12.144)
2 2
1
( ) ( )
2 2
( ) 12.144 ft/s (1 ft) (0.3)
( )(12.144)
, 0.3
2 2
B B
B
B B
B B
B k B B
B
f k B
W W
T v
g g
W W
T v v
g g
v U W W
v
F W T U T
g g
µ
µ
−
−
′= =
′ ′= =
′ = = − =
′
= + = − =
 
 4( ) 11.321 ft/sBv′ = 
 Conservation of momentum as B hits C: 
 
0.8
2.4
B
C
m
g
m
g
=
=
 
 4( ) 11.321 ft/sBv′ = 
 4 4
4
4
( ) ( )
0.8 0.8 (2.4)
(11.321) 0 ( ) ( )
11.321 ( ) 3
B B C C B B C C
B C
B C
m v m v m v m v
v v
g g g
v v
′ ′′ ′+ = +
′′ ′+ = +
′′ ′= +
 
 
 
 
 
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792 
PROBLEM 13.178 (Continued) 
 
 4Velocity of after hits , ( ) 0.BB B C v′′ = 
 (Compare Figures (1) and (2).) 
 3.774 ft/sCv′ = 
 Relative velocities B and C: 
 4 4
(( ) ) ( )
(11.321 0) 3.774 0
B C BC C B
BC
v v e v v
e
′ ′ ′′− = −
− = −
 
 0.333BCe = W 
(b) Work and energy, Block C: 
 
2
4 5 4 5
2
4 4 5 5
1
( ) 0 ( )
2
1
(3.774) (0.3) ( ) 0
2
C
C k C
C
C
W
T v T U W x
g
W
T U T W x
g
µ−
−
= = = −
+ = − =
 
 
2(3.774)
0.737 ft
2(32.2)(0.3)
x = = 
 8.84 in.x = W 
 
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793 
 
 
PROBLEM 13.179 
A 0.5-kg sphere A is dropped from a height of 0.6 m onto a 1.0 kg 
plate B, which is supported by a nested set of springs and is initially 
at rest. Knowing that the coefficient of restitution between the sphere 
and the plate is 0.8,e = determine (a) the height h reached by the 
sphere after rebound, (b) the constant k of the single spring 
equivalent to the given set if the maximum deflection of the plate is 
observed to be equal to 3h. 
 
SOLUTION 
Velocity of A and B after impact. 
 
0.5 kg
1.0 kg
A
B
m
m
=
=
 
Sphere A falls. Use conservation of energy to find ,Av the speed just before impact. Use the plate surface as 
the datum. 
 
2
1 1 0 2 2
2
1 1 2 2 0
1
0, , , 0
2
1
0 0
2
A A A
A A A
T V m gh T m v V
T V T V m gh m v
= = = =
+ = + + = +
 
With 0
0
0.6 m,
2 (2)(9.81)(0.6)A
h
v gh
=
= =
 
 3.4310 m/sA = ↓v 
Analysis of the impact. Conservation of momentum. 
 withA A B B A A B Bm m m m′ ′+ = +v v v v 0B =v 
Dividing by Am and using y-components with ( / 2)B Am m = 
 3.4310 0 ( ) 2( )A y B yv v′ ′− + = + (1) 
Coefficient of restitution. ( ) ( ) [( ) ( ) ]B y A y A y B yv v e v v′ ′− = − 
 ( ) ( ) ( ) 3.4310B y A y A yv v e v e′ ′− = = − (2) 
 
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794 
PROBLEM 13.179 (Continued) 
 
Solving Eqs. (1) and (2) simultaneously with 0.8e = gives 
 
( ) 0.68621 m/s
( ) 2.0586 m/s
A y
B y
v
v
′ =
′ = −
 
 0.68621 m/sA′ =v 
 2.0586 m/sB′ =v 
(a) Sphere A rises. Use conservation of energy to find h. 
 
2
1 1 2 2
2
1 1 2 2
1
( ) , 0, 0,
2
1
: ( ) 0 0
2
A A A
A A A
T m v V T V m gh
T V T V m v m gh
′= = = =
′+ = + + = +
 
 
2 2( ) (0.68621)
2 (2)(9.81)
Avh
g
′
= = 0.0240 mh = W 
(b) Plate B falls and compresses the spring. Use conservation of energy. 
 Let 0δ be the initial compression of the spring and ∆ be the additional compression of the spring after 
impact. In the initial equilibrium state, 
 0 00:0 ory B BF k W k Wδ δΣ = − = = (3) 
 Just after impact: 2 21 1 0
1 1
( ) ,
2 2B B
T m v V kδ′= = 
 At maximum deflection of the plate, 2 0T = 
 22 2 2 0
1
( ) ( ) ( )
2g e B
V V V W k δ= + = − ∆ + + ∆ 
 Conservation of energy: 1 1 2 2T V T V+ = + 
 2 2 2 20 0 0
1 1 1 1
( ) 0
2 2 2 2B B B
m v k W k k kδ δ δ′ + = − ∆ + + ∆ + ∆ 
 Invoking the result of Eq. (3) gives 
 2 2
1 1
( )
2 2B B
m v k′ = ∆ (4) 
 Data: 1.0 kg, 2.0586 m/sB Bm v′= = 
 3 (3)(0.024) 0.072 mh∆ = = = 
 
2 2
2 2
( ) (1.0)(2.0586)
(0.072)
B Bm vk
′
= =
∆
 817 N/mk = W 
 
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795 
 
 
PROBLEM 13.180 
A 0.5-kg sphere A is dropped from a height of 0.6 m onto a 1.0-kg 
plate B, which is supported by a nested set of springs and is initially 
at rest. Knowing that the set of springs is equivalent to a single 
spring of constant 900 N/mk = , determine (a) the value of the 
coefficient of restitution between the sphere and the plate for which 
the height h reached by the sphere after rebound is maximum, 
(b) the corresponding value of h, (c) the corresponding value of the 
maximum deflection of the plate. 
 
SOLUTION 
 
0.5 kg
1.0 kg
900 N/m
A
B
m
m
k
=
=
=
 
Sphere A falls. Use conservation of energy to find ,Av the speed just before impact. Use the plate surface as 
the datum. 
 
1 1 0
2
2 2
0
1
, 0
2
A
A A
T V m gh
T m v V
= =
= =
 
With 0 0.6 m,h = 02 (2)(9.81)(0.6)Av gh= = 
 3.4310 m/sA =v 
Analysis of impact. Conservation of momentum. 
 A A B B A A B Bm m m m v′ ′+ = +v v v with 0B =v 
Dividing by Am and using y components with ( / 2)B Am m = 
 3.4310 ( ) 2( )A y B yv v′ ′− = + (1) 
Coefficient of restitution. ( ) ( ) [( ) ( ) ]B y A y A y B yv v e v v′ ′− = − 
 
( ) ( ) ( ) 3.4310
( ) 3.4310 ( )
B y A y A y
B y A y
v v e v e
v v
′ ′− = = −
′ ′= − + (2) 
 
 
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796 
PROBLEM 13.180 (Continued) 
 
Substituting into Eq. (1), 
 
3.4310 ( ) (2)[ 3.4310 ( ) ]
( ) 1.1437(2 1)
A y A y
A y
v e v
v e
′ ′− = + − +
′ = − (3) 
From Eq. (2), ( ) 1.1437(1 )B yv e′ = − + (4) 
(a) Sphere A rises. Use conservation of energy to find h. 
 
2
1 1
2 2
2
1 1 2 2
2 2 2
1
( ) , 0
2
0,
1
: ( ) 0 0
2
( ) (1.1437) (2 1)
2 (2)(9.81)
A A
A
A A A
A
T m v V
T V m gh
T V T V m v m gh
v e
h
g
′= =
= =
′+ = + + = +
′ −= =
 
 Since h is to be maximum, e must be as large as possible. 
 Coefficient of restitution for maximum h: 1.000e = W 
(b) Corresponding value of h. ( ) 1.1437[(2)(1) 1] 1.1437 m/sAv′ = − = 
 
2 2( ) (1.1437)
2 (2)(9.81)
Avh
g
′
= = 0.0667 mh = W 
(c) Plate B falls and compresses the spring. Use conservation of energy. 
 Let 0δ be the initial compression of the spring and ∆ be the additional compression of the spring after 
 impact. In the initial equilibrium state, 
 0 00 0 ory B BF k W k Wδ δΣ = − = = (3) 
 Just after impact: 2 21 1 0
1 1
( ) ,
2 2B B
T m v V kδ′= = 
 At maximum deflection of the plate, 2 0T = 
 22 2 2 0
1
( ) ( ) ( )
2g e B
V V V W k δ= + = − ∆ + + ∆ 
 Conservation of energy: 1 1 2 2T V T V+ = + 
 2 2 2 20 0 0
1 1 1 1
( ) 0
2 2 2 2B B B
m v k W k k kδ δ δ′ + = − ∆ + + ∆ + ∆ 
 
 
 
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797 
PROBLEM 13.180 (Continued) 
 
Invoking the result of Eq. (3) gives 
 2 2
1 1
( )
2 2B B
m v k′ = ∆ 
Data: 1.0 kg,
( ) 1.1437(1 1) 2.2874 m/s.
B
B y
m
v
=
′ = − + = −
 
 2.2874 m/sB′ =v , 900 N/mk = 
 
2 2
2 2( ) (1.0)(2.2874) 0.0058133 m
900
B Bm v
k
′
∆ = = = 0.0762 m∆ = W 
 
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798 
 
 
PROBLEM 13.181 
The three blocks shown are identical. Blocks B and C are at 
rest when block B is hit by block A, which is moving with a 
velocity Av of 3 ft/s. After the impact, which is assumed to 
be perfectly plastic ( 0),e = the velocity of blocks A and B 
decreases due to friction, while block C picks up speed, 
until all three blocks are moving with the same velocity .v 
Knowing that the coefficient of kinetic friction between all 
surfaces is 0.20,kµ = determine (a) the time required for 
the three blocks to reach the same velocity, (b) the total 
distance traveled by each block during that time. 
 
SOLUTION 
 
(a) Impact between A and B, conservation of momentum 
 A B C A B Cmv mv mv mv mv mv′ ′ ′+ + = + + 
 3 0 0A Bv v′ ′+ = + + 
 Relative velocities ( 0)e = 
 
( ) 3 2
0 1.5 ft/s
A B B A B
B A B
A B
v v e v v v
v v v
v v
′ ′ ′− = − =
′ ′ ′= − =
′ ′=
 
 Final (common) velocityv = 
 Block C: Impulse and momentum 
 
 CC C f f k C
W
W v F t v F W
g
µ+ = = 
 0 (0.2) (0.2)
v
t v gt
g
+ = = (1) 
 
 
 
 
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799 
PROBLEM 13.181 (Continued) 
 
 Blocks A and B: Impulse and momentum 
 
 
2 (1.5) 4(0.2) 2
1.5 0.4
W W
Wt v
g g
gt v
− =
− = (2) 
 Substitute v from Eq. (1) into Eq. (2) 
 1.5 0.4 0.2gt gt− = 
 
2
(1.5 ft/s)
0.6(32.2 ft/s )
t = 0.0776 st = W�
(b) Work and energy: 
 From Eq. (1) (0.2)(32.2)(0.0776) 0.5 ft/sv = = 
 Block C: 
 2 21 2
1
0 ( ) (0.5)
2 2
W W
T T v
g g
= = = 
 1 2 0.2f C k C CU F x Wx Wxµ− = = = 
 21 1 2 2
1
0 (0.2)( )
2C
W
T U T W x v
g−
+ = + = 
 
2
2
(0.5 ft/s)
0.01941 ft
0.2(2)(32.2 ft/s )
Cx = = 
� 0.01941 ftCx = W�
 Blocks A and B: 
 
 2 21 2
1 1
2 (1.5) 2.25 2 (0.5) 0.25
2 2
W W
T W T W
g g
§ · § ·= = = =¨ ¸ ¨ ¸
© ¹ © ¹
 
 1 2 4 0.8k A AU Wgx Wgxµ− = − = − 
 1 1 2 2T U T−+ = 
 2.25 4(0.2) (32.2) 0.25AW W x W− = 
� 0.07764 ftAx = 0.0776 ftAx = W 
 
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800 
 
PROBLEM 13.182 
Block A is released from rest and slides down 
the frictionless surface of B until it hits a 
bumper on the right end of B. Block A has a 
mass of 10 kg and object B has a mass of 30 kg 
and B can roll freely on the ground. Determine 
the velocities of A and B immediately after 
impact when (a) 0,e = (b) 0.7.e = 
 
SOLUTION 
Let the x-direction be positive to the right and the y-direction vertically upward. 
Let ( ) , ( ) , ( )A x A y A xv v v and ( )B yv be velocity components just before impact and ( ) , ( ) , ( ) ,A x A y B xv v v′ ′ ′ ′ and 
( )B yv′ those just after impact. By inspection, 
 ( ) ( ) ( ) ( ) 0A y B y A y B yv v v v′ ′= = = = 
Conservation of momentum for x-direction: 
While block is sliding down: 0 0 ( ) ( ) ( ) ( )A A x B B x B x A xm v m v v vβ+ = + = − (1) 
Impact: 0 0 ( ) ( ) ( ) ( )A A x B B x B x A xm v m v v vβ′ ′ ′ ′+ = + = − (2) 
where /A Bm mβ = 
Conservation of energy during frictionless sliding: 
Initial potential energies: Am gh for A, 0 for B. 
Potential energy just before impact: 1 0V = 
Initial kinetic energy: 0 0T = (rest) 
Kinetic energy just before impact: 2 21
1 1
2 2A A B B
T m v m v= + 
 
0 0 1 1
2 2 2 2
2
1 1 1
( )
2 2 2
1
(1 )
2
A A A B B A B A
A A
T V T V
m gh m v m v m m v
m v
β
β
+ = +
= + = +
= + 
 2 2
2 2
( )
1 1A A x A
gh gh
v v v
β β
= = =
+ +
 (3) 
 
 
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801 
PROBLEM 13.182 (Continued) 
 
Velocities just before impact: 
2
1A
gh
β
=
+
v 
 
2
1B
ghβ
β
=
+
v 
Analysis of impact. Use Eq. (2) together with coefficient of restitution. 
 
( ) ( ) [( ) ( ) ]
( ) ( ) [( ) ( ) ]
( ) ( )
B x A x A x B x
A x A x A x A x
A x A x
v v e v v
v v e v v
v e v
β β
′ ′− = −
′ ′− − = +
′ = − (4) 
Data: 
2
10 kg
30 kg
0.2 m
9.81 m/s
A
B
m
m
h
g
=
=
=
=
 
 
10kg
0.33333
30kg
β = = 
From Eq. (3), 
(2)(9.81)(0.2)
1.33333
1.71552 m/s
Av =
=
 
(a) 0:e = ( ) 0 ( ) 0A x B xv v′ ′= = 0A′ =v W 
 0B′ =v W 
(b) 0.7:e = ( ) (0.7)(1.71552)
1.20086 m/s
( ) (0.33333)(1.20086)
0.40029 m/s
A x
B x
v
v
′ = −
= −
′ = −
=
 
 1.201 m/sA′ =v W 
 0.400 m/sB′ =v W 
 
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802 
 
 
PROBLEM 13.183 
A 20-g bullet fired into a 4-kg wooden block suspended from 
cords AC and BD penetrates the block at Point E, halfway 
between C and D, without hitting cord BD. Determine (a) the 
maximum height h to which the block and the embedded bullet 
will swing after impact, (b) the total impulse exerted on the block 
by the two cords during the impact. 
 
SOLUTION 
 
Total momentum in x is conserved: 
 
 bl bl bu bu bl bl bu bu bl bucos 20 ( )m v m v m v m v v v′ ′ ′ ′+ ° = + = 
 bl0 (0.02 kg)( 600 m/s)(cos 20) (4.02 kg)( )v′+ − = 
 bl 2.805 m/sv′ = − 
Conservation of energy: 
 
2
1 bl bu bl
2
1
1
1
( )( )
2
4.02 kg
(2.805 m/s)
2
15.815 J
T m m v
T
T
′= +
§ ·= ¨ ¸
© ¹
=
 
 
1
2 2 bl bu
2
2
1 1 2 2
0
0 ( )
(4.02 kg)(9.81 m/s )( ) 39.44
V
T V m m gh
V h h
T V T V
=
= = +
= =
+ = +
 
 
15.815 0 0 39.44
0.401 m
h
h
+ = +
= 401 mmh = W 
(b) Refer to figure in part (a). 
 Impulse-momentum in y-direction: 
 bu bu bl bu blsin 20 ( )( ) ym v F t m m v′+ ∆ = + 
 bl( ) 0yv = 
 (0.02 kg) ( 600 m/s) (sin 20 ) 0F t− ° + ∆ = 4.10 N sF t∆ = ⋅ W 
 
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803 
 
 
PROBLEM 13.184 
A 2-lb ball A is suspended from a spring of constant 10 lb/in and is initially at rest when it 
is struck by 1-lb ball B as shown. Neglecting friction and knowing the coefficient of 
restitution between the balls is 0.6, determine (a) the velocities of A and B after the impact, 
(b) the maximum height reached by A. 
 
SOLUTION 
Masses: 2 2
2 2
2 lb 1 lb
0.062112 lb s /ft 0.031056 lb s /ft
32.2 ft/s 32.2 ft/sA B
m m= = ⋅ = = ⋅ 
Other data: (10 lb/in.)(12 in./ft.) 120 lb/ft, 0.6k e= = = 
 0, 2 ft/sA Bv v= = 
For analysis of the impact use the principle of impulse and momentum. 
 1 1 2 2m mΣ + Σ = Σv Imp vX 
 
t-direction for ball A: 
 0 0 ( )A A tm v′+ = ( ) 0A tv′ = 
t-direction for ball B: sin 20 0 ( )B B B B tm v m v′° + = 
 ( ) sin 20 (2)(sin 20 ) 0.6840 ft/sB t Bv v′ = ° = ° = 
 
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804 
PROBLEM 13.184 (Continued) 
 
n-direction for balls A and B: 
 
cos 20 0 ( ) ( )
( ) ( ) cos20
B B B B n A A n
A
B n A n B
B
m v m v m v
m
v v v
m
′ ′° + = +
′ ′+ = °
 
 ( ) 2( ) (2)cos 20B n A nv v′ ′+ = ° (1) 
Coefficient of restitution: ( ) ( ) [( ) ( ) ]
[0 ( cos20 )]
B n A n A n B n
B
v v e v v
e v
′ ′− = −
= − °
 
 (0.6)(2)cos20= − ° (2) 
Solving Eqs. (1) and (2) simultaneously, 
 ( ) 1.00234 ft/s ( ) 0.12529 ft/sA n B nv v′ ′= = − 
(a) Velocities after the impact: 
 1.00234 ft/sA′ =v 1.002 ft/sA′ =v W 
 (0.6840 ft/s ) (0.12529 ft/sB′ = → +v ) 
 
2 2(0.6840) (0.12529) 0.695 ft/s
0.12529
tan 10.4
0.6840
Bv
β β
= + =
= = °
 
 0.695 ft/sB′ =v 10.4° W 
(b) Maximum height reached by A: 
 Use conservation of energy for ball A after the impact. 
 Position 1: Just after impact. 
 2 21
1 1
( ) (0.062112)(1.00234) 0.0312021 ft lb
2 2A A
T m v′= = = ⋅ 
 Force in spring = weight of A 
 
1
2 2
2
1 1
2
1
2 lb
0.016667 ft
120 lb/ft
1 1
( )
2 2 2
(2 lb)
0.016667 ft lb
(2)(120)
( ) 0 (datum)
A
B A
e
g
WF
x
k k
W W
V kx k
k k
V
= − = − = − = −
§ ·= = =¨ ¸
© ¹
= = ⋅
=
 
 
 
 
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805 
PROBLEM 13.184 (Continued) 
 Position 2: Maximum height h. 
 
2
2
2 2
2 1
2
2
0
0
1 1
( ) ( ) (120)( 0.016667)
2 2
60 2 0.016667
( ) (2 lb) 2
e
g A
V
T
V k h x h
h h
V W h h h
=
=
= + = −
= − +
= = =
 
 Conservation of energy: 1 1 2 2T V T V+ = + 
 20.031202 0.016667 0 60 2 0.016667 2h h h+ = + − + + 
 260 0.031202 0.022804 fth h= = ± 
 Using the positive root,