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Problem 16.41 The masses mA=mB. The surface
is smooth. At t=0,A is stationary, the spring is
unstretched, and Bis given a velocity v0toward the
right.
(a) In the subsequent motion, what is the velocity of
the common center of mass of Aand B?
(b) What are the velocities of Aand Bwhen the spring
is unstretched?
mA
k
mB
Problem 16.42 In Problem 16.41, mA=40 kg, mB=
30 kg, and k=400 N/m. The two masses are released
from rest on the smooth surface with the spring stretched
1 m. What are the magnitudes of the velocities of the
masses when the spring is unstretched?
274
Problem 16.43 A girl weighing 80 lb stands at rest on
a 325-lb floating platform. She starts running at 10 ft/s
relative to the platform and runs off the end. Neglect the
horizontal force exerted on the platform by the water.
(a) After she starts running, what is her velocity rela-
tive to the water?
(b) While she is running, what is the velocity of the
common center of mass of the girl and the platform
relative to the water? (See Active Example 16.4.)
Problem 16.44 Two railroad cars with weights WA=
120,000 lb and WB=70,000 lb collide and become cou-
pled together. Car Ais full, and car Bis half full, of
carbolic acid. When the cars collide, the acid in B sloshes
back and forth violently.
(a) Immediately after the impact, what is the velocity
of the common center of mass of the two cars?
(b) When the sloshing in B has subsided, what is the
velocity of the two cars?
2 ft/s
A1 ft/s
B
Solution:
vcenter of mass =(120,000 lb)(2 ft/s)+(70,000 lb)(1 ft/s)
Problem 16.45 Suppose that the railroad track in Prob-
lem 16.44 has a constant slope of 0.2 degrees upward
toward the right. If the cars are 6 ft apart at the instant
shown, what is the velocity of their common center of
mass immediately after they become coupled together?
32.2 ft/s2vcenter of mass
vcenter of mass =0.957 ft/s
Problem 16.46 The 400-kg satellite Straveling at
7 km/s is hit by a 1-kg meteor Mtraveling at 12 km/s.
The meteor is embedded in the satellite by the impact.
Determine the magnitude of the velocity of their
common center of mass after the impact and the angle β
between the path of the center of mass and the original
path of the satellite.
45
⬚
M
S7 km/s
b
Problem 16.47 The 400-kg satellite Straveling at
7 km/s is hit by a 1-kg meteor M. The meteor is
embedded in the satellite by the impact. What would
the magnitude of the velocity of the meteor need to
be to cause the angle βbetween the original path of
the satellite and the path of the center of mass of the
combined satellite and meteor after the impact to be
0.5◦? What is the magnitude of the velocity of the center
of mass after the impact?
vm=34.26 km/s : v=6.92 km/s.
7 km/s
β
= 0.5°
M
vM
276
c
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Problem 16.48 A 68-kg astronaut is initially station-
ary at the left side of an experiment module within an
orbiting space shuttle. The 105,000-kg shuttle’s center of
Solution: Consider the motion of the astronaut (A) and shuttle (S)
relative to a reference frame that is stationary with respect to their
common center of mass. During the astronaut’s motion,
Problem 16.49 An 80-lb boy sitting in a stationary
20-lb wagon wants to simulate rocket propulsion by
throwing bricks out of the wagon. Neglect horizontal
forces on the wagon’s wheels. If the boy has three bricks
weighing 10 lb each and throws them with a horizontal
velocity of 10 ft/s relative to the wagon, determine the
velocity he attains (a) if he throws the bricks one at a
time and (b) if he throws them all at once.
Solving,v
w1=0.769 ft/s.
Second brick:
(mB+mw+2mb)vw1=(mB+mw+mb)vw2+mbvb2,
vb2−vw2=−10.
Solving,v
w2=1.603 ft/s.
Third brick :
(mB+mw+mb)vw2=(mB+mw)vw3+mbvb3,
vb3−vw3=−10.
Problem 16.50 A catapult designed to throw a line to
ships in distress throws a 2-kg projectile. The mass of
the catapult is 36 kg, and it rests on a smooth surface.
It the velocity of the projectile relative to the earth as
it leaves the tube is 50 m/s at θ0=30◦relative to the
horizontal, what is the resulting velocity of the catapult
toward the left?
θ
0
278
Problem 16.51 The catapult, which has a mass of 36
as it leaves the tube is 50 m/s at θ0=30◦relative to the
horizontal. What is the resulting velocity of the catapult
toward the left?
Solution:
where
Problem 16.52 A bullet with a mass of 3.6 grams is
moving horizontally with velocity vand strikes a 5-kg
block of wood, becoming embedded in it. After the
impact, the bullet and block slide 24 mm across the floor.
The coefficient of kinetic friction between the block and
the floor is µk=0.4. Determine the velocity v.
v
Solution: Momentum is conserved through the collision and then
Problem 16.53 A 0.12-ounce bullet hits a suspended
15-lb block of wood and becomes embedded in it. The
angle through which the wires supporting the block rotate
as a result of the impact is measured and determined to
be 7◦. What was the bullet’s velocity? 2 ft 7⬚
Problem 16.54 The overhead conveyor drops the 12-kg
package Ainto the 1.6-kg carton B. The package is tacky
and sticks to the bottom of the carton. If the coefficient
of friction between the carton and the horizontal con-
veyor is µk=0.2, what distance does the carton slide
after impact?
1 m/s
A
26°
Let vbe the velocity of the combined object relative to the belt.
Use work and energy to determine the sliding distance d:
Problem 16.55 A 12,000-lb bus collides with a 2800-
lb car. The velocity of the bus before the collision
is vB=18i(ft/s) and the velocity of the car is vC=
33j(ft/s). The two vehicles become entangled and
remain together after the collision. The coefficient of
kinetic friction between the vehicles’ tires and the road
is µk=0.6.
(a) What is the velocity of the common center of mass
of the two vehicles immediately after the collision?
(b) Determine the approximate final position of the
common center of mass of the vehicles relative to
its position when the collision occurred. (Assume
that the tires skid, not roll, on the road.)
vC
y
x
vB
Solution:
(a) The collision (impulse - momentum).
280
Problem 16.56 The velocity of the 200-kg astronaut
Arelative to the space station is 40i+30j(mm/s). The
velocity of the 300-kg structural member Brelative to
the station is −20i+30j(mm/s). When they approach
each other, the astronaut grasps and clings to the struc-
tural member.
(a) What is the velocity of their common center of
mass when they arrive at the station?
(b) Determine the approximate position at which they
contact the station.
A
B
y
x
9 m
6 m
Solution:
Problem 16.57 The weights of the two objects are
WA=5 lb and WB=8 lb. Object Ais moving at vA=
2 ft/s and undergoes a perfectly elastic impact with the
stationary object B. Determine the velocities of the
objects after the impact.
vA
AB
Solution: Momentum is conserved and the coefficient of restitu-
Problem 16.58 The weights of the two objects are
WA=5 lb and WB=8 lb. Object Ais moving at vA=
2 ft/s and undergoes a direct central impact with the sta-
tionary object B. The coefficient of restitution is e=0.8.
Determine the velocities of the objects after the impact.
vA
AB
Solution: Momentum is conserved and the coefficient of restitu-
Problem 16.59 The objects Aand Bwith velocities
vA=20 m/s and vB=4 m/s undergo a direct central
impact. Their masses are mA=8 kg and mB=12 kg.
After the impact, the object Bis moving to the right at
16 m/s. What is the coefficient of restitution?
v
A
AB
vB
Problem 16.60 The 8-kg mass Aand the 12-kg mass
Bslide on the smooth horizontal bar with the velocities
shown. The coefficient of restitution is e=0.2. Deter-
mine the velocities of the masses after they collide. (See
Active Example 16.5).
3 m/s2 m/s
AB
Solution: Momentum is conserved, and the coefficient of restitu-
282
Problem 16.61 In a study of the effects of an accident
on simulated occupants, the 1900-lb car with velocity
vA=30 mi/h collides with the 2800-lb car with velocity
vB=20 mi/h. The coefficient of restitution of the
impact is e=0.15. What are the velocities of the cars
immediately after the collision?
vAvB
Problem 16.62 In a study of the effects of an accident
on simulated occupants, the 1900-lb car with velocity
vA=30 mi/h collides with the 2800-lb car with velocity
vB=20 mi/h. The coefficient of restitution of the impact
is e=0.15. The duration of the collision is 0.22 s. Deter-
mine the magnitude of the average acceleration to which
the occupants of each car are subjected.
vAvB
Solution: The velocities before the collision are converted into
Problem 16.63 The balls are of equal mass m. Balls
Band Care connected by an unstretched spring and are
stationary. Ball Amoves toward ball Bwith velocity
vA. The impact of Awith Bis perfectly elastic (e =1).
(a) What is the velocity of the common center of mass
of Band Cimmediately after the impact?
(b) What is the velocity of the common center of mass
of Band Cat time tafter the impact?
A
ABC
k
x=mxB+mxC
m+m=xB+xC
2,
so
dx
dt =1
2dxB
dt +dxC
dt .
Immediately after the impact dxB/dt =vAand dxC/dt =0, so
dx
dt =1
2vA.
Problem 16.64 In Problem 16.63, what is the maxi-
mum compressive force in the spring as a result of the
impact?
284
Problem 16.65* The balls are of equal mass m. Balls
Band Care connected by an unstretched spring and are
stationary. Ball Amoves toward ball Bwith velocity
vA. The impact of Awith Bis perfectly elastic (e =1).
Suppose that you interpret this as an impact between ball
Aand an “object” Dconsisting of the connected balls
Band C.
(a) What is the coefficient of restitution of the impact
between Aand D?
(b) If you consider the total energy after the impact
to be the sum of the kinetic energies, 1
2m(v′)2
A+
1
2(2m)(v′
D)2, where v′
Dis the velocity of the center
of mass of Dafter the impact, how much energy
is “lost” as a result of the impact?
(c) How much energy is actually lost as a result of the
impact? (This problem is an interesting model for
one of the mechanisms for energy loss in impacts
between objects. The energy “loss” calculated in
part (b) is transformed into “internal energy” — the
vibrational motions of Band Crelative to their
common center of mass.)
Solution: See the solution of Problem 16.135. Just after the impact
of A and B, A is stationary and the center of mass D of B and C is
AB D C
Problem 16.66 Suppose that you investigate an acci-
dent in which a 3400-lb car Astruck a parked 1960-lb
car B. All four of car B’s wheels were locked, and skid
marks indicate that it slid 20 ft after the impact. If you
estimate the coefficient of kinetic friction between B’s
tires and the road to be µk=0.8 and the coefficient of
restitution of the impact to be e=0.2, what was A’s
velocity vAjust before the impact? (Assume that only
one impact occurred.)
AB
vA
Solution: We can use work-energy to find the velocity of car B
just after the impact. Then we use conservation of momentum and the
Problem 16.67 When the player releases the ball from
rest at a height of 5 ft above the floor, it bounces to a
height of 3.5 ft. If he throws the ball downward, releas-
ing it at 3 ft above the floor, how fast would he need to
throw it so that it would bounce to a height of 12 ft?
17.94 ft/s =0.837
0.837 =
33.23 ft/s before it hits the floor. To find the original velocity when it
286
Problem 16.68 The 0.45-kg soccer ball is 1 m above
the ground when it is kicked upward at 12 m/s. If the
coefficient of restitution between the ball and the ground
is e=0.6, what maximum height above the ground does
the ball reach on its first bounce?
12 m/s
1 m
Solution: We must first find the velocity with which the ball strikes
Post Impact:
Problem 16.69 The 0.45-kg soccer ball is stationary
just before it is kicked upward at 12 m/s. If the impact
lasts 0.02 s, what average force is exerted on the ball by
the player’s foot?
Problem 16.70 By making measurements directly
from the photograph of the bouncing golf ball, estimate
the coefficient of restitution.
Solution: For impact on a stationary surface, the coefficient of
Problem 16.71 If you throw the golf ball in Prob-
lem 16.70 horizontally at 2 ft/s and release it 4 ft above
the surface, what is the distance between the first two
bounces?
Solution: The normal velocity at impact is vAn =−
√2g(4)=
Problem 16.72 In a forging operation, the 100-lb
weight is lifted into position 1 and released from rest. It
falls and strikes a workpiece in position 2. If the weight
is moving at 15 ft/s immediately before the impact and
the coefficient of restitution is e=0.3, what is the
velocity of the weight immediately after impact?
1
288
Problem 16.73 The 100-lb weight is released from
rest in position 1. The spring constant is k=120 lb/ft,
and the springs are unstretched in position 2. If the
coefficient of restitution of the impact of the weight
with the workpiece in position 2 is e=0.6, what is
the magnitude of the velocity of the weight immediately
after the impact? 16 in
2
1
Problem 16.74* A bioengineer studying helmet design
uses an experimental apparatus that launches a 2.4-kg
helmet containing a 2-kg model of the human head
against a rigid surface at 6 m/s. The head, suspended
within the helmet, is not immediately affected by the
impact of the helmet with the surface and continues to
move to the right at 6 m/s, so the head then undergoes
an impact with the helmet. If the coefficient of restitu-
tion of the helmet’s impact with the surface is 0.85 and
the coefficient of restitution of the subsequent impact
of the head with the helmet is 0.15, what is the velocity
of the head after its initial impact with the helmet?
6 m/s
Problem 16.75*
Solution: See the solution to Problem 16.74
Problem 16.76 Two small balls, each of 1-lb weight,
hang from strings of length L=3 ft. The left ball is
released from rest with θ=35◦. The coefficient of resti-
tution of the impact is e=0.9. Through what maximum
angle does the right ball swing? L
L
m
m
u
2[32.2 ft/s2][3 ft] =33.2◦.
290
Problem 16.77 In Example 16.6, if the Apollo
command-service module approaches the Soyuz space-
craft with velocity 0.25i+0.04j+0.01k(m/s) and the
docking is successful, what is the velocity of the center
of mass of the combined vehicles afterward?
x
B
A
v
A
z
y
Problem 16.78 The 3-kg object Aand 8-kg object B
undergo an oblique central impact. The coefficient of
restitution is e=0.8. Before the impact, vA=10i+
4j+8k(m/s) and vB=−2i−6j+5k(m/s). What are
the velocities of Aand Bafter the impact?
AB
y
x
Problem 16.79 A baseball bat (shown with the bat’s
axis perpendicular to the page) strikes a thrown base-
ball. Before their impact, the velocity of the baseball is
vb=132(cos 45◦i+cos 45◦j)(ft/s) and the velocity of
the bat is vB=60(−cos 45◦i−cos 45◦j)(ft/s). Neglect
the change in the velocity of the bat due to the direct
central impact. The coefficient of restitution is e=0.2.
What is the ball’s velocity after the impact? Assume that
the baseball and the bat are moving horizontally. Does
the batter achieve a potential hit or a foul ball?
baseball
Bat
y
vb
vB
Problem 16.80 The cue gives the cue ball Aa velocity
parallel to the yaxis. The cue ball hits the eight ball
Band knocks it straight into the corner pocket. If the
magnitude of the velocity of the cue ball just before the
impact is 2 m/s and the coefficient of restitution is e=1,
what are the velocity vectors of the two balls just after
the impact? (The balls are of equal mass.)
x
y
45°
A
B
292
Problem 16.81 In Problem 16.80, what are the veloc-
ity vectors of the two balls just after impact if the coef-
ficient of restitution is e=0.9?
