Solutions: Chapter 3 Exercises
1. Supertankers are so massive, that even at modest speeds their motional inertia, or momenta,
are enormous. This means enormous impulses are needed for changing motion. How can large
impulses be produced with modest forces? By applying modest forces over long periods of time.
Hence the force of the water resistance over the time it takes to coast 25 kilometers sufficiently
reduces the momentum.
2. Air bags lengthen the time of impact thereby reducing the force of impact.
3. This illustrates the same point as Exercise 4. The time during which momentum decreases is
lengthened, thereby decreasing the jolting force of the rope. Note that in all of these examples,
bringing a person to a stop more gently does not reduce the impulse. It only reduces the force.
4. Crumpling allows more time for reducing the momentum of the car, resulting in a smaller
force of impact on the occupants.
5. Its momentum is the same (its weight might change, but not its mass).
6. Although the impulses may be the same for the two cases, the times of impact are not. When
the egg strikes the wall, impact time is short and impact force correspondingly large. The egg
breaks. But when the egg strikes the sagging sheet, impact time is long and the force
correspondingly small. Doing this makes a nice demonstration of impulse-momentum.
7. When a boxer hits his opponent, the opponent contributes to the impulse that changes the
momentum of the punch. When punches miss, no impulse is supplied by the opponent—all effort
that goes into reducing the momentum of the punches is supplied by the boxer himself. This tires
the boxer. This is very evident to a boxer who can punch a heavy bag in the gym for hours and
not tire, but who finds by contrast that a few minutes in the ring with an opponent is a tiring
experience.
8. Without this slack, a locomotive might simply sit still and spin its wheels. The loose coupling
enables a longer time for the entire train to gain momentum, requiring less force of the
locomotive wheels against the track. In this way, the overall required impulse is broken into a
series of smaller impulses. (This loose coupling can be very important for braking as well.)
9. If no momentum is imparted to the ball, no oppositely directed momentum will be imparted to
the thrower. Going through the motions of throwing has no net effect. If at the beginning of the
throw you begin recoiling backward, at the end of the throw when you stop the motion of your
arm and hold onto the ball, you stop moving too. Your position may change a little, but you end
up at rest. No momentum given to the ball means no recoil momentum gained by you.
10. In jumping, you impart the same momentum to both you and the canoe. This means you
jump from a canoe that is moving away from the dock, reducing your speed relative to the dock,
so you don’t jump as far as you expected to.
11. To get to shore, the person may throw keys, coins or an item of clothing. The momentum of
what is thrown will be accompanied by the thrower’s oppositely-directed momentum. In this
way, one can recoil towards shore. (One can also inhale facing the shore and exhale facing away
from the shore.)
12. Regarding Exercise 9, if you don’t exert a net force on the ball, it won’t exert a force on you.
So no net force, no change in motion. Regarding Exercise 10, you exert a force on the boat, call
it action, and the boat exerts a force on you, call it reaction. If your jump lacks sufficient force,
you end up in the water. Regarding Exercise 11, if one throws clothing, the force that accelerates
the clothes will be paired with an equal and opposite force on the thrower. This force can provide
recoil toward shore.
13. If the rocket and its exhaust gases are treated as a single system, the forces between rocket
and exhaust gases are internal, and momentum in the rocket-gases system is conserved. So any
momentum given to the gases is equal and opposite to momentum given to the rocket. A rocket
attains momentum by giving momentum to the exhaust gases.
14. Momentum is not conserved for the ball itself because an impulse is exerted on it
(gravitational force × time). So the ball gains momentum. It is in the absence of an external force
that momentum doesn’t change. If the whole Earth and the rolling ball are taken together as a
system, then the gravitational interaction between the Earth and the ball are internal forces and
no external impulse acts. Then the momentum of the ball is accompanied by an equal and
opposite momentum of the Earth, which results in no change in momentum.
15. For the system comprised of ball + Earth, momentum is conserved because the impulses
acting are internal impulses. The momentum of the falling apple is equal in magnitude to the
momentum of the Earth toward the apple.
16. If the system is the stone only, its momentum certainly changes as it falls. If the system is
enlarged to include the stone plus the Earth, then the downward momentum of the stone is
cancelled by the equal but opposite momentum of the Earth “racing” up to meet the stone.
17. This exercise is similar to the previous one. If we consider Bronco to be the system, then a
net force acts and momentum changes. In the system composed of Bronco alone, momentum is
not conserved. If, however we consider the system to be Bronco and the world (including the
air), then all the forces that act are internal forces and momentum is conserved. Momentum is
conserved only in systems not subject to external forces.
18. The craft moves to the right. This is because there are two impulses that act on the craft: One
is that of the wind against the sail, and the other is that of the fan recoiling from the wind it
produces. These impulses are oppositely directed, but are they equal in magnitude? No, because
of bouncing. The wind bounces from the sail and produces a greater impulse than if it merely
stopped. This greater impulse on the sail produces a net impulse in the forward direction, toward
the right. We can see this in terms of forces as well. Note in the sketch there are two force pairs
to consider: (1) the fan-air force pair, and (2) the air-sail force pair. Because of bouncing, the airsail pair is greater. Solid vectors show forces exerted on the craft; dashed vectors show forces
exerted on the air. The net force on the craft is forward, to the right. The principle described here
is applied in thrust reversers used to slow jet planes after they land. Also, you can see that after
the fan is turned on, there is a net motion of air to the left, so the boat, to conserve momentum,
will move to the right.
19. If the air is brought to a halt by the sail, then the impulse against the sail will be equal and
opposite to the impulse on the fan. There will be no net impulse and no change in momentum.
The boat will remain motionless. Bouncing counts!
20. Removing the sail and turning the fan around is the best means of propelling the boat! Then
maximum impulse is exerted on the craft. If the fan is not turned around, the boat is propelled
backward, to the left. (Such propeller-driven boats are used where the water is very shallow, as
in the Florida Everglades.)
21. Yes, you exert an impulse on a ball that you throw. You also exert an impulse on the ball
when you catch it. Since you change its momentum by the same amount in both cases, the
impulse you exert in both cases is the same. To catch the ball and then throw it back again at the
same speed requires twice as much impulse. On a skateboard, you’d recoil and gain momentum
when throwing the ball, you’d also gain the same momentum by catching the ball, and you’d
gain twice the momentum if you did both—catch and then throw the ball at its initial speed in the
opposite direction.
22. In terms of force: When the sand lands on the cart it is brought up to the cart’s speed. This
means a horizontal force provided by the cart acts on the sand. By action-reaction, the sand
exerts a force on the cart in the opposite direction—which slows the cart. In terms of momentum
conservation: Since no external forces act in the horizontal direction, the momentum after the
cart catches sand equals the momentum before. Since mass is added, velocity must decrease.
23. Momentum conservation is being violated. The momentum of the boat before the hero lands
on it will be the same as the momentum of boat + hero after. The boat will slow down. If, for
example, the masses of the hero and boat were the same, the boat should be slowed to half speed;
mvbefore = 2m(v/2)after. From an impulse-momentum point of view, when the hero makes contact
with the boat, he is moved along with the boat by a friction force between his feet and the boat
surface. The equal and opposite friction force on the boat surface provides the impulse that slows
the boat. (Here we consider only horizontal forces and horizontal component of momentum.)
24. We assume the equal strengths of the astronauts means that each throws with the same speed.
Since the masses are equal, when the first throws the second, both the first and second move
away from each other at equal speeds. Say the thrown astronaut moves to the right with velocity
V, and the first recoils with velocity -V. When the third makes the catch, both she and the second
move to the right at velocity V/2 (twice the mass moving at half the speed, like the freight cars in
Figure 3.13). When the third makes her throw, she recoils at velocity V (the same speed she
imparts to the thrown astronaut) which is added to the V/2 she acquired in the catch. So her
velocity is V + V/2 = 3V/2, to the right—too fast to stay in the game. Why? Because the velocity
of the second astronaut is V/2 – V = -V/2, to the left—too slow to catch up with the first astronaut
who is still moving at -V. The game is over. Both the first and the third got to throw the second
astronaut only once!
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25. Your friend does twice as much work (4 × ⁄2 > 1 × 1).
26. More force is required to stretch the strong spring, so more work is done in stretching it the
same distance as a weaker spring.
27. Work done by each is the same, for they reach the same height. The one who climbs in 30 s
uses more power because work is done in a shorter time.
28. When a rifle with a long barrel is fired, more work is done as the bullet is pushed through the
longer distance. A greater KE is the result of the greater work, so of course, the bullet emerges
with a greater velocity. (It might be mentioned that the force acting on the bullet is not constant,
but decreases with increasing distance inside the barrel.)
29. KE depends on the square of speed, so the faster one, the lighter golf ball, has the greater KE.
30. The KE of a pendulum bob is maximum where it moves fastest, at the lowest point; PE is
maximum at the uppermost points. When the pendulum bob swings by the point that marks half
its maximum height, it has half its maximum KE, and its PE is halfway between its minimum
and maximum values. If we define PE = 0 at the bottom of the swing, the place where KE is half
its maximum value is also the place where PE is half its maximum value, and KE = PE at this
point. (In accordance with energy conservation: Total energy = KE + PE.)
31. If the ball is given an initial KE, it will return to its starting position with that KE (moving in
the other direction!) and hit the instructor. (The usual classroom procedure is to release the ball
from the nose at rest. Then when it returns it will have no KE and will stop short of bumping the
nose.)
32. On the hill there is a component of gravitational force in the direction of the car’s motion.
This component of force does work on the car. But on the level, there is no component of
gravitational force along the direction of the car’s motion, so the force of gravity does no work in
this case.
33. The 100 J of potential energy that doesn’t go into increasing her kinetic energy goes into
thermal energy—heating her bottom and the slide.
34. Both will have the same speed. This is easier to see here because both balls convert the same
PE to KE. Which gets to the end first, however, is a different question! (Ball B wins due to its
greater average speed.)
35. If KEs are the same but masses differ, then the ball with the smaller mass has the greater
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speed. That is, ⁄2Mv = ⁄2mV . Likewise with molecules, where lighter ones move faster on the
average than more massive ones. (We will see in Chapter 6 that temperature is a measure of
average molecular KE—lighter molecules in a gas move faster than same-temperature heavier
molecules.)
36. Yes, a car burns more gasoline when its lights are on. The overall consumption of gasoline
does not depend on whether or not the engine is running. Lights and other devices run off the
battery, which “run down” the battery. The energy used to recharge the battery ultimately comes
from the gasoline.
37. You need to know the distance the rock penetrates into the ground, or the time to do so. The
work that the rock does on the ground is equal to its PE before being dropped, mgh = 100 joules.
The force of impact, however, depends on the distance of rock penetration into the ground. If we
do not know this distance we cannot calculate the force. If we knew the time during which the
impulse occurs we could calculate the force from the impulse-momentum relationship—but not
knowing the distance or time of the rock’s penetration into the ground, we cannot calculate the
force.
38. When air resistance is a factor, the ball will return with less speed (air resistance never
increases speed!). It therefore will have less KE. You can see this directly from the fact that the
ball loses mechanical energy to the air molecules it encounters, so when it returns to its starting
point and to its original PE, it will have less KE. This does not contradict energy conservation,
for energy is dissipated, not destroyed.
39. The ball strikes the ground with the same speed, whether thrown upward or downward. The
ball starts with the same energy at the same place, so they will have the same energy when they
reach the ground. This means they will strike with the same speed. This is assuming negligible
air resistance, for if air resistance is a factor, then the ball thrown upward will dissipate more
energy in its longer path and strike with somewhat less speed. Both hit the ground at the same
speed (but at different times).
40. In a conventional car, braking converts KE to heat. In a hybrid car, braking charges up the
batteries. In this way, braking energy can soon be transformed to KE.
41. Yes, and object can have energy without momentum if we consider PE (if an object has KE,
then it must have momentum—for it is moving). But it can have potential energy without being
in motion, and therefore without having momentum. And every object has “energy of being”—
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stated in the celebrated equation E = mc . So whether an object moves or not, it has some form of
energy. If it has KE, then with respect to the frame of reference in which its KE is measured, it
also has momentum.
42. When the mass is doubled with no change in speed, both momentum and KE are doubled.
43. When the velocity is doubled, the momentum is doubled and the KE is increased by a factor
of 4. Momentum is proportional to speed, KE to speed squared.
44. Both have the same momentum, but the 1-kg one, the faster one, has the greater KE.
45. Net momentum before the lumps collide is zero and is also zero after collision. Momentum is
indeed conserved. Kinetic energy after is zero, but was greater than zero before collision. The
lumps are warmer after colliding because the initial kinetic energy of the lumps transforms into
thermal energy. Momentum has only one form. There is no way to “transform” momentum from
one form to another, so it is conserved. But energy comes in various forms and can easily be
transformed. No single form of energy such as KE need be conserved.
46. An engine that is 100% efficient would not be warm to the touch, nor would its exhaust heat
the air, nor would it make any noise, nor would it vibrate. This is because all these are transfers
of energy, which cannot happen if all the energy given to the engine is transformed to useful
work.
47. In the popular sense, conserving energy means not wasting energy. In the physics sense
energy conservation refers to a law of nature that underlies natural processes. Although energy
can be wasted (which really means transforming it from a more useful to a less useful form), it
cannot be destroyed. Nor can it be created. Energy is transferred or transformed, without gain or
loss. That’s what a physicist means in saying energy is conserved.
48. Your friend is correct, for changing KE requires work, which means more fuel consumption
and decreased air quality.
49. In accord with energy conservation, a person who takes in more energy than is expended
stores what’s left over as added chemical energy in the body—which in practice means more fat.
One who expends more energy than is taken in gets extra energy by “burning” body fat. An
undernourished person who performs extra work does so by consuming stored chemical energy
in the body—something that cannot long occur without losing health—and life.
50. There is more to the “swinging balls” problem than momentum conservation, which is why
the problem wasn’t posed in the previous chapter. Momentum is certainly conserved if two balls
strike with momentum 2mv and one ball pops out with momentum m(2v). That is, 2mv = m2v.
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We must also consider KE. Two balls would strike with 2( ⁄2 mv ) = mv . The single ball
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popping out with twice the speed would carry away twice as much energy as was put in: ⁄2
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m(2v) = ⁄2 m(4 v ) = 2mv . This is clearly a conservation of energy no-no!
Solutions: Chapter 3 Problems