# Homework, Chapter 5: Momentum

Ch 5; Ex 1, 2, 4, 18, 22, 31, 32, 38, 40

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## Exercises (Discussion Questions)

Ex 5.1To bring a supertanker to a stop, its engines are typically cut off about 25 km from port. Why is it so difficult to stop or turn a supertanker?

A supertanker has an enormous mass so, with any velocity at all, its momentumwill be very large and it will require a force acting over a very long time to bring it to a stop.

Ex 5.2 In terms of impulse and momentum, why are padded dashboards safer in automobiles?

Impulse = (force) (time)

Impulse = change in momentum

At the beginning of a car crash you have some momentum. At the end, your momentum is zero. This change in momentum is brought about by a force acting over some time. If you run into a solid steel dashboard, you will be brought to rest is a very short time; this means the force exerted on you by the solid steel dashboard will be very large. If you run into a padded dashboard, you will be brought to rest in a longer time; this means the force exerted on you by a padded dashboard will be smaller.

Ex 5.4 In terms of impulse and momentum, why are nylon ropes, which stretch considerably under stress, favord by mountain climbers?

Consider a mountain climber who falls and is brought to a stop by a safety rope. Initially the falling climber has some momentum (mass x velocity). After the rope brings the climber to a stop, the climber's momentum is zero. As nylon ropes stretch, they exert a force over a greater time so that force will be smaller than with a hemp rope which exerts a force over a shorter time.

Ex 5.18 If only an external force can change the state of motion of a body, how can the internal force of the brakes bring a car to rest?

A car moves forward by exerting a (backward) force on the road which then exerts a (forward) force on the car. The brakes of a car cause the tires to exert a force on the road and the road then exerts a force on the tires or the car. This is another example of action-reaction pairs of forces described by Newton's Third Law of Motion.

Ex 5.22 In Chapter 4, rocket propulsion was explained in terms of Newton's third law. That is, the force that propels a rocket is from the exhaust gases pushing against the rocket, the reaction to the force the rocket exerts on the exhaust gases. Explain rocket propulsion in terms of momentum conservation.

Momentum conservation and Newton's third law (action and reaction) are very closely related. In fact, they are two ways of describing or explaining the same idea.

Rocket exhaust gases carry some momentum in one direction and the rocket must carry momentum in the opposite direction to keep the total momentum conserved. Remember, momentum is a vector, so the direction is very important.

Ex 5.31 When you are traveling in your car at highway speed, the momentum of a bug is suddenly changed as it splatters on your windshield. Compared to the change in momentum of the bug, by how much does the momentum of your car change?

The change in momentum of your car is the same as the change in the momentum of the bug -- but in the opposite direction. Of course, the bug's velocity changes enormously because of its small mass while your velocity change is very difficult to measure for it is so small because of the large mass of your car.

Ex 5.32 If a Mac truck and a Volkswagen have a head-on collision, which vehicle will experience the greater force of impact? The greater impulse? The greater change in its momentum? The greater acceleration?

The force of impact is the same.

The impulse is the same.

The change in momentum is the same.

Remember, tho', momentum = (mass) (velocity). The Volkswagen has a small mass so its change in velocity will be large; this means its acceleration will be large. The Mac truck has a very large mass so its change in velocity will be small; this means its acceleration will be small.

Ex 5.38 A deuteron is a nuclear particle of unique mass made up of one proton and one neutron. Suppose it is accelerated up to a certain very high speed in a cyclotron and directed into an observation chamber, where it collides and sticks to a target particle that is initially at rest and then is observed to move at exactly half the speed of the incident deutron. Why do observers state that the target particle is itself a deuteron?

momentum before = momentum after

momentum before = (mass before) (velocity before) = (mass after) (velocity after) = momentum after

(mass before) (velocity before) = (mass after) (velocity after)

(1/2) (velocity before) = velocity after

mass before = (1/2) (mass after)

2 (mass before) = mass after

That is, the mass after the collision - after the incoming deuteron sticks to the target - is exactly twice the mass of the deuteron. Only another deuteron has the mass of exactly the mass of a deuteron.

Ex 5.40 A billiard ball will stop short when it collides head-on with a ball at rest. The ball cannot stop short, however, if the collision is not exactly head-on -- that is, if the second ball moves at an angle to the path of the first. Why? (Hint: Consider momentum before and after the collision along the initial direction of the first ball and also in a direction perpendicular to this initial direction.)

Momentum is a vector. Momentum is conserved along any direction and the direction perpendicular to that direction. Consider the direction perpendicular to the initial velocity of the incomming billiard ball. Initially, there is no momentum along that direction. But, after the collision, if the second billiard ball moves at some angle, it does have momentum along that direction. If the incoming billiard ball simply stopped then momentum would not be conserved. We would, in that case, go from having no momentum in this perpendicular direction to having momentum in that direction. This can not happen. So that means such a collision can not happen.

* * * Extra Exercises * * *

Ex 5.(**extra**) Why is it difficult for a fire fighter to hold a hose that ejects large amounts of water at a high speed?

Again, think of Newton's Third Law. As water moves from the hose at high speed, it carries great momentum with it and has great forces acting upon it. That means it will respond by putting great forces on the hose.

Ex 5. (**extra**) Would you care to fire a gun that has a bullet ten times as massive as the gun?

The recoil of a gun is an example of the conservation of momentum. Initially, the gun and bullet are at rest and their total momentum is zero. Ordinarily the bullet will have a small mass and a large velocity while the gun, with a much greater mass, will have a small velocity. In this question, however, the bullet is more massive than the gun. That means the velocity of the gun will be greater than the velocity of the bullet. In fact, the velocity of this gun will be ten times the velocity of the bullet.

Ex 5. (**extra**) The momentum of the rifle and bullet shown in Figure 4.8 is conserved. Why do we not say the velocities are conserved? What would happen if the mass of the bullet were equal to the mass of the rifle?

From playing with Newton's Third Law, the meaning of acceleration, and the meaning of momentum, we found that the total momentum is conserved -- the momentum before an interaction, like the explosion in firing a bullet from a gun, is the same as the momentum after the interaction. It is the momentum that is conserved, not the velocity. In this particular case, if the masses are equal, then having equal momentum carried to the left and to the right does mean that the speed to the left will be equal to the speed to the right. But this is only true for this special case of equal masses.

Typical or possible multiple-guess questions over this material:

1. Which of the following has the largest momentum relative to Earth?

A) a tightrope walker crossing Niagara Falls.

B) a pickup truck speeding along a highway.

C) a Mack truck sitting in the parking lot.

D) the Science building on campus.

2. A moving object on which no forces are acting will continue to move with constant

A) acceleration

B) impulse

C) momentum

D) all of these

3. Impulse is equal to

A) kinetic energy

B) momentum

C) change in momentum

D) change in force

4. Conservation of momentum is directly related to

A) Newton's First Law of Motion

B) Newton's Second Law of Motion

C) Newton's Third Law of Motion

D) International shortages of momentum

5. A rifle recoils from firing a bullet. The speed of the rifle's recoil is small because the

A) force against the rifle is smaller than against the bullet.

B) momentum is mainly concentrated in the bullet.

C) rifle has more mass than the bullet.

D) momentum of the rifle is smaller.

6. Two objects, A and B, have the same size and shape, but A is twice as heavy as B. When they are dropped simultaneously from a tower, they reach the ground at the same time, but A has a greater

A) speed

B) acceleration

C) momentum

D) all of the above

7. A car traveling along the highway needs a certain amount of force exerted on it to stop. More stopping force may be required when the car has

A) more mass

B) more momentum

C) less stopping distance

D) all of the above

8. A 4 kg ball has a momentum of 12 kg m/s. What is the ball's speed?

A) 3 m/s

B) 4 m/s

C) 12 m/s

D) 48 m/s

9. A ball is moving at 4 m/s and has a momentum of 48 kg m/s. What is the ball's mass?

A) 4 kg

B) 8 kg

C) 12 kg

D) 192 kg

Answers to the typical or possible multiple-guess questions:

1. Which of the following has the largest momentum relative to Earth?

A) a tightrope walker crossing Niagara Falls.

B) a pickup truck speeding along a highway.

momentum = mass x velocity

The pickup has both more mass and a greater speed than the tightrope walker.

While the Mack truck and the Science building each have greater mass than the pickup, their velocity is zero.

C) a Mack truck sitting in the parking lot.

D) the Science building on campus.

2. A moving object on which no forces are acting will continue to move with constant

A) acceleration

B) impulse

C) momentum

momentum = mass x velocity

Newton's First Law of Motion, the Law of Inertia, is usually stated something like "a moving object on which no forces act will continue to move with constant velocity" and constant velocity requires constant momentum.

D) all of these

3. Impulse is equal to

A) kinetic energy

B) momentum

C) change in momentum

impulse = F t = (m a) t = (m) (a t) = (m) (v) = (m v) = momentum

D) change in force

4. Conservation of momentum is directly related to

A) Newton's First Law of Motion

B) Newton's Second Law of Motion

C) Newton's Third Law of Motion

F12 = - F21

F12 t = - F21 t

p1 = - p2

p1 + p2 = 0

( p1 + p2 ) = 0

( p1 + p2 ) = constant

D) International shortages of momentum

5. A rifle recoils from firing a bullet. The speed of the rifle's recoil is small because the

A) force against the rifle is smaller than against the bullet.

B) momentum is mainly concentrated in the bullet.

C) rifle has more mass than the bullet.

p1 = - p2

p1 = - p2

m1 v1 = - m2 v2

D) momentum of the rifle is smaller.

6. Two objects, A and B, have the same size and shape, but A is twice as heavy as B. When they are dropped simultaneously from a tower, they reach the ground at the same time, but A has a greater

A) speed

B) acceleration

C) momentum

If they reach the ground at the same time, we know they have the same speed (or velocity) so the one with the greater mass or weight must also have more momentum.

D) all of the above

7. A car traveling along the highway needs a certain amount of force exerted on it to stop. More stopping force may be required when the car has

A) more mass

B) more momentum

C) less stopping distance

D) all of the above

8. A 4 kg ball has a momentum of 12 kg m/s. What is the ball's speed?

A) 3 m/s
momentum = mass x velocity

12 kg-m/s = (4 kg) x (3 m/s)

B) 4 m/s

C) 12 m/s

D) 48 m/s

9. A ball is moving at 4 m/s and has a momentum of 48 kg m/s. What is the ball's mass?

A) 4 kg

B) 8 kg

C) 12 kg

momentum = mass x velocity

48 kg-m/s = (12 kg) x (4 m/s)

D) 192 kg

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