Content check — College Physics (AP), Chapter 8: Linear Momentum and Collisions

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 8: Linear Momentum and Collisions

58 problems · 2 figures

(a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of

7 . 50 m/s

. (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of

600 m/s

. (c) What is the momentum of the 90.0-kg hunter running at

7 . 40 m/s

after missing the elephant?

Answer

(a) $1.50 \times 10^{4} kg \cdot m/s$

(b) 625 to 1

8-3 §8.1 source ↗

(a) At what speed would a

2 . 00 \times 10^{4} -kg

airplane have to fly to have a momentum of

1 . 60 \times 10^{9} kg \cdot m/s

(the same as the ship’s momentum in the problem above)? (b) What is the plane’s momentum when it is taking off at a speed of

60.0 m/s

? (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship.

Answer

(a) $8 . 00 \times 10^{4} m/s$

(b) $1 . 20 \times 10^{6} kg \cdot m/s$

(c) Because the momentum of the airplane is 3 orders of magnitude smaller than of the ship, the ship will not recoil very much. The recoil would be $- 0 . 0100 m/s$ , which is probably not noticeable.

8-5 §8.1 source ↗

A runaway train car that has a mass of 15,000 kg travels at a speed of

5 .4 m/s

down a track. Compute the time required for a force of 1500 N to bring the car to rest.

Answer

54 s

8-7 §8.2 source ↗

A bullet is accelerated down the barrel of a gun by hot gases produced in the combustion of gun powder. What is the average force exerted on a 0.0300-kg bullet to accelerate it to a speed of 600.0 m/s in a time of 2.00 ms (milliseconds)?

Answer

9 . 00 \times 10^{3} N

8-9 §8.2 source ↗

A person slaps her leg with her hand, bringing her hand to rest in 2.50 milliseconds from an initial speed of 4.00 m/s. (a) What is the average force exerted on the leg, taking the effective mass of the hand and forearm to be 1.50 kg? (b) Would the force be any different if the woman clapped her hands together at the same speed and brought them to rest in the same time? Explain why or why not.

Answer

a) $2 . 40 \times 10^{3} N$ toward the leg

b) The force on each hand would have the same magnitude as that found in part (a) (but in opposite directions by Newton’s third law) because the change in momentum and the time interval are the same.

8-11 §8.2 source ↗

Professional Application

Suppose a child drives a bumper car head on into the side rail, which exerts a force of 4000 N on the car for 0.200 s. (a) What impulse is imparted by this force? (b) Find the final velocity of the bumper car if its initial velocity was 2.80 m/s and the car plus driver have a mass of 200 kg. You may neglect friction between the car and floor.

Answer

a) $800 kg \cdot m/s$ away from the wall

b) $1 . 20 m/s$ away from the wall

8-13 §8.2 source ↗

Professional Application

A 75.0-kg person is riding in a car moving at 20.0 m/s when the car runs into a bridge abutment. (a) Calculate the average force on the person if he is stopped by a padded dashboard that compresses an average of 1.00 cm. (b) Calculate the average force on the person if he is stopped by an air bag that compresses an average of 15.0 cm.

Answer

(a) $1 . 50 \times 10^{6} N$ away from the dashboard

(b) $1 . 00 \times 10^{5} N$ away from the dashboard

8-15 §8.2 source ↗

A cruise ship with a mass of

1 . 00 \times 10^{7} kg

strikes a pier at a speed of 0.750 m/s. It comes to rest 6.00 m later, damaging the ship, the pier, and the tugboat captain’s finances. Calculate the average force exerted on the pier using the concept of impulse. (Hint: First calculate the time it took to bring the ship to rest.)

Answer

4 . 69 \times 10^{5} N

in the boat’s original direction of motion

8-17 §8.2 source ↗

Water from a fire hose is directed horizontally against a wall at a rate of 50.0 kg/s and a speed of 42.0 m/s. Calculate the magnitude of the force exerted on the wall, assuming the water’s horizontal momentum is reduced to zero.

Answer

2 . 10 \times 10^{3} N

away from the wall

8-19 §8.2 source ↗

Starting with the definitions of momentum and kinetic energy, derive an equation for the kinetic energy of a particle expressed as a function of its momentum.

Answer

\begin{array}{l} p = m v \Rightarrow p^{2} = m^{2} v^{2} \Rightarrow \frac{p^{2}}{m} = {mv}^{2} \\ \Rightarrow \frac{p^{2}}{2 m} = \frac{1}{2} {mv}^{2} = KE \\ KE = \frac{p^{2}}{2 m} \end{array}

8-21 §8.2 source ↗

When serving a tennis ball, a player hits the ball when its velocity is zero (at the highest point of a vertical toss). The racquet exerts a force of 540 N on the ball for 5.00 ms, giving it a final velocity of 45.0 m/s. Using these data, find the mass of the ball.

Answer

60.0 g

8-23 §8.3 source ↗

Professional Application

Train cars are coupled together by being bumped into one another. Suppose two loaded train cars are moving toward one another, the first having a mass of 150,000 kg and a velocity of 0.300 m/s, and the second having a mass of 110,000 kg and a velocity of $- 0 . 120 m/s$ . (The minus indicates direction of motion.) What is their final velocity?

Answer

0.122 m/s

8-25 §8.3 source ↗

Professional Application

Consider the following question: A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seatbelt exerts on a passenger in the car to bring him to a halt. The mass of the passenger is 70 kg. Would the answer to this question be different if the car with the 70-kg passenger had collided with a car that has a mass equal to and is traveling in the opposite direction and at the same speed? Explain your answer.

Answer

In a collision with an identical car, momentum is conserved. Afterwards

v_{f} = 0

for both cars. The change in momentum will be the same as in the crash with the tree. However, the force on the body is not determined since the time is not known. A padded stop will reduce injurious force on body.

8-27 §8.3 source ↗

A 1.80-kg falcon catches a 0.650-kg dove from behind in midair. What is their velocity after impact if the falcon’s velocity is initially 28.0 m/s and the dove’s velocity is 7.00 m/s in the same direction?

Answer

22.4 m/s in the same direction as the original motion

8-29 §8.4 source ↗

Professional Application

Two piloted satellites approach one another at a relative speed of 0.250 m/s, intending to dock. The first has a mass of $4 . 00 \times 10^{3} kg$ , and the second a mass of $7 . 50 \times 10^{3} kg$ . If the two satellites collide elastically rather than dock, what is their final relative velocity?

Answer

0.250 m/s

8-31 §8.5 source ↗

A 0.240-kg billiard ball that is moving at 3.00 m/s strikes the bumper of a pool table and bounces straight back at 2.40 m/s (80% of its original speed). The collision lasts 0.0150 s. (a) Calculate the average force exerted on the ball by the bumper. (b) How much kinetic energy in joules is lost during the collision? (c) What percent of the original energy is left?

Answer

(a) 86.4 N perpendicularly away from the bumper

(b) 0.389 J

8-33 §8.5 source ↗

Professional Application

Using mass and speed data from Example 8.1 and assuming that the football player catches the ball with his feet off the ground with both of them moving horizontally, calculate: (a) the final velocity if the ball and player are going in the same direction and (b) the loss of kinetic energy in this case. (c) Repeat parts (a) and (b) for the situation in which the ball and the player are going in opposite directions. Might the loss of kinetic energy be related to how much it hurts to catch the pass?

Example 8.1 View it in the book ↗

Answer

(a) 8.06 m/s

(b) -56.0 J

(c)(i) 7.88 m/s; (ii) -223 J

8-35 §8.5 source ↗

Professional Application

Two piloted satellites approaching one another, at a relative speed of 0.250 m/s, intending to dock. The first has a mass of $4 . 00 \times 10^{3} kg$ , and the second a mass of $7.50 \times 10^{3} kg$ . (a) Calculate the final velocity (after docking) by using the frame of reference in which the first satellite was originally at rest. (b) What is the loss of kinetic energy in this inelastic collision? (c) Repeat both parts by using the frame of reference in which the second satellite was originally at rest. Explain why the change in velocity is different in the two frames, whereas the change in kinetic energy is the same in both.

Answer

(a) 0.163 m/s in the direction of motion of the more massive satellite

(b) 81.6 J

(c) $8 . 70 \times 10^{- 2} m/s$ in the direction of motion of the less massive satellite, 81.5 J. Because there are no external forces, the velocity of the center of mass of the two-satellite system is unchanged by the collision. The two velocities calculated above are the velocity of the center of mass in each of the two different individual reference frames. The loss in KE is the same in both reference frames because the KE lost to internal forces (heat, friction, etc.) is the same regardless of the coordinate system chosen.

8-37 §8.5 source ↗

Professional Application

Space probes may be separated from their launchers by exploding bolts. (They bolt away from one another.) Suppose a 4800-kg satellite uses this method to separate from the 1500-kg remains of its launcher, and that 5000 J of kinetic energy is supplied to the two parts. What are their subsequent velocities using the frame of reference in which they were at rest before separation?

Answer

0.704 m/s

–2.25 m/s

8-38 §8.5 source ↗

A 0.0250-kg bullet is accelerated from rest to a speed of 550 m/s in a 3.00-kg rifle. The pain of the rifle’s kick is much worse if you hold the gun loosely a few centimeters from your shoulder rather than holding it tightly against your shoulder. (a) Calculate the recoil velocity of the rifle if it is held loosely away from the shoulder. (b) How much kinetic energy does the rifle gain? (c) What is the recoil velocity if the rifle is held tightly against the shoulder, making the effective mass 28.0 kg? (d) How much kinetic energy is transferred to the rifle-shoulder combination? The pain is related to the amount of kinetic energy, which is significantly less in this latter situation. (e) Calculate the momentum of a 110-kg football player running at 8.00 m/s. Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s. Discuss its relationship to this problem.

Answer

(a) 4.58 m/s away from the bullet

(b) 31.5 J

(d) 3.38 J

8-40 §8.5 source ↗

Professional Application

The Moon’s craters are remnants of meteorite collisions. Suppose a fairly large asteroid that has a mass of $5 . 00 \times 10^{12} kg$ (about a kilometer across) strikes the Moon at a speed of 15.0 km/s. (a) At what speed does the Moon recoil after the perfectly inelastic collision (the mass of the Moon is $7 . 36 \times 10^{22} kg$ ) ? (b) How much kinetic energy is lost in the collision? Such an event may have been observed by medieval English monks who reported observing a red glow and subsequent haze about the Moon. (c) In October 2009, NASA crashed a rocket into the Moon, and analyzed the plume produced by the impact. (Significant amounts of water were detected.) Answer part (a) and (b) for this real-life experiment. The mass of the rocket was 2000 kg and its speed upon impact was 9000 km/h. How does the plume produced alter these results?

Answer

(a) $1 . 02 \times 10^{- 6} m/s$

(b) $5 . 63 \times 10^{20} J$ (almost all KE lost)

(c) Recoil speed is $6 . 79 \times 10^{- 17} m/s$ , energy lost is $6 . 25 \times 10^{9} J$ . The plume will not affect the momentum result because the plume is still part of the Moon system. The plume may affect the kinetic energy result because a significant part of the initial kinetic energy may be transferred to the kinetic energy of the plume particles.

8-42 §8.5 source ↗

What is the speed of a garbage truck that is

1 . 20 \times 10^{4} kg

and is initially moving at 25.0 m/s just after it hits and adheres to a trash can that is 80.0 kg and is initially at rest?

Answer

24.8 m/s

8-44 §8.5 source ↗

(a) During an ice skating performance, an initially motionless 80.0-kg clown throws a fake barbell away. The clown’s ice skates allow her to recoil frictionlessly. If the clown recoils with a velocity of 0.500 m/s and the barbell is thrown with a velocity of 10.0 m/s, what is the mass of the barbell? (b) How much kinetic energy is gained by this maneuver? (c) Where does the kinetic energy come from?

Answer

(a) 4.00 kg

(b) 210 J

(c) The clown does work to throw the barbell, so the kinetic energy comes from the muscles of the clown. The muscles convert the chemical potential energy of ATP into kinetic energy.

8-45 §8.6 source ↗

Two identical pucks collide on an air hockey table. One puck was originally at rest. (a) If the incoming puck has a speed of 6.00 m/s and scatters to an angle of

30 . 0º

,what is the velocity (magnitude and direction) of the second puck? (You may use the result that

θ_{1} - θ_{2} = 90º

for elastic collisions of objects that have identical masses.) (b) Confirm that the collision is elastic.

Answer

(a) 3.00 m/s, $60º$ below $x$ -axis

(b) Find speed of first puck after collision: $0 =$ $m v_{1}^{'} sin 30º -$ $m v_{2}^{'} sin 60º \Rightarrow v_{1}^{'} =$ $v_{2}^{'} \frac{sin 60º}{sin 30º} =$ $5.196 m/s$

Verify that ratio of initial to final KE equals one: $\begin{array}{l} KE = \frac{1}{2} {mv}_{1}^{2} = 18 m J \\ KE = \frac{1}{2} {mv'}_{1}^{2} + \frac{1}{2} {mv'}_{2}^{2} = 18 m J \end{array} \frac{KE}{KE′} = 1.00$

8-47 §8.6 source ↗

A 3000-kg cannon is mounted so that it can recoil only in the horizontal direction. (a) Calculate its recoil velocity when it fires a 15.0-kg shell at 480 m/s at an angle of

20 . 0º

above the horizontal. (b) What is the kinetic energy of the cannon? This energy is dissipated as heat transfer in shock absorbers that stop its recoil. (c) What happens to the vertical component of momentum that is imparted to the cannon when it is fired?

Answer

(a) $- 2 . 26 m/s$

(b) $7 . 63 \times 10^{3} J$

(c) The ground will exert a normal force to oppose recoil of the cannon in the vertical direction. The momentum in the vertical direction is transferred to the earth. The energy is transferred into the ground, making a dent where the cannon is. After long barrages, cannon have erratic aim because the ground is full of divots.

8-49 §8.6 source ↗

Professional Application

Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei $(^{4} He)$ from gold-197 nuclei $(^{197} Au)$ . The energy of the incoming helium nucleus was $8.00 \times 10^{- 13} J$ , and the masses of the helium and gold nuclei were $6.68 \times 10^{- 27} kg$ and $3.29 \times 10^{- 25} kg$ , respectively (note that their mass ratio is 4 to 197). (a) If a helium nucleus scatters to an angle of $120º$ during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the gold nucleus. (b) What is the final kinetic energy of the helium nucleus?

Answer

(a) $5 . 36 \times 10^{5} m/s$ at $- 29.5º$

(b) $7 . 52 \times 10^{- 13} J$

8-51 §8.6 source ↗

Starting with equations $m_{1} v_{1} =$ $m_{1} v_{1}^{'} cos θ_{1} +$ $m_{2} v_{2}^{'} cos θ_{2}$ and $0 =$ $m_{1} v_{1}^{'} sin θ_{1} +$ $m_{2} v_{2}^{'} sin θ_{2}$ for conservation of momentum in the $x$ - and $y$ -directions and assuming that one object is originally stationary, prove that for an elastic collision of two objects of equal masses,

\begin{matrix} \frac{1}{2} {mv}_{1}^{2} = \\ \frac{1}{2} {mv'}_{1}^{2} + \frac{1}{2} {mv'}_{2}^{2} \\ + {mv'}_{1} {v'}_{2} cos (θ_{1} - θ_{2}) \end{matrix}

as discussed in the text.

Answer

We are given that $m_{1} = m_{2} \equiv m$ . The given equations then become:

v_{1} = v_{1} cos θ_{1} + v_{2} cos θ_{2}

and

0 = v_{1}^{'} sin θ_{1} + v_{2}^{'} sin θ_{2} .

Square each equation to get

\begin{array}{l} {v_{1}}^{2} & = & {v'}_{1}^{2} {cos}^{2} θ_{1} + {v'}_{2}^{2} {cos}^{2} θ_{2} + 2 {v'}_{1} {v'}_{2} cos θ_{1} cos θ_{2} \\ 0 & = & {v'}_{1}^{2} {sin}^{2} θ_{1} + {v'}_{2}^{2} {sin}^{2} θ_{2} + 2 {v'}_{1} {v'}_{2} sin θ_{1} sin θ_{2} . \end{array}

Add these two equations and simplify:

\begin{array}{l} v_{1}^{2} & = & {v'}_{1}^{2} + {v'}_{2}^{2} + 2 {v'}_{1} {v'}_{2} cos θ_{1} cos θ_{2} + sin θ_{1} sin θ_{2} \\ = & {v'}_{1}^{2} + {v'}_{2}^{2} + 2 {v'}_{1} {v'}_{2} \frac{1}{2} cos θ_{1} - θ_{2} + \frac{1}{2} cos θ_{1} + θ_{2} + \frac{1}{2} cos θ_{1} - θ_{2} - \frac{1}{2} cos θ_{1} + θ_{2} \\ = & {v'}_{1}^{2} + {v'}_{2}^{2} + 2 {v'}_{1} {v'}_{2} cos θ_{1} - θ_{2} . \end{array}

Multiply the entire equation by $\frac{1}{2} m$ to recover the kinetic energy:

\frac{1}{2} {mv}_{1}^{2} = \frac{1}{2} m {v'}_{1}^{2} + \frac{1}{2} m {v'}_{2}^{2} + m {v'}_{1} {v'}_{2} cos θ_{1} - θ_{2}

8-53 §8.7 source ↗

Professional Application

Antiballistic missiles (ABMs) are designed to have very large accelerations so that they may intercept fast-moving incoming missiles in the short time available. What is the takeoff acceleration of a 10,000-kg ABM that expels 196 kg of gas per second at an exhaust velocity of $2 . 50 \times 10^{3} m/s?$

Answer

39.2 {m/s}^{2}

8-55 §8.7 source ↗

Professional Application

Calculate the increase in velocity of a 4000-kg space probe that expels 3500 kg of its mass at an exhaust velocity of $2 . 00 \times 10^{3} m/s$ . You may assume the gravitational force is negligible at the probe’s location.

Answer

4 . 16 \times 10^{3} m/s

8-57 §8.7 source ↗

Derive the equation for the vertical acceleration of a rocket.

Answer

The force needed to give a small mass

Δ m

an acceleration

a_{Δ m}

F = Δ {ma}_{Δ m}

. To accelerate this mass in the small time interval

Δ t

requires

v_{e} = a_{Δ m} Δ t

, so

F = v_{e} \frac{Δ m}{Δ t}

. By Newton’s third law, this force is equal in magnitude to the thrust force acting on the rocket, so

F_{thrust} =

v_{e} \frac{Δ m}{Δ t}

, where all quantities are positive. Applying Newton’s second law to the rocket gives

F_{thrust} -

mg =

ma \Rightarrow a =

\frac{v_{e}}{m} \frac{Δ m}{Δ t} -

g

, where

m

is the mass of the rocket and unburnt fuel.

8-60 §8.7 source ↗

How much of a single-stage rocket that is 100,000 kg can be anything but fuel if the rocket is to have a final speed of

8.00 km/s

, given that it expels gases at an exhaust velocity of

2 . 20 \times 10^{3} m/s?

Answer

2 . 63 \times 10^{3} kg

8-61 §8.7 source ↗

Professional Application

(a) A 5.00-kg squid initially at rest ejects 0.250-kg of fluid with a velocity of 10.0 m/s. What is the recoil velocity of the squid if the ejection is done in 0.100 s and there is a 5.00-N frictional force opposing the squid’s movement. (b) How much energy is lost to work done against friction?

Answer

(a) 0.421 m/s away from the ejected fluid.

(b) $0 .237 J$ .

AP Test Prep 8.1 §8.1 source ↗

A boy standing on a frictionless ice rink is initially at rest. He throws a snowball in the +x-direction, and it travels on a ballistic trajectory, hitting the ground some distance away. Which of the following is true about the boy while he is in the act of throwing the snowball?

a.
He feels an upward force to compensate for the downward trajectory of the snowball.
b.
He feels a backward force exerted by the snowball he is throwing.
c.
He feels no net force.
d.
He feels a forward force, the same force that propels the snowball.

Answer

B. He feels a backward force exerted by the snowball he is throwing.

AP Test Prep 8.3 §8.2 source ↗

A 1.0-kg ball of putty is released from rest and falls vertically 1.5 m until it strikes a hard floor, where it comes to rest in a 0.045-s time interval. What is the magnitude and direction of the average force exerted on the ball by the floor during the collision?

a.
33 N, up
b.
120 N, up
c.
120 N, down
d.
240 N, down

Answer

B. 120 N, up

AP Test Prep 8.5 §8.2 source ↗

A 2.4-kg ceramic bowl falls to the floor. During the 0.018-s impact, the bowl experiences an average force of 750 N from the floor. The bowl is at rest after the impact. From what initial height did the bowl fall?

a.
1.6 m
b.
2.8 m
c.
3.2 m
d.
5.6 m

Answer

A. 1.6 m

AP Test Prep 8.7 §8.2 source ↗

A 2.5-kg block slides across a frictionless table toward a horizontal spring.As the block bounces off the spring, a probe measures the velocity of the block (initially negative, moving away from the probe) over time as follows:

What is the average force exerted on the block by the spring over the entire 0.60-s time interval of the collision?

(a) 50 N

(b) 60 N

(d) 120 N

Table 8.1 in the book ↗

Velocity (m/s)	Time (s)
−12.0	0
−10.0	0.10
−6.0	0.20
0	0.30
6.0	0.40
10.0	0.50
12.0	0.60

Answer

F = \frac{m Δ v}{Δ t}

)

AP Test Prep 8.9 §8.2 source ↗

A test car is driving toward a solid crash-test barrier with a speed of 45 mi/h. Two seconds prior to impact, the car begins to brake, but it is still moving when it hits the wall. After the collision with the wall, the car crumples somewhat and comes to a complete stop. In order to estimate the average force exerted by the wall on the car, what information would you need to collect?

a.
The (negative) acceleration of the car before it hits the wall and the distance the car travels while braking.
b.
The (negative) acceleration of the car before it hits the wall and the velocity of the car just before impact.
c.
The velocity of the car just before impact and the duration of the collision with the wall.
d.
The duration of the collision with the wall and the distance the car travels while braking.

Answer

C. The velocity of the car just before impact and the duration of the collision with the wall.

AP Test Prep 8.11 §8.2 source ↗

A 22-g puck hits the wall of an air hockey table perpendicular to the wall with an initial speed of 14 m/s.The puck is in contact with the wall for 0.0055 s, and it rebounds from the wall with a speed of 14 m/s in the opposite direction.What is the magnitude of the average force exerted by the wall on the puck?

a.
0.308 N
b.
0.616 N
c.
56 N
d.
112 N

Answer

D. 112 N

AP Test Prep 8.13 §8.2 source ↗

The graph in Figure 8.15 represents the force exerted on a particle during a collision. What is the magnitude of the change in momentum of the particle as a result of the collision?

a.
1.2 kg • m/s
b.
2.4 kg • m/s
c.
3.6 kg • m/s
d.
4.8 kg • m/s

The force exerted by the wall on the object is a constant 15 N from t =0.080 s until t=0.24 s, at which time the force drops to zero. — Figure 8.15 in the book ↗

Answer

B. 2.4 kg • m/s

AP Test Prep 8.15 §8.3 source ↗

Which of the following is an example of an open system?

a.
Two air cars colliding on a track elastically.
b.
Two air cars colliding on a track and sticking together.
c.
A bullet being fired into a hanging wooden block and becoming embedded in the block, with the system then acting as a ballistic pendulum.
d.
A bullet being fired into a hillside and becoming buried in the earth.

Answer

D. A bullet being fired into a hillside and becoming buried in the earth.

AP Test Prep 8.17 §8.3 source ↗

A student sets up an experiment to measure the momentum of a system of two air cars, A and B, of equal mass, moving on a linear, frictionless track. Before the collision, car A has a certain speed, and car B is at rest. Which of the following will be true about the total momentum of the two cars?

a.
It will be greater before the collision.
b.
It will be equal before and after the collision.
c.
It will be greater after the collision.
d.
The answer depends on whether the collision is elastic or inelastic.

Answer

B. It will be equal before and after the collision.

AP Test Prep 8.19 §8.3 source ↗

Cart A is moving with an initial velocity +v (in the positive direction) toward cart B, initially at rest. Both carts have equal mass and are on a frictionless surface. Which of the following statements correctly characterizes the velocity of the center of mass of the system before and after the collision?

a.
$\frac{+ v}{2}$ before, $\frac{- v}{2}$ after
b.
$\frac{+ v}{2}$ before, 0 after
c.
$\frac{+ v}{2}$ before, $\frac{+ v}{2}$ after
d.
0 before, 0 after

Answer

\frac{+ v}{2}

before,

\frac{+ v}{2}

after

AP Test Prep 8.21 §8.4 source ↗

Two cars (A and B) of mass 1.5 kg collide. Car A is initially moving at 12 m/s, and car B is initially moving in the same direction with a speed of 6 m/s. The two cars are moving along a straight line before and after the collision. What will be the change in momentum of this system after the collision?

a.
−27 kg • m/s
b.
zero
c.
+27 kg • m/s
d.
It depends on whether the collision is elastic or inelastic.

Answer

B. zero

AP Test Prep 8.23 §8.4 source ↗

Puck A (200 g) slides across a frictionless surface to collide with puck B (800 g), initially at rest. The velocity of each puck is measured during the experiment as follows:

What is the change in momentum of the center of mass of the system as a result of the collision?

a.
+1.6 kg•m/s
b.
+0.8 kg•m/s
c.
0
d.
−1.6 kg•m/s

Table 8.2 in the book ↗

Time	Velocity A	Velocity B
0	+8.0 m/s	0
1.0 s	+8.0 m/s	0
2.0 s	−2.0 m/s	+2.5 m/s
3.0 s	−2.0 m/s	+2.5 m/s

Answer

C. 0

AP Test Prep 8.25 §8.4 source ↗

Two cars (A and B) of equal mass have an elastic collision. Prior to the collision, car A is moving at 15 m/s in the +x-direction, and car B is moving at 10 m/s in the –x-direction. Assuming that both cars continue moving along the x-axis after the collision, what will be the velocity of car A after the collision?

a.
same as the original 15 m/s speed, opposite direction
b.
equal to car B’s velocity prior to the collision
c.
equal to the average of the two velocities, in its original direction
d.
equal to the average of the two velocities, in the opposite direction

Answer

B. equal to car B’s velocity prior to the collision

AP Test Prep 8.27 §8.4 source ↗

A rubber ball is dropped from rest at a fixed height. It bounces off a hard floor and rebounds upward, but it only reaches 90% of its original fixed height. What is the best way to explain the loss of kinetic energy of the ball during the collision?

a.
Energy was required to deform the ball’s shape during the collision with the floor.
b.
Energy was lost due to work done by the ball pushing on the floor during the collision.
c.
Energy was lost due to friction between the ball and the floor.
d.
Energy was lost due to the work done by gravity during the motion.

Answer

A. Energy was required to deform the ball’s shape during the collision with the floor.

AP Test Prep 8.29 §8.4 source ↗

Two objects, A and B, have equal mass. Prior to the collision, mass A is moving 10 m/s in the +x-direction, and mass B is moving 4 m/s in the +x-direction. Which of the following results represents an inelastic collision between A and B?

a.
After the collision, mass A is at rest, and mass B moves 14 m/s in the +x-direction.
b.
After the collision, mass A moves 4 m/s in the –x-direction, and mass B moves 18 m/s in the +x-direction.
c.
After the collision, the two masses stick together and move 7 m/s in the +x-direction.
d.
After the collision, mass A moves 4 m/s in the +x-direction, and mass B moves 10 m/s in the +x-direction.

Answer

C. After the collision, the two masses stick together and move 7 m/s in the +x-direction.

AP Test Prep 8.31 §8.4 source ↗

Two objects (A and B) of equal mass collide elastically. Mass A is initially moving 5.0 m/s in the +x-direction prior to the collision. Mass B is initially moving 3.0 m/s in the –x-direction prior to the collision. After the collision, mass A will be moving with a velocity of 3.0 m/s in the –x-direction. What will be the velocity of mass B after the collision?

a.
3.0 m/s in the +x-direction
b.
5.0 m/s in the +x-direction
c.
3.0 m/s in the –x-direction
d.
5.0 m/s in the –x-direction

Answer

B. 5.0 m/s in the +x-direction

AP Test Prep 8.33 §8.4 source ↗

Two objects of equal mass collide. Object A is initially moving in the +x-direction with a speed of 12 m/s, and object B is initially at rest. After the collision, object A is at rest, and object B is moving away with some unknown velocity. There are no external forces acting on the system of two masses. What statement can we make about this collision?

a.
Both momentum and kinetic energy are conserved.
b.
Momentum is conserved, but kinetic energy is not conserved.
c.
Neither momentum nor kinetic energy is conserved.
d.
More information is needed in order to determine which is conserved.

Answer

A. Both momentum and kinetic energy are conserved.

AP Test Prep 8.35 §8.4 source ↗

Which of the following statements is true about an inelastic collision?

a.
Momentum is conserved, and kinetic energy is conserved.
b.
Momentum is conserved, and kinetic energy is not conserved.
c.
Momentum is not conserved, and kinetic energy is conserved.
d.
Momentum is not conserved, and kinetic energy is not conserved.

Answer

B. Momentum is conserved, and kinetic energy is not conserved.

AP Test Prep 8.37 §8.4 source ↗

The figure shows a graph of the position in meters as a function of time in seconds for two objects, A and B. The time axis runs from 0 to 1.2 seconds. The position axis runs from 0 to 2.5 meters. Object A starts at position 0 at time 0. Its position increases linearly to reach position 1.5 meters at time 0.6 seconds. Its position then decreases linearly to reach position 0 meters at time 1.2 seconds. Object B starts at position 2.0 meters at time 0. Its position decreases linearly to reach position 1.5 meters at time 0.6 seconds. Its position then increases linearly to reach position 2.0 meters at time 1.2 seconds.

This figure shows the positions of two colliding objects measured before, during, and after a collision. Mass A is 1.0 kg. Mass B is 3.0 kg. Which of the following statements is true?

a.
This is an elastic collision, with a total momentum of 0 kg • m/s.
b.
This is an elastic collision, with a total momentum of 1.67 kg • m/s.
c.
This is an inelastic collision, with a total momentum of 0 kg • m/s.
d.
This is an inelastic collision, with a total momentum of 1.67 kg • m/s.

Answer

A. This is an elastic collision, with a total momentum of 0 kg • m/s.

AP Test Prep 8.39 §8.4 source ↗

Mass A (1.0 kg) slides across a frictionless surface with a velocity of 8 m/s in the positive direction. Mass B (3.0 kg) is initially at rest. The two objects collide and stick together. What will be the change in the center-of-mass velocity of the system as a result of the collision?

a.
There will be no change in the center-of-mass velocity.
b.
The center-of-mass velocity will decrease by 2 m/s.
c.
The center-of-mass velocity will decrease by 6 m/s.
d.
The center-of-mass velocity will decrease by 8 m/s.

Answer

A. There will be no change in the center-of-mass velocity.

AP Test Prep 8.41 §8.5 source ↗

Mass A (2.0 kg) has an initial velocity of 4 m/s in the +x-direction. Mass B (2.0 kg) has an initial velocity of 5 m/s in the –x-direction. If the two masses have an elastic collision, what will be the final velocities of the masses after the collision?

a.
Both will move 0.5 m/s in the –x-direction.
b.
Mass A will stop; mass B will move 9 m/s in the +x-direction.
c.
Mass B will stop; mass A will move 9 m/s in the –x-direction.
d.
Mass A will move 5 m/s in the –x-direction; mass B will move 4 m/s in the +x-direction.

Answer

D. Mass A will move 5 m/s in the –x-direction; mass B will move 4 m/s in the +x-direction.

AP Test Prep 8.43 §8.5 source ↗

Mass A (2.0 kg) is moving with an initial velocity of 15 m/s in the +x-direction, and it collides with mass B (5.0 kg), initially at rest. After the collision, the two objects stick together and move as one. What is the change in kinetic energy of the system as a result of the collision?

(a) no change

(b) decrease by 225 J

(d) decrease by 64 J

Answer

(c). Because of conservation of momentum, the final velocity of the combined mass must be 4.286 m/s. The initial kinetic energy is

(0.5) (2.0) {(15)}^{2} =

225 J

. The final kinetic energy is

(0.5) (7.0) {(4.286)}^{2} =

64 J

, so the difference is −161 J.

AP Test Prep 8.45 §8.5 source ↗

Mass A slides across a rough table with an initial velocity of 12 m/s in the +x-direction. By the time mass A collides with mass B (a stationary object with equal mass), mass A has slowed to 10 m/s. After the collision, the two objects stick together and move as one. Immediately after the collision, the velocity of the system is measured to be 5 m/s in the +x-direction, and the system eventually slides to a stop. Which of the following statements is true about this motion?

a.
Momentum is conserved during the collision, but it is not conserved during the motion before and after the collision.
b.
Momentum is not conserved at any time during this analysis.
c.
Momentum is conserved at all times during this analysis.
d.
Momentum is not conserved during the collision, but it is conserved during the motion before and after the collision.

Answer

A. Momentum is conserved during the collision, but it is not conserved during the motion before and after the collision.

AP Test Prep 8.47 §8.5 source ↗

Mass A is initially moving with a velocity of 24 m/s in the +x-direction. Mass B is twice as massive as mass A and is initially at rest. The two objects experience a totally inelastic collision. What is the final speed of both objects after the collision?

a.
A is not moving; B is moving 24 m/s in the +x-direction.
b.
Neither A nor B is moving.
c.
A is moving 24 m/s in the –x-direction. B is not moving.
d.
Both A and B are moving together 8 m/s in the +x-direction.

Answer

D. Both A and B are moving together 8 m/s in the +x-direction.

AP Test Prep 8.49 §8.5 source ↗

Mass A is initially moving with a velocity of 2 m/s in the +x-direction. Mass B is initially moving with a velocity of 6 m/s in the –x-direction. The two objects have equal masses. After they collide, mass A moves with a speed of 4 m/s in the –x-direction. What is the final velocity of mass B after the collision?

a.
6 m/s in the +x-direction
b.
4 m/s in the +x-direction
c.
zero
d.
4 m/s in the –x-direction

Answer

C. zero

AP Test Prep 8.51 §8.6 source ↗

Two cars of equal mass approach an intersection. Car A is moving east at a speed of 45 m/s. Car B is moving south at a speed of 35 m/s. They collide inelastically and stick together after the collision, moving as one object. Which of the following statements is true about the center-of-mass velocity of this system?

a.
The center-of-mass velocity will decrease after the collision as a result of lost energy (but not drop to zero).
b.
The center-of-mass velocity will remain the same after the collision since momentum is conserved.
c.
The center-of-mass velocity will drop to zero since the two objects stick together.
d.
The magnitude of the center-of-mass velocity will remain the same, but the direction of the velocity will change.

Answer

B. The center-of-mass velocity will remain the same after the collision since momentum is conserved.

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Problems and answers are adapted from College Physics for AP® Courses 2e by Gregg Wolfe, Erika Gasper, John Stoke, Julie Kretchman, David Anderson and others, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/college-physics-ap-courses-2e/pages/1-connection-for-ap-r-courses. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.