College Physics

Roger A. Freedman; Todd Ruskell; Philip R. Kesten

Chapter 7

Momentum, Collisions, and the Center of Mass - all with Video Answers

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Chapter Questions

Problem 1

If the mass of a basketball is 18 times that of a tennis ball, can they ever have the same momentum? Explain your answer. SSM

Paul Gabriel

Paul Gabriel

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Problem 2

Starting from Newton's second law explain how a collision that is free from external forces conserves momentum. In other words, explain how the momentum of the system remains constant with time.

Ma Ednelyn Lim

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Problem 3

The classic collision problem involves two hard spheres colliding. When you define the system as both spheres, momentum is conserved, but when you define the system as only one of the spheres, momentum is not conserved. Explain why momentum is conserved in the first case but not the second.

Ma Ednelyn Lim

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Problem 4

Two objects have equal kinetic energies. Are the magnitudes of their momenta equal? Explain your answer.

Ma Ednelyn Lim

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Problem 5

How would you determine if a collision is elastic or inelastic?

Paul Gabriel

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Problem 6

Cite two examples of totally inelastic collisions that occur in your daily life.

Paul Gabriel

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Problem 7

Why is conservation of energy alone not sufficient to explain the motion of a Newton's cradle toy, shown in Figure 7-23? Consider the case when two balls are raised and released together.

Ma Ednelyn Lim

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Problem 8

An arrow shot into a straw target penetrates a distance that depends on the speed with which it strikes the target. How does the penetration distance change if the arrow's speed is doubled? Be sure to list all the assumptions you make while arriving at your answer.

Paul Gabriel

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Problem 9

Using the common definition of the word impulse, comment on physicists' choice to define impulse as the change in momentum.

Ma Ednelyn Lim

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Problem 10

A recent U.S. patent application describes a "damage avoidance system" for cell phones, which, upon detecting an impending, uncontrolled impact with a surface, deploys an air-filled bag around the phone. Explain how this could protect the cell phone from damage.

Paul Gabriel

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Problem 11

A glass will break if it falls onto a hardwood floor but not if it falls from the same height onto a padded, carpeted floor. Describe the different outcomes in the collision between a glass and the floor in terms of fundamental physical quantities.

Paul Gabriel

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Problem 12

A child stands on one end of a long wooden plank that rests on a frictionless icy surface. (a) Describe the motion of the plank when she runs to the other end of the plank. (b) Describe the motion of the center of mass of the system. (c) How would your answers change if she had walked the plank rather than run down it?

Hubert Agamasu

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Problem 13

Based on what you know about center of mass, why is it potentially dangerous to step off of a small boat before it is secured to the dock?

Paul Gabriel

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Problem 14

A man and his large dog sit at opposite ends of a rowboat, floating on a still pond. You notice from shore that the boat moves as the dog walks toward his owner. Describe the motion of the boat from your perspective.

Paul Gabriel

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Problem 15

After being thrown into the air, a lit firecracker explodes at the apex of its parabolic flight. (a) Is momentum conserved before or after the explosion? (b) Is the mechanical energy conserved? (c) What is the path of the center of mass? Neglect the effects of air resistance and explain your answers.

Hubert Agamasu

Numerade Educator

Problem 16

A large semitrailer truck and a small car have equal momentum. How do their speeds compare?
A. The truck has a much higher speed than the car.
B. The truck has only a slightly higher speed than the car.
C. Both have the same speed.
D. The truck has only a slightly lower speed than the car.
E. The truck has a much lower speed than the car.

Paul Gabriel

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Problem 17

A tennis player smashes a ball of mass $m$ horizontally at a vertical wall. The ball rebounds at the same speed $v$ with which it struck the wall. Has the momentum of the ball changed, and if so, what is the magnitude of the change?
A. $m v$
B. 0
C. $m v / 2^{m v / 2}$
D. $2 m v$
E. $4 m v$

Paul Gabriel

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Problem 18

You throw a bouncy rubber ball and a wet lump of clay, both of mass $m$, at a wall. Both strike the wall at speed $v$, but while the ball bounces off with no loss of speed, the clay sticks. What is the change in momentum of the clay and ball, respectively, assuming that toward the wall is the positive direction?
A. $0 ; m v$
B. $m v ; 0$
C. $0 ;-2 m v$
D. $-m v ;-m v$
E. $-m v ;-2 m v$

Paul Gabriel

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Problem 19

Consider a completely inelastic, head-on collision between two particles that have equal masses and equal speeds. Describe the velocities of the particles after the collision.
A. The velocities of both particles are zero.
B. Both of their velocities are reversed.
C. One of the particles continues with the same velocity and the other comes to rest.
D. One of the particles continues with the same velocity, and the other reverses direction at twice the speed.
E. More information is required to determine the final velocities.

Paul Gabriel

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Problem 20

An object is traveling in the positive $x$ direction with speed $v$. A second object that has half the mass of the first is traveling in the opposite direction with the same speed. The two experience a completely inelastic collision. The final $x$ component of the velocity is
A. 0
B. $\mathrm{v} / 2^{v / 2}$
C. $\mathrm{v} / 3^{v / 3}$
D. $2 \mathrm{v} / 3^{2 v / 3}$
E. $v$

Paul Gabriel

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Problem 21

21. - Consider a completely elastic head-on collision between two particles that have the same mass and the same speed. What are the velocities after the collision?
A. Both are zero.
B. The magnitudes of the velocities are the same, but the directions are reversed.
C. One of the particles continues with the same velocity, and the other comes to rest.
D. One of the particles continues with the same velocity, and the other reverses direction at twice the speed.
E. More information is required to determine the final velocities.

Paul Gabriel

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Problem 22

Two small, identical steel balls collide completely elastically. Initially ball 1 is moving with velocity $v_{1}$, and ball 2 is stationary. After the collision, the final velocities of ball 1 and ball 2 are
A. $12 \mathrm{v} 1 ; 12 \mathrm{v} 1 \frac{1}{2} v_{1} ; \frac{1}{2} v_{1}$
B. $v_{1} ; 2 v_{1}$
C. $0 ; v_{1}$
D. $-v_{1}$; 0
E. $-v_{1} ; 2 v_{1}$

Paul Gabriel

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Problem 23

Two ice skaters, Lilly and John, face each other while stationary and push against each other's hands. John's mass is twice that of Lilly. How do their speeds compare after the push-off?
A. Lilly's speed is one-fourth of John's speed.
B. Lilly's speed is one-half of John's speed.
C. Lilly's speed is the same as John's speed.
D. Lilly's speed is twice John's speed.
E. Lilly's speed is four times John's speed.

Paul Gabriel

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Problem 24

A friend throws a heavy ball toward you while you are standing on smooth ice. You can either catch the ball or deflect it back toward your friend. Which of the following options will maximize your speed right after your interaction with the ball?
A. You should catch the ball.
B. You should deflect the ball back toward your friend at the same speed with which it hit your hand.
C. You should let the ball go past you without touching it.
D. It doesn't matter-your speed is the same regardless of what you do.
E. You should deflect the ball back toward your friend at half the speed with which it hit your hand.

Paul Gabriel

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Problem 25

Two blocks are released from rest on either side of a frictionless half-pipe (Figure 7-24). Block B is more massive than block A. The height $H_{\mathrm{B}}$ from which block B is released is less than $H_{\mathrm{A}}$, the height from which block A is released. The blocks collide elastically on the flat section. After the collision, which is correct?
A. Block A rises to a height greater than $H_{\mathrm{A}}$, and block B rises to a height less than $H_{\mathrm{B}}$.
B. Block A rises to a height less than $H_{\mathrm{A}}$, and block B rises to a height greater than $H_{\mathrm{B}}$.
C. Block A rises to height $H_{\mathrm{A}}$, and block B rises to height $H_{\mathrm{B}}$.
D. Block A rises to height $H_{\mathrm{B}}$, and block B rises to height $H_{\mathrm{A}}$.
E. The heights to which the blocks rise depends on where along the flat section they collide.

Manish Jain

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Problem 26

A 50-m wide asteroid collides with Mars. Estimate the ratio of the magnitude of the momentum transfer from the asteroid impact to the momentum of Mars's orbital motion.

Hubert Agamasu

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Problem 27

Estimate the momentum of a car driving the speed limit on a freeway.

Paul Gabriel

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Problem 28

Sports Estimate the momentum of a fastball thrown by a major league pitcher.

Hubert Agamasu

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Problem 29

Sports Estimate the momentum of a tennis ball served by a professional tennis player.

Paul Gabriel

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Problem 30

Medical, Biology Estimate the location of the center of mass of your body.

Hubert Agamasu

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Problem 31

• Estimate the momentum of a bumblebee when it strikes a motorcycle

Paul Gabriel

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Problem 32

Estimate the momentum of Earth as it orbits the Sun.

Hubert Agamasu

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Problem 33

Compare the momentum of a fast-pitch softball to a major league fastball.

Paul Gabriel

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Problem 34

Estimate the impulse delivered to a tennis ball that rebounds from a practice wall.

Hubert Agamasu

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Problem 35

A car moving at $20 \mathrm{~m} / \mathrm{s}^{20} \mathrm{~m} / \mathrm{s}$ slams into the back end of a car stopped at a red light. After the collision, the two cars stick together. If the cars have the same mass, estimate the distance the two cars travel before coming to rest. Assume that neither driver applies his brakes during the collision.

Ma Ednelyn Lim

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Problem 36

The following gives the force (in newtons) acting on a 2-kg object as a function of time. (a) Make a graph of force versus time. (b) If the object starts from rest, what is its speed after 25 s?
$$
\begin{array}{cccccc}
\boldsymbol{t}(\mathrm{s}) & \boldsymbol{F}(\mathbf{N}) & \boldsymbol{t}(\mathbf{s}) & \boldsymbol{F}(\mathbf{N}) & \boldsymbol{t}(\mathbf{s}) & \boldsymbol{F}(\mathbf{N}) \\
\hline 0 & -20 & 9 & 25 & 18 & 25 \\
1 & -20 & 10 & 25 & 19 & 25 \\
2 & -10 & 11 & 25 & 20 & 25 \\
3 & 0 & 12 & 25 & 22 & 15 \\
4 & 10 & 13 & 25 & 22 & 15 \\
5 & 15 & 14 & 25 & 23 & 10 \\
6 & 18 & 15 & 25 & 24 & 5 \\
7 & 20 & 16 & 25 & 25 & 0 \\
8 & 25 & 17 & 25 & & \\
\hline
\end{array}
$$

Manish Jain

Numerade Educator

Problem 37

A $1.00 \times 104-\mathrm{kg} 1.00 \times 10^{4}$ -kg train car moves east at $15 \mathrm{~m} / \mathrm{s}^{15 \mathrm{~m} / \mathrm{s}}$. Determine the momentum of the train car.

Paul Gabriel

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Problem 38

Sports The magnitude of the instantaneous momentum of a 57 -g tennis ball is $2.6 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}^{2} \cdot 6 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$. What is its speed? SSM Example $7-1$

Paul Gabriel

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Problem 39

Determine the initial momentum, final momentum, and change in momentum of a $1250-\mathrm{kg}$ car initially backing up at $5.00 \mathrm{~m} / \mathrm{s}, 5.00 \mathrm{~m} / \mathrm{s}$, then moving forward at $14.0 \mathrm{~m} / \mathrm{s}^{14.0 \mathrm{~m} / \mathrm{s}} . \underline{\text { Example } 7-1}$

Paul Gabriel

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Problem 40

Sports What is the magnitude of momentum of a 135 -kg defensive lineman running at $7.00 \mathrm{~m} / \mathrm{s}^{7} .00 \mathrm{~m} / \mathrm{s}$ ? Example $7-1$

Paul Gabriel

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Problem 41

One ball has four times the mass and twice the speed of another. (a) How does the magnitude of momentum of the more massive ball compare to the magnitude of momentum of the less massive one? (b) How does the kinetic energy of the more massive ball compare to the kinetic energy of the less massive one? Example 7-1

Paul Gabriel

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Problem 42

A girl who has a mass of $55.0$ kg rides the skateboard to class at a speed of $6.00 \mathrm{~m} / \mathrm{s}^{6.00 \mathrm{~m} / \mathrm{s}}$. (a) What is the magnitude of her momentum? (b) If the magnitude of momentum of the skateboard itself is $30.0 \mathrm{~kg} \mathrm{~m} / \mathrm{s}, 30.0 \mathrm{~kg} \mathrm{~m} / \mathrm{s}$, what is its mass? SSM Example 7-1

Paul Gabriel

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Problem 43

Blythe and Geoff are ice skating together. Blythe has a mass of $50.0 \mathrm{~kg}$, and Geoff has a mass of $80.0 \mathrm{~kg}$. Blythe pushes Geoff in the chest when both are at rest, causing him to move away at a speed of $4.00 \mathrm{~m} / \mathrm{s}^{4} .00 \mathrm{~m} / \mathrm{s}$. (a) Determine Blythe's speed after she pushes Geoff. (b) In what direction does she move? Example 7-1

Justin Swantek

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Problem 44

A $2.00-\mathrm{kg}$ object is moving east at $4.00 \mathrm{~m} / \mathrm{s}^{4} .00 \mathrm{~m} / \mathrm{s}$ when it collides with a $6.00$ -kg object that is initially at rest. After the collision the larger object moves east at $1.00 \mathrm{~m} / \mathrm{s}^{1.00 \mathrm{~m} / \mathrm{s}}$. What is the final velocity of the smaller object after the collision? Assume no external forces act on the objects. Example $7-2$

Paul Gabriel

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Problem 45

A $3.00-\mathrm{kg}$ object is moving toward the right at $6.00 \mathrm{~m} / \mathrm{s}^{6.00 \mathrm{~m} / \mathrm{s}}$. A 5.00-kg object moves to the left at $4.00 \mathrm{~m} / \mathrm{s}^{4.00 \mathrm{~m} / \mathrm{s}}$. After the two objects collide the $3.00$ -kg object moves toward the left at $2.00 \mathrm{~m} / \mathrm{s}^{2.00 \mathrm{~m} / \mathrm{s}}$. What is the final velocity of the $5.00$ -kg object? Assume no external forces act on the objects. Example 7-2

Paul Gabriel

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Problem 46

Two hockey players collide on the ice and go down together in a tangled heap. Player 1 has a mass of $105 \mathrm{~kg}$, and player 2 has a mass of $92 \mathrm{~kg}$. Before the collision player 1 had a velocity of $\mathrm{v} 1=6.3 \mathrm{~m} / \mathrm{s}^{v_{1}}=6.3 \mathrm{~m} / \mathrm{s}$ in the $+x$ direction, and player 2 had a velocity of $\mathrm{v} 2=5.6 \mathrm{~m} / \mathrm{s}^{v_{2}}=5.6 \mathrm{~m} / \mathrm{s}$ at an angle of $72^{\circ}$ with respect to the $+x$ axis as shown in Figure $7-25$. At what speed and in what direction do they slide together on the ice after the collision? Example 7-7

Nicholas Mogoi

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Problem 47

An object of mass $3 M$, moving in the $+x$ direction at speed $v_{0}$, breaks into two pieces of mass $M$ and $2 M$ as shown in Figure 7-26. If $\theta 1=45^{\circ}$ $\theta_{1}=45^{\circ}$ and $\theta 2=30^{\circ}, \theta_{2}=30^{\circ}$, determine the final velocities of the resulting pieces in terms of $v_{0}$. SSM Example 7-4

Hubert Agamasu

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Problem 48

In a game of pool the cue ball is rolling at $2.00 \mathrm{~m} / \mathrm{s}^{2.00 \mathrm{~m} / \mathrm{s}}$ in a direction $30.0^{\circ}$ north of east when it collides with the eight ball (initially at rest). The mass of the cue ball is $170 \mathrm{~g}$, but the mass of the eight ball is only 156 g. After the collision the cue ball heads off at $10.0^{\circ}$ north of east, and the eight ball moves off due north. What are the final speeds of each ball after the collision? Example 7-3

Hubert Agamasu

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Problem 49

During mating season, male bighorn sheep establish dominance with head-butting contests which can be heard up to a mile away. When two males butt heads, the "winner" is the one that knocks the other backward. In one contest a sheep of mass $95.0 \mathrm{~kg}$ moving at $10.0 \mathrm{~m} / \mathrm{s}^{10.0 \mathrm{~m} / \mathrm{s}}$ runs directly into a sheep of mass $80.0 \mathrm{~kg}$ moving at $12.0 \mathrm{~m} / \mathrm{s}^{12.0 \mathrm{~m} / \mathrm{s}}$. Which ram wins the head-butting contest? SSM Example $7-2$

Paul Gabriel

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Problem 50

A superhero and a supervillain are having a battle. The hero is flying parallel to the ground at a speed of $51.8 \mathrm{~m} / \mathrm{s}^{51.8 \mathrm{~m} / \mathrm{s}}$ when the villain on the ground picks up and hurls a car at him. The 1860 -kg car is moving parallel to the ground at $19.1 \mathrm{~m} / \mathrm{s}^{19.1 \mathrm{~m} / \mathrm{s}}$ when it strikes the $102-\mathrm{kg}$ hero at an angle of $-46^{\circ}$ with respect to the hero's initial direction of motion, as shown in Figure 7-27. The hero and the car are stuck together after the collision. What is their speed, and in what direction relative to the hero's initial velocity do they travel after the collision? Example 7-7

Hubert Agamasu

Numerade Educator

Problem 51

A $1.00 \times 104-\mathrm{kg} 1.00 \times 10^{4}-\mathrm{kg}$ train car moving due east at $20.0 \mathrm{~m} / \mathrm{s}$
$20.0 \mathrm{~m} / \mathrm{s}$ collides with and couples to a $2.00 \times 104-\mathrm{kg}^{2} .00 \times 10^{4}-\mathrm{kg}$ train car
that is initially at rest. (a) What is the common velocity of the two-car train after the collision? (b) What is the total kinetic energy of the two train cars before and after the collision? Example 7-6

Ma Ednelyn Lim

Numerade Educator

Problem 52

A large fish has a mass of $25.0 \mathrm{~kg}$ and swims at $1.00 \mathrm{~m} / \mathrm{s}$ toward and then swallows a smaller fish that is not moving. If the smaller fish has a mass of $1.00 \mathrm{~kg}$, what is the speed of the larger fish immediately after it finishes lunch? Example 7-6

Paul Gabriel

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Problem 53

A $5.00$ -kg howler monkey is swinging due east on a vine. It overtakes and grabs onto a 6.00-kg monkey also moving east on a second vine. The first monkey is moving at $12.0 \mathrm{~m} / \mathrm{s}$ at the instant it grabs the second, which is moving at $8.00 \mathrm{~m} / \mathrm{s}$. (a) After they join on the same vine, what is their common speed? (b) What is the total kinetic energy of the two monkeys before and after they join the same vine? SSM Example 7-6

Ma Ednelyn Lim

Numerade Educator

Problem 54

A $1200-\mathrm{kg}$ car is moving at $20.0 \mathrm{~m} / \mathrm{s}$ due north. A $1500-\mathrm{kg}$ car is moving at $18.0 \mathrm{~m} / \mathrm{s}$ due east. The two cars simultaneously approach an icy intersection where, with no brakes or steering, they collide and stick together. Determine the speed and direction of the combined two-car wreck immediately after the collision. Example $7-7$

Paul Gabriel

Numerade Educator

Problem 55

Sports An 85.0-kg linebacker is running at $8.00 \mathrm{~m} / \mathrm{s}$ directly toward the sideline of a football field. He tackles a 75.0-kg running back moving at $9.00$ $\mathrm{m} / \mathrm{s}$ straight toward the goal line (perpendicular to the original direction of the linebacker). Determine their common speed and direction immediately after they collide. Example 7-7

Paul Gabriel

Numerade Educator

Problem 56

One way that scientists measure the mass of an unknown particle is to bounce a known particle, such as a proton or an electron, off the unknown particle in a bubble chamber. The initial and rebound velocities of the known particle are measured from photographs of the bubbles it creates as it moves; the information is used to determine the mass of the unknown particle. (a) If a known particle of mass $m$ and initial speed $v_{0}$ collides elastically, head-on with a stationary unknown particle and then rebounds with speed $v$, find an expression for the mass of the unknown particle in terms of $m, v$, and $v_{0}$. (b) If the known particle is a proton and the unknown particle is a neutron, what will be the recoil speed of the proton and the final speed of the neutron?

Hubert Agamasu

Numerade Educator

Problem 57

A $2.00-\mathrm{kg}$ ball moves at $3.00 \mathrm{~m} / \mathrm{s}$ toward the right. It collides elastically with a $4.00-\mathrm{kg}$ ball that is initially at rest. Determine the velocities of the balls after the collision. Example $7-8$

Paul Gabriel

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Problem 58

A $10.0$ -kg block of ice is sliding due east at $8.00 \mathrm{~m} / \mathrm{s}$ when it collides elastically with a $6.00$ -kg block of ice that is sliding in the same direction at $4.00 \mathrm{~m} / \mathrm{s}$. Determine the velocities of the blocks of ice after the collision. SSM Example 7-8

Ma Ednelyn Lim

Numerade Educator

Problem 59

A 0.170-kg ball moves at $4.00 \mathrm{~m} / \mathrm{s}$ toward the right. It collides elastically with a 0.155-kg ball moving at $2.00 \mathrm{~m} / \mathrm{s}$ toward the left. Determine the final velocities of the balls after the collision. Example $7-8$

Ma Ednelyn Lim

Numerade Educator

Problem 60

A neutron traveling at $2.00 \times 105 \mathrm{~m} / \mathrm{s}^{2} .00 \times 10^{5} \mathrm{~m} / \mathrm{s}$ collides elastically with a deuteron that is initially at rest. Determine the final speeds of the two particles after the collision. The mass of a neutron is $1.67 \times 10-27 \mathrm{~kg}$, $1.67 \times 10^{-27} \mathrm{~kg}$, and the mass of a deuteron is $3.34 \times 10-27 \mathrm{~kg}^{3.34 \times 10^{-27} \mathrm{~kg} \text { . }}$
Example 7-8

Ma Ednelyn Lim

Numerade Educator

Problem 61

A sudden gust of wind exerts a force of $20.0 \mathrm{~N}$ for $1.20 \mathrm{~s}$ on a bird that had been flying at $5.00 \mathrm{~m} / \mathrm{s}$. As a result the bird ends up moving in the opposite direction at $7.00 \mathrm{~m} / \mathrm{s}$. What is the mass of the bird? Example $7-9$

Paul Gabriel

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Problem 62

Determine the average force exerted on your hand as you catch a $0.200$ kg ball moving at $20.0 \mathrm{~m} / \mathrm{s}$. Assume the time of contact is $0.0250 \mathrm{~s}$. SSM Example 7-9

Paul Gabriel

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Problem 63

An expert boxer delivers a $3.20 \times 103 \mathrm{~N} 3.20 \times 10^{3} \mathrm{~N}$ punch to the head of his opponent. If the punch contacts the 4.82-kg head for $0.35 \mathrm{~s}$, and the unfortunate boxer's head has a mass of $4.82 \mathrm{~kg}$, what is the speed of the head when the punch loses contact? Assume no other forces act on the opponent's head. (In reality, the force of the neck on the head plays an important role.) Example 7-9

Paul Gabriel

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Problem 64

Sports A baseball of mass $0.145 \mathrm{~kg}$ is thrown at a speed of $40.0 \mathrm{~m} / \mathrm{s}$. The batter strikes the ball with a force of $25,000 \mathrm{~N} ;$ the bat and ball are in contact for $0.500 \mathrm{~ms}$. Assuming that the force is exactly opposite to the original direction of the ball, determine the final speed of the ball. Example $\frac{7-9}{2}$

Paul Gabriel

Numerade Educator

Problem 65

Bean bag rounds used by police are nonlethal projectiles fired from shotguns. If a $40.0$ -g round strikes at $70.0 \mathrm{~m} / \mathrm{s}$ and delivers all its momentum over a time of $0.15 \mathrm{~s}$, what is the average force of impact? Example $7-9$

Paul Gabriel

Numerade Educator

Problem 66

A baseball bat strikes a ball when both are moving at $31.3 \mathrm{~m} / \mathrm{s}$ (relative to the ground) toward each other. The bat and ball are in contact for $1.20 \mathrm{~ms}$, after which the ball is traveling opposite its initial direction at a speed of $42.5 \mathrm{~m} / \mathrm{s}$. The mass of the bat and the ball are 850 and $145 \mathrm{~g}$, respectively. Calculate the magnitude and direction of the impulse given to
(a) the ball by the bat and (b) the bat by the ball. (c) What average force does the bat exert on the ball? (d) Why doesn't the force shatter the bat? Example $7-9$

Hubert Agamasu

Numerade Educator

Problem 67

Find the coordinates of the center of mass of the three objects shown in Figure 7-28 if $\mathrm{m} 1=4.00 \mathrm{~kg}, m_{1}=4.00 \mathrm{~kg}, \mathrm{~m} 2=2.00 \mathrm{~kg}, m_{2}=2.00 \mathrm{~kg}$, and
$\mathrm{m} 3=3.00 \mathrm{~kg} m_{3}=3.00 \mathrm{~kg}$. Distances are in meters. Example 7-10

Paul Gabriel

Numerade Educator

Problem 68

What are the coordinates of the center of mass for the combination of the three objects shown in Figure 7-29? The uniform rod has a mass of $10.0$ $\mathrm{kg}$, has a length of $30.0 \mathrm{~cm}$, and is located at $\mathrm{x}=50.0 \mathrm{~cm} . x=50.0 \mathrm{~cm}$. The oval football has a mass of $2.00 \mathrm{~kg}$, a semimajor axis of $15.0 \mathrm{~cm}$, has a semiminor axis of $8.00 \mathrm{~cm}$, and is centered at $\mathrm{x}=-50.0 \mathrm{~cm} . x=-50.0 \mathrm{~cm}$. The spherical volleyball has a mass of $1.00 \mathrm{~kg}$, has a radius of $10.0 \mathrm{~cm}$, and is centered at $\mathrm{y}=-30 \mathrm{~cm} . y=-30 \mathrm{~cm}$. Assume both balls are of uniform mass density. SSM Example 7-10

Manish Jain

Numerade Educator

Problem 69

Four beads each of mass M are attached at various locations to a hoop of mass $M$ and radius $R$ (Figure $7-30$ ). Find the center of mass of the hoop and beads. Example $7-10$

Manish Jain

Numerade Educator

Problem 70

A $3.0$ -kg block (A) is attached to a $1.0$ -kg block (B) by a massless spring that is compressed and locked in place, as shown in Figure 7-31. The blocks slide without friction along the $x$ direction at an initial constant speed of $2.0 \mathrm{~m} / \mathrm{s}$. At time $t=0$, the positions of blocks $\mathrm{A}$ and $\mathrm{B}$ are $\mathrm{x}=1.0 \mathrm{~m}$ $x=1.0 \mathrm{~m}$ and $\mathrm{x}=1.2 \mathrm{~m}, x=1.2 \mathrm{~m}$, respectively, at which point a mechanism releases the spring, and the blocks begin to oscillate as they slide. If $2.0 \mathrm{~s}$ later block $\mathrm{B}$ is located at $\mathrm{x}=6.0 \mathrm{~m}, x=6.0 \mathrm{~m}$, where will block $\mathrm{A}$ be located? $\underline{\text { Example } 7-10}$

Manish Jain

Numerade Educator

Problem 71

A major league baseball has a mass of $0.145 \mathrm{~kg}$. Neglecting the effects of air resistance, determine the momentum of the ball when it hits the ground if it falls from rest on the roof of the Willis Tower in Chicago, Illinois, a height of $442.0 \mathrm{~m} .$ Example $7-1$

Paul Gabriel

Numerade Educator

Problem 72

Forensic scientists can determine the speed at which a rifle fires a bullet by shooting into a heavy block hanging by a wire. As the bullet embeds itself in the block, the block and embedded bullet swing up; the impact speed is determined from the maximum angle of the swing. (a) Which would make the block swing higher, a $0.204$ Ruger bullet of mass $2.14 \mathrm{~g}$ and muzzle speed $1290 \mathrm{~m} / \mathrm{s}$ or a 7-mm Remington Magnum bullet of mass $9.71 \mathrm{~g}$ and muzzle speed $948 \mathrm{~m} / \mathrm{s}$ ? Assume the bullets enter the block right after leaving the muzzle of the rifle. (b)Using your answer in part (a), determine the mass of the block so that when hit by the bullet it will swing through a $60.0^{\circ}$ angle. The block hangs from wire of length $1.25 \mathrm{~m}$ and negligible mass. (c) What is the speed of an 8.41-g bullet that causes the block to swing upward through a $30.0^{\circ}$ angle? Example 7-6

Manish Jain

Numerade Educator

Problem 73

A friend suggests that if all the people in the United States dropped down from a 1-m-high table at the same time, Earth would move in a noticeable way. To test the credibility of this proposal, (a) determine the momentum imparted to Earth by 300 million people, of an average mass of $65.0 \mathrm{~kg}$, dropping from $1.00 \mathrm{~m}$ above the surface. Assume no one bounces. (b) What change in Earth's speed would result? SSM Example 7-6

Paul Gabriel

Numerade Educator

Problem 74

Sally finds herself stranded on a frozen pond so slippery that she can't stand up or walk on it. To save herself, she throws one of her heavy boots horizontally, directly away from the closest shore. Sally's mass is $60 \mathrm{~kg}$, the boot's mass is $5 \mathrm{~kg}$, and Sally throws the boot with speed equal to $30.0 \mathrm{~m} / \mathrm{s}$.
(a) What is Sally's speed immediately after throwing the boot?
(b) Where is the center of mass of the Sally-boot system, relative to where she threw the boot, after $10.0$ s? (c) How long does it take Sally to reach the shore, a distance of $30.0 \mathrm{~m}$ away from where she threw the boot? For all parts, assume the ice is frictionless. Example $7-1$

Hubert Agamasu

Numerade Educator

Problem 75

You have been called to testify as an expert witness in a trial involving a head-on collision. Car A weighs $680 \mathrm{~kg}$ and was traveling eastward. Car $\mathrm{B}$ weighs $500 \mathrm{~kg}$ and was traveling westward at $72.0 \mathrm{~km} / \mathrm{h}$. The cars locked bumpers and slid eastward with their wheels locked for $6.00 \mathrm{~m}$ before stopping. You have measured the coefficient of kinetic friction between the tires and the pavement to be $0.750$. How fast (in miles per hour) was car A traveling just before the collision? Example $7-6$

Manish Jain

Numerade Educator

Problem 76

A 5000 -kg open train car is rolling at a speed of $20.0 \mathrm{~m} / \mathrm{s}$ when it begins to rain heavily, and $200 \mathrm{~kg}$ of water collects quickly in the car. If only the total mass has changed, what is the speed of the flooded train car? For simplicity, assume that all of the water collects at one instant and that the train tracks are frictionless. Example $7-2$

Paul Gabriel

Numerade Educator

Problem 77

An $8000-\mathrm{kg}$ open train car is rolling at a speed of $20.0 \mathrm{~m} / \mathrm{s}$ when it begins to rain heavily. After water has collected in the car, it slows to $19.0$ $\mathrm{m} / \mathrm{s}$. What mass of water has collected in the car? For simplicity, assume that all of the water collects at one instant and that the train tracks are frictionless.

Paul Gabriel

Numerade Educator

Problem 78

An open rail car of initial mass $10,000 \mathrm{~kg}$ is moving at $5.00 \mathrm{~m} / \mathrm{s}$ when rocks begin to fall into it from a conveyor belt. The rate at which the mass of rocks increases is $500 \mathrm{~kg} / \mathrm{s}$. Find the speed of the train car after rocks have fallen into the car for a total of $3.00 \mathrm{~s}$. Example $7-2$

Paul Gabriel

Numerade Educator

Problem 79

A 65 -kg novice skier stops to rest partway down a slope. An inattentive snowboarder with the same 65 -kg mass is barreling down the same hill at $9.6$ $\mathrm{m} / \mathrm{s}$ and crashes right into the back of the skier. Miraculously, the collision is perfectly elastic and nobody falls over. What are the velocities of the skier and snowboarder after the collision? Example 7-8

Paul Gabriel

Numerade Educator

Problem 80

The sport of curling is quite popular in Canada. A curler slides a 19.1-kg stone so that it strikes a competitor's stationary stone at $6.40 \mathrm{~m} / \mathrm{s}$ before moving at an angle of $120^{\circ}$ from its initial direction. The competitor's stone moves off at $5.60 \mathrm{~m} / \mathrm{s}$. Determine the final speed of the first stone and the final direction of the second one. Example $7-3$

Anand Jangid

Numerade Educator

Problem 81

Lions can run at speeds up to approximately $80.0 \mathrm{~km} / \mathrm{h}$. A hungry $135-\mathrm{kg}$ lion running northward at top speed attacks and holds onto a 29.0-kg Thomson's gazelle running eastward at $60.0 \mathrm{~km} / \mathrm{h}$. Find the speed and direction of travel of the lion-gazelle system just after the lion attacks.

Hubert Agamasu

Numerade Educator

Problem 82

The mass of a pigeon hawk is twice that of the pigeons it hunts. Suppose a pigeon is gliding north at a speed of $\mathrm{vP}=23.0 \mathrm{~m} / \mathrm{s}$ $v_{\mathrm{P}}=23.0 \mathrm{~m} / \mathrm{s}_{\text {when a hawk swoops down, grabs the pigeon, and flies off }}$ (Figure 7-32). The hawk was flying north at speed of $\mathrm{vH}=35.0 \mathrm{~m} / \mathrm{s}$, $v_{\mathrm{H}}=35.0 \mathrm{~m} / \mathrm{s}$, at an angle $\theta=45^{\circ} \theta=45^{\circ}$ below the horizontal, at the instant of the attack. Find the final velocity vector of the birds just after the attack. Example 7-7

Manish Jain

Numerade Educator

Problem 83

A $12.0$ -g bullet is fired into a block of wood with speed $\mathrm{v}=250 \mathrm{~m} / \mathrm{s}$ $v=250 \mathrm{~m} / \mathrm{s}$ (Figure 7-33). The block is attached to a spring that has a spring constant of $200 \mathrm{~N} / \mathrm{m}$. The block with the embedded bullet compresses the spring a distance $\mathrm{d}=30.0 \mathrm{~cm} d=30.0 \mathrm{~cm}$ to the right, before momentarily coming to a stop. Determine the mass of the wooden block. SSM Example 7$\underline{6}$

Manish Jain

Numerade Educator

Problem 84

In a ballistic pendulum experiment, a small marble is fired into a cup attached to the end of a pendulum. If the mass of the marble is $0.00750 \mathrm{~kg}$ and the mass of the pendulum is $0.250 \mathrm{~kg}$, how high will the pendulum swing if the marble has an initial speed of $6.00 \mathrm{~m} / \mathrm{s}$ ? Assume that the mass of the pendulum is concentrated at its end. Example 7-6

Paul Gabriel

Numerade Educator

Problem 85

A $0.0750$ - $\mathrm{kg}$ ball is thrown at $25.0 \mathrm{~m} / \mathrm{s}$ toward a brick wall. (a) Determine the impulse that the wall imparts to the ball when it hits and rebounds at $25.0 \mathrm{~m} / \mathrm{s}$ in the opposite direction. (b) Determine the impulse that the wall imparts to the ball when it hits and rebounds at $25.0 \mathrm{~m} / \mathrm{s}$ at an angle of $45.0^{\circ} .$ (c)If the ball thrown in part (b) contacts the wall for $0.0100 \mathrm{~s}$, determine the magnitude and direction of the average force that the wall exerts on the ball. Example $7-9$

Manish Jain

Numerade Educator