Physics

James S. Walker

Chapter 9

Linear Momentum and Collisions - all with Video Answers

Educators

Chapter Questions

Problem 1

Referring to Exercise $9-1,$ what speed must the baseball have if its momentum is to be equal in magnitude to that of the car? Give your result in miles per hour.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 2

Find the total momentum of the birds in Example $9-1$ if the goose reverses direction.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 3

A 26.2-kg dog is running northward at 2.70 m/s, while a $5.30-\mathrm{kg}$ cat is running eastward at $3.04 \mathrm{m} / \mathrm{s} .$ Their $74.0-\mathrm{kg}$ owner has the same momentum as the two pets taken together. Find the direction and magnitude of the owner's velocity.

Nick Auwerda

Numerade Educator

Problem 4

Two air-track carts move toward one another on an air track. Cart 1 has a mass of $0.35 \mathrm{kg}$ and a speed of $1.2 \mathrm{m} / \mathrm{s}$. Cart 2 has a mass of $0.61 \mathrm{kg}$. (a) What speed must cart 2 have if the total momentum of the system is to be zero? (b) Since the momentum of the system is zero, does it follow that the kinetic energy of the system is also zero? (c) Verify your answer to part (b) by calculating the system's kinetic energy.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 5

A 0.150-kg baseball is dropped from rest. If the magnitude of the baseball's momentum is $0.780 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ just before it lands on the ground, from what height was it dropped?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 6

A 285-g ball falls vertically downward, hitting the floor with a speed of $2.5 \mathrm{m} / \mathrm{s}$ and rebounding upward with a speed of $2.0 \mathrm{m} / \mathrm{s}$ (a) Find the magnitude of the change in the ball's momentum. (b) Find the change in the magnitude of the ball's momentum. (c) Which of the two quantities calculated in parts (a) and (b) is more directly related to the net force acting on the ball during its collision with the floor? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

Object 1 has a mass $m_{1}$ and a velocity $\overrightarrow{\mathbf{v}}_{1}=(2.80 \mathrm{m} / \mathrm{s}) \hat{\mathrm{x}}$. Object 2 has a mass $m_{2}$ and a velocity $\overrightarrow{\mathbf{v}}_{2}=(3.10 \mathrm{m} / \mathrm{s}) \hat{\mathrm{y}} .$ The total momentum of these two objects has a magnitude of $17.6 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ and points in a direction $66.5^{\circ}$ above the positive $x$ axis. Find $m_{1}$ and $m_{2}$

Nick Auwerda

Numerade Educator

Problem 8

Your car rolls slowly in a parking lot and bangs into the metal base of a light pole. In terms of safety, is it better for your collision with the light pole to be elastic, inelastic, or is the safety risk the same for either case? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

Predict/Explain A net force of $200 \mathrm{N}$ acts on a $100-\mathrm{kg}$ boulder, and a force of the same magnitude acts on a $100-\mathrm{g}$ pebble. (a) Is the change of the boulder "s momentum in one second greater than, less than, or equal to the change of the pebble's momentum in the same time period? (b) Choose the best explanation from among the following:
I. The large mass of the boulder gives it the greater momentum.
II. The force causes a much greater speed in the 100 -g pebble, resulting in more momentum.
III. Equal force means equal change in momentum for a given time.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 10

Predict/Explain Referring to the previous question, (a) is the change in the boulder's speed in one second greater than, less than, or equal to the change in speed of the pebble in the same time period? (b) Choose the best explanation from among the following:
I. The large mass of the boulder results in a small acceleration.
II. The same force results in the same change in speed for a given time.
III. Once the boulder gets moving it is harder to stop than the pebble.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

A friend tosses a ball of mass $m$ to you with a speed o. When you catch the ball, you feel a noticeable sting in your hand, due to the force required to stop the ball. (a) If you now catch a second ball, with a mass $2 m$ and speed $v / 2,$ is the sting you feel greater than, less than, or equal to the sting you felt when you caught the first ball? The time required to stop the two balls is the same. (b) Choose the best explanation from among the following:
I. The second ball has less kinetic energy, since kinetic energy depends on $v^{2}$, and hence it produces less sting.
II. The two balls have the same momentum, and hence they produce the same sting.
III. The second ball has more mass, and hence it produces the greater sting.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 12

Force A has a magnitude $F$ and acts for the time $\Delta t$ force B has a magnitude $2 F$ and acts for the time $\Delta t / 3,$ force $C$ has a magnitude $5 F$ and acts for the time $\Delta t / 10,$ and force $D$ has a magnitude $10 F$ and acts for the time $\Delta t / 100 .$ Rank these forces in order of increasing impulse. Indicate ties where appropriate.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 13

Find the magnitude of the impulse delivered to a soccer ball when a player kicks it with a force of $1250 \mathrm{N}$. Assume that the player's foot is in contact with the ball for $5.95 \times 10^{-3} \mathrm{s}$.

Nick Auwerda

Numerade Educator

Problem 14

In a typical golf swing, the club is in contact with the ball for about $0.0010 \mathrm{s}$. If the 45-g ball acquires a speed of $67 \mathrm{m} / \mathrm{s}$, estimate the magnitude of the force exerted by the club on the ball.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 15

A $0.50-\mathrm{kg}$ croquet ball is initially at rest on the grass. When the ball is struck by a mallet, the average force exerted on it is $230 \mathrm{N}$. If the ball's speed after being struck is $3.2 \mathrm{m} / \mathrm{s}$, how long was the mallet in contact with the ball?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

When spiking a volleyball, a player changes the velocity of the ball from $4.2 \mathrm{m} / \mathrm{s}$ to $-24 \mathrm{m} / \mathrm{s}$ along a certain direction. If the impulse delivered to the ball by the player is $-9.3 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ what is the mass of the volleyball?

Nick Auwerda

Numerade Educator

Problem 17

A 15.0-g marble is dropped from rest onto the floor $1.44 \mathrm{m}$ below. (a) If the marble bounces straight upward to a height of $0.640 \mathrm{m},$ what are the magnitude and direction of the impulse delivered to the marble by the floor? (b) If the marble had bounced to a greater height, would the impulse delivered to it have been greater or less than the impulse found in part (a)? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 18

To make a bounce pass, a player throws a $0.60-\mathrm{kg}$ basketball toward the floor. The ball hits the floor with a speed of $5.4 \mathrm{m} / \mathrm{s}$ at an angle of $65^{\circ}$ to the vertical. If the ball rebounds with the same speed and angle, what was the impulse delivered to it by the floor?

Nick Auwerda

Numerade Educator

Problem 19

A 0.14-kg baseball moves toward home plate with a velocity $\overrightarrow{\mathbf{v}}_{i}=(-36 \mathrm{m} / \mathrm{s}) \hat{\mathrm{x}}$. After striking the bat, the ball moves vertically upward with a velocity $\overrightarrow{\mathbf{v}}_{\mathrm{f}}=(18 \mathrm{m} / \mathrm{s}) \hat{\mathrm{y}} .$ (a) Find the direction and magnitude of the impulse delivered to the ball by the bat. Assume that the ball and bat are in contact for $1.5 \mathrm{ms}$. (b) How would your answer to part (a) change if the mass of the ball were doubled? (c) How would your answer to part (a) change if the mass of the bat were doubled instead?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

A player bounces a $0.43-\mathrm{kg}$ soccer ball off her head, changing the velocity of the ball from $\overrightarrow{\mathbf{v}}_{i}=(8.8 \mathrm{m} / \mathrm{s}) \hat{\mathrm{x}}+(-2.3 \mathrm{m} / \mathrm{s}) \hat{\mathrm{y}}$ to $\overrightarrow{\mathbf{v}}_{f}=(5.2 \mathrm{m} / \mathrm{s}) \hat{\mathrm{x}}+(3.7 \mathrm{m} / \mathrm{s}) \hat{y} .$ If the ball is in contact with the player's head for $6.7 \mathrm{ms},$ what are (a) the direction and (b) the magnitude of the impulse delivered to the ball?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 21

In a situation similar to Example $9-3,$ suppose the speeds of the two canoes after they are pushed apart are $0.58 \mathrm{m} / \mathrm{s}$ for canoe 1 and $0.42 \mathrm{m} / \mathrm{s}$ for canoe $2 .$ If the mass of canoe 1 is $320 \mathrm{kg}$ what is the mass of canoe $2 ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

Two ice skaters stand at rest in the center of an ice rink. When they push off against one another the $45-\mathrm{kg}$ skater acquires a speed of $0.62 \mathrm{m} / \mathrm{s}$. If the speed of the other skater is $0.89 \mathrm{m} / \mathrm{s}$ what is this skater's mass?

Nick Auwerda

Numerade Educator

Problem 23

Suppose the bee in Active Example $9-2$ has a mass of 0.175 g. If the bee walks with a speed of $1.41 \mathrm{cm} / \mathrm{s}$ relative to the still water, what is the speed of the $4.75-g$ stick relative to the water?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 24

An object initially at rest breaks into two pieces as the result of an explosion. One piece has twice the kinetic energy of the other piece. What is the ratio of the masses of the two pieces? Which piece has the larger mass?

Nick Auwerda

Numerade Educator

Problem 25

A 92-kg astronaut and a 1200-kg satellite are at rest relative to the space shuttle. The astronaut pushes on the satellite, giving it a speed of $0.14 \mathrm{m} / \mathrm{s}$ directly away from the shuttle. Seven and a half seconds later the astronaut comes into contact with the shuttle. What was the initial distance from the shuttle to the astronaut?

Nick Auwerda

Numerade Educator

Problem 26

An $85-k g$ lumberjack stands at one end of a $380-\mathrm{kg}$ floating log, as shown in Figure $9-15$. Both the log and the lumberjack are at rest initially. (a) If the lumberjack now trots toward the other end of the log with a speed of $2.7 \mathrm{m} / \mathrm{s}$ relative to the log, what is the lumberjack's speed relative to the shore? Ignore friction between the log and the water. (b) If the mass of the log had been greater, would the lumberjack's speed relative to the shore be greater than, less than, or the same as in part (a)? Explain. (c) Check your answer to part (b) by calculating the lumberjack's speed relative to the shore for the case of a $450-\mathrm{kg}$ log.

Mark J

Numerade Educator

Problem 27

A plate drops onto a smooth floor and shatters into three pieces of equal mass. Two of the pieces go off with equal speeds $v$ at right angles to one another. Find the speed and direction of the third piece.

Nick Auwerda

Numerade Educator

Problem 28

A cart of mass $m$ moves with a speed $v$ on a frictionless air track and collides with an identical cart that is stationary. If the two carts stick together after the collision, what is the final kinetic energy of the system?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

Suppose the car in Example $9-6$ has an initial speed of $20.0 \mathrm{m} / \mathrm{s}$ and that the direction of the wreckage after the collision is $40.0^{\circ}$ above the $x$ axis. Find the initial speed of the minivan and the final speed of the wreckage.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 30

Two $72.0-\mathrm{kg}$ hockey players skating at $5.45 \mathrm{m} / \mathrm{s}$ collide and stick together. If the angle between their initial directions was $115^{\circ},$ what is their speed after the collision?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 31

(a) Referring to Exercise $9-2,$ is the final kinetic energy of the car and truck together greater than, less than, or equal to the sum of the initial kinetic energies of the car and truck separately? Explain. (b) Verify your answer to part (a) by calculating the initial and final kinetic energies of the system.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 32

A bullet with a mass of 4.0 g and a speed of $650 \mathrm{m} / \mathrm{s}$ is fired at a block of wood with a mass of $0.095 \mathrm{kg}$. The block rests on a frictionless surface, and is thin enough that the bullet passes completely through it. Immediately after the bullet exits the block, the speed of the block is $23 \mathrm{m} / \mathrm{s}$. (a) What is the speed of the bullet when it exits the block? (b) Is the final kinetic energy of this system equal to, less than, or greater than the initial kinetic energy? Explain. (c) Verify your answer to part (b) by calculating the initial and final kinetic energies of the system.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

A $0.420-\mathrm{kg}$ block of wood hangs from the ceiling by a string, and a $0.0750-\mathrm{kg}$ wad of putty is thrown straight upward, striking the bottom of the block with a speed of $5.74 \mathrm{m} / \mathrm{s}$. The wad of putty sticks to the block. (a) Is the mechanical energy of this system conserved? (b) How high does the putty-block system rise above the original position of the block?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 34

A $0.430-\mathrm{kg}$ block is attached to a horizontal spring that is at its equilibrium length, and whose force constant is $20.0 \mathrm{N} / \mathrm{m}$. The block rests on a frictionless surface. $A 0.0500-\mathrm{kg}$ wad of putty is thrown horizontally at the block, hitting it with a speed of $2.30 \mathrm{m} / \mathrm{s}$ and sticking. How far does the putty-block system compress the spring?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 35

Two objects moving with a speed $v$ travel in opposite directions in a straight line. The objects stick together when they collide, and move with a speed of $v / 4$ after the collision. (a) What is the ratio of the final kinetic energy of the system to the initial kinetic energy? (b) What is the ratio of the mass of the more massive object to the mass of the less massive object?

Nick Auwerda

Numerade Educator

Problem 36

The collision between a hammer and a nail can be considered to be approximately elastic. Calculate the kinetic energy acquired by a 12 -g nail when it is struck by a 550 -g hammer moving with an initial speed of $4.5 \mathrm{m} / \mathrm{s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 37

A 732-kg car stopped at an intersection is rear-ended by a $1720-\mathrm{kg}$ truck moving with a speed of $15.5 \mathrm{m} / \mathrm{s}$. If the car was in neutral and its brakes were off, so that the collision is approximately elastic, find the final speed of both vehicles after the collision.

Nick Auwerda

Numerade Educator

Problem 38

Suppose you throw a rubber ball at an elephant that is charging directly at you (not a good idea). When the ball bounces back toward you, is its speed greater than, less than, or equal to the speed with which you threw it? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 39

A charging bull elephant with a mass of $5240 \mathrm{kg}$ comes directly toward you with a speed of $4.55 \mathrm{m} / \mathrm{s}$. You toss a $0.150-\mathrm{kg}$ rubber ball at the elephant with a speed of $7.81 \mathrm{m} / \mathrm{s}$. (a) When the ball bounces back toward you, what is its peed?(b) How do you account for the fact that the ball's kinetic energy has increased?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 40

Moderating a Neutron In a nuclear reactor, neutrons released by nuclear fission must be slowed down before they can trigger additional reactions in other nuclei. To see what sort of material is most effective in slowing (or moderating) a neutron, calculate the ratio of a neutron's final kinetic energy to its initial kinetic energy, $K_{f} / K_{i},$ for a head-on elastic collision with each of the following stationary target particles. (Note: The mass of a neutron is $m=1.009 \mathrm{u},$ where the atomic mass unit, $u,$ is defined as follows: $\left.1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{kg} \cdot\right)$ (a) An electron $\left(M=5.49 \times 10^{-4} \mathrm{u}\right)$ (b) A proton $(M=1.007 \mathrm{u})$ (c) The nucleus of a lead atom $(M=207.2 \mathrm{u})$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 41

In the apple-orange collision in Example $9-7$, suppose the final velocity of the orange is $1.03 \mathrm{m} / \mathrm{s}$ in the negative $y$ direction. What are the final speed and direction of the apple in this case?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 42

The three air carts shown in Figure $9-16$ have masses, reading from left to right, of $4 m, 2 m,$ and $m,$ respectively. The most massive cart has an initial speed of $v_{0}$; the other two carts are at rest initially. All carts are equipped with spring bumpers that give elastic collisions. (a) Find the final speed of each cart. (b) Verify that the final kinetic energy of the system is equal to the initial kinetic energy. (Assume the air track is long enough to accommodate all collisions.)

Nick Auwerda

Numerade Educator

Problem 43

In this problem we show that when one ball is pulled to the left in the photo on page $275,$ only a single ball recoils to the right-under ideal elastic-collision conditions. To begin, suppose that each ball has a mass $m,$ and that the ball coming in from the left strikes the other balls with a speed $v_{0}$. Now, consider the hypothetical case of two balls recoiling to the right. Determine the speed the two recoiling balls must have in order to satisfy (a) momentum conservation and (b) energy conservation. since these speeds are not the same, it follows that momentum and energy cannot be conserved simultaneously with a recoil of two balls.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 44

Predict/Explain A stalactite in a cave has drops of water falling from it to the cave floor below. The drops are equally spaced in time and come in rapid succession, so that at any given moment there are many drops in midair. (a) Is the center of mass of the midair drops higher than, lower than, or equal to the halfway distance between the tip of the stalactite and the cave floor? (b) Choose the best explanation from among the following:
I. The drops bunch up as they near the floor of the cave.
II. The drops are equally spaced as they fall, since they are released at equal times.
III. Though equally spaced in time, the drops are closer together higher up.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 45

Find the $x$ coordinate of the center of mass of the bricks shown in Figure $9-17$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 46

You are holding a shopping basket at the grocery store with two $0.56-\mathrm{kg}$ cartons of cereal at the left end of the basket. The basket is $0.71 \mathrm{m}$ long. Where should you place a $1.8-\mathrm{kg}$ half gallon of milk, relative to the left end of the basket, so that the center of mass of your groceries is at the center of the basket?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 47

Earth-Moon Center of Mass The Earth has a mass of $5.98 \times 10^{24} \mathrm{kg},$ the Moon has a mass of $7.35 \times 10^{22} \mathrm{kg},$ and their center-to-center distance is $3.85 \times 10^{8} \mathrm{m} .$ How far from the center of the Earth is the Earth-Moon center of mass? Is the Earth-Moon center of mass above or below the surface of the Earth? By what distance? (As the Earth and Moon orbit one another, their centers orbit about their common center of mass.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 48

A piece of sheet metal of mass $M$ is cut into the shape of a right triangle, as shown in Figure $9-18$. A vertical dashed line is drawn on the sheet at the point where the mass to the left of the line $(M / 2)$ is equal to the mass to the right of the line (also $M / 2$ ). The sheet is now placed on a fulcrum just under the dashed line and released from rest. (a) Does the metal sheet remain level, tip to the left, or tip to the right? (b) Choose the best explanation from among the following:
I. Equal mass on either side will keep the metal sheet level.
II. The metal sheet extends for a greater distance to the left, which shifts the center of mass to the left of the dashed line.
III. The center of mass is to the right of the dashed line because the metal sheet is thicker there.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 49

A pencil standing upright on its eraser end falls over and lands on a table. As the pencil falls, its eraser does not slip. The following questions refer to the contact force exerted on the pencil by the table. Let the positive $x$ direction be in the direction the pencil falls, and the positive $y$ direction be vertically upward. (a) During the pencil's fall, is the $x$ component of the contact force positive, negative, or zero? Explain. (b) Is the $y$ component of the contact force greater than, less than, or equal to the weight of the pencil? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 50

A cardboard box is in the shape of a cube with each side of length $L$. If the top of the box is missing, where is the center of mass of the open box? Give your answer relative to the geometric center of the box.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

The location of the center of mass of the partially eaten, 12-inch-diameter pizza shown in Figure $9-19$ is $X_{c m}=-1.4$ in. and $Y_{\mathrm{Cm}}=-1.4$ in. Assuming each quadrant of the pizza to be the same, find the center of mass of the uneaten pizza above the $x$ axis (that is, the portion of the pizza in the second quadrant).

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 52

The Center of Mass of Sulfur Dioxide Sulfur dioxide $\left(\mathrm{SO}_{2}\right)$ consists of two oxygen atoms (each of mass $16 \mathrm{u},$ where $\mathrm{u}$ is defined in Problem 40 ) and a single sulfur atom (of mass $32 \mathrm{u}$ ). The center-to-center distance between the sulfur atom and either of the oxygen atoms is $0.143 \mathrm{nm},$ and the angle formed by the three atoms is $120^{\circ},$ as shown in Figure $9-20$. Find the $x$ and $y$ coordinates of the center of mass of this molecule.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 53

Three uniform metersticks, each of mass $m$, are placed on the floor as follows: stick 1 lies along the $y$ axis from $y=0$ to $y=1.0 \mathrm{m},$ stick 2 lies along the $x$ axis from $x=0$ to $x=1.0 \mathrm{m}$ stick 3 lies along the $x$ axis from $x=1.0 \mathrm{m}$ to $x=2.0 \mathrm{m}$ (a) Find the location of the center of mass of the metersticks. (b) How would the location of the center of mass be affected if the mass of the metersticks were doubled?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 54

A 0.726-kg rope 2.00 meters long lies on a floor. You grasp one end of the rope and begin lifting it upward with a constant speed of $0.710 \mathrm{m} / \mathrm{s}$. Find the position and velocity of the rope's center of mass from the time you begin lifting the rope to the time the last piece of rope lifts off the floor. Plot your results. (Assume the rope occupies negligible volume directly below the point where it is being lifted.)

Nick Auwerda

Numerade Educator

Problem 55

Repeat the previous problem, this time lowering the rope onto a floor instead of lifting it.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 56

Consider the system shown in Figure $9-21$. Assume that after the string breaks the ball falls through the liquid with constant speed. If the mass of the bucket and the liquid is $1.20 \mathrm{kg}$, and the mass of the ball is $0.150 \mathrm{kg}$, what is the reading on the scale (a) before and (b) after the string breaks?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

A metal block of mass $m$ is attached to the ceiling by a a spring. Connected to the bottom of this block is a string that supports a second block of the same mass $m,$ as shown in Figure $9-22$. The string connecting the two blocks is now cut. (a) What is the net force acting on the two-block system immediately after the string is cut? (b) What is the acceleration of the center of mass of the two-block system immediately after the string is cut?

Nick Auwerda

Numerade Educator

Problem 58

Helicopter Thrust During a rescue operation, a $5300-\mathrm{kg}$ helicopter hovers above a fixed point. The helicopter blades send air downward with a speed of $62 \mathrm{m} / \mathrm{s}$. What mass of air must pass through the blades every second to produce enough thrust for the helicopter to hover?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 59

Rocks for a Rocket Engine A child sits in a wagon with a pile of $0.65-\mathrm{kg}$ rocks. If she can throw each rock with a speed of $11 \mathrm{m} / \mathrm{s}$ relative to the ground, causing the wagon to move, how many rocks must she throw per minute to maintain a constant average speed against a $3.4-\mathrm{N}$ force of friction?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 60

A 57.8-kg person holding two 0.880-kg bricks stands on a $2.10-\mathrm{kg}$ skateboard. Initially, the skateboard and the person are at rest. The person now throws the two bricks at the same time so that their speed relative to the person is $17.0 \mathrm{m} / \mathrm{s}$. What is the recoil speed of the person and the skateboard relative to the ground, assuming the skateboard moves without friction?

Nick Auwerda

Numerade Educator

Problem 61

In the previous problem, calculate the final speed of the person and the skateboard relative to the ground if the person throws the bricks one at a time. Assume that each brick is thrown with a speed of $17.0 \mathrm{m} / \mathrm{s}$ relative to the person.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 62

A 0.540-kg bucket rests on a scale. Into this bucket you pour sand at the constant rate of $56.0 \mathrm{g} / \mathrm{s}$. If the sand lands in the bucket with a speed of $3.20 \mathrm{m} / \mathrm{s},$ (a) what is the reading of the scale when there is $0.750 \mathrm{kg}$ of sand in the bucket? (b) What is the weight of the bucket and the $0.750 \mathrm{kg}$ of sand?

Nick Auwerda

Numerade Educator

Problem 63

Holding a long rope by its upper end, you lower it onto a scale. The rope has a mass of $0.13 \mathrm{kg}$ per meter of length, and is lowered onto the scale at the constant rate of $1.4 \mathrm{m} / \mathrm{s}$. (a) Calculate the thrust exerted by the rope as it lands on the scale. (b) At the instant when the amount of rope at rest on the scale has a weight of $2.5 \mathrm{N}$, does the scale read $2.5 \mathrm{N}$, more than $2.5 \mathrm{N}$, or less than $2.5 \mathrm{N}$ ? Explain. (c) Check your answer to part (b) by calculating the reading on the scale at this time.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 64

Object A has a mass $m$, object B has a mass $2 m,$ and object C has a mass $m / 2$. Rank these objects in order of increasing kinetic energy, given that they all have the same momentum. Indicate ties where appropriate.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 65

Object A has a mass $m$, object B has a mass $4 m,$ and object C has a mass $m / 4 .$ Rank these objects in order of increasing momentum, given that they all have the same kinetic energy. Indicate ties where appropriate.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 66

A block of wood is struck by a bullet. (a) Is the block more likely to be knocked over if the bullet is metal and embeds itself in the wood, or if the bullet is rubber and bounces off the wood? (b) Choose the best explanation from among the following:
I. The change in momentum when a bullet rebounds is larger than when it is brought to rest.
II. The metal bullet does more damage to the block.
III. since the rubber bullet bounces off, it has little effect.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 67

A juggler performs a series of tricks with three bowling balls while standing on a bathroom scale. Is the average reading of the scale greater than, less than, or equal to the weight of the juggler plus the weight of the three balls? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 68

A 72.5 -kg tourist climbs the stairs to the top of the Washington Monument, which is $555 \mathrm{ft}$ high. How far does the Earth move in the opposite direction as the tourist climbs?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 68

A 72.5 -kg tourist climbs the stairs to the top of the Washington Monument, which is $555 \mathrm{ft}$ high. How far does the Earth move in the opposite direction as the tourist climbs?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 69

Figure 9-23 shows a block of mass 2 $m$ at rest on a horizontal, frictionless table. Attached to this block by a string that passes over a pulley is a second block, with a mass $m$.\
The initial position of the center of mass of the blocks is indicated by the point i. The blocks are now released and allowed to accelerate; a short time later their center of mass is at the point
f. (a) Did the center of mass follow the red path, the green path, or the blue path? (b) Choose the best explanation from among the following:
I. The center of mass must always be closer to the $2 m$ block than to the $m$ block.
II. The center of mass starts at rest, and moves in a straight line in the direction of the net force.
III. The masses are accelerating, which implies parabolic motion.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

A car moving with an initial speed $v$ collides with a second stationary car that is one-half as massive. After the collision the first car moves in the same direction as before with a speed $v / 3$. (a) Find the final speed of the second car. (b) Is this collision elastic or inelastic?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 71

A 1.35-kg block of wood sits at the edge of a table, $0.782 \mathrm{m}$ above the floor. A 0.0105-kg bullet moving horizontally with a speed of $715 \mathrm{m} / \mathrm{s}$ embeds itself within the block. What horizontal distance does the block cover before hitting the ground?

Nick Auwerda

Numerade Educator

Problem 72

The carton of eggs shown in Figure $9-24$ is filled with a dozen eggs, each of mass $m$. Initially, the center of mass of the eggs is at the center of the carton.
(a) Does the location of the center of mass of the eggs change more if egg 1 is removed or if egg 2 is removed? Explain. (b) Find the center of mass of the eggs when egg 1 is removed.
(c) Find the center of mass of the eggs if egg 2 is removed instead.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

The Force of a Storm During a severe storm in Palm Beach, $\mathrm{FL}$, on January 2,1999,31 inches of rain fell in a period of nine hours. Assuming that the raindrops hit the ground with a speed of $10 \mathrm{m} / \mathrm{s},$ estimate the average upward force exerted by one square meter of ground to stop the falling raindrops during the storm. (Note: One cubic meter of water has a mass of $1000 \mathrm{kg}$

Nick Auwerda

Numerade Educator

Problem 74

An apple that weighs $2.7 \mathrm{N}$ falls vertically downward from rest for 1.4 s. (a) What is the change in the apple's momentum per second? (b) What is the total change in its momentum during the 1.4 -second fall?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 75

To balance a 35.5 -kg automobile tire and wheel, a mechanic must place a $50.2-g$ lead weight $25.0 \mathrm{cm}$ from the center of the wheel. When the wheel is balanced, its center of mass is exactly at the center of the wheel. How far from the center of the wheel was its center of mass before the lead weight was added?

Nick Auwerda

Numerade Educator

Problem 76

A hoop of mass $M$ and radius $R$ rests on a smooth, level surface. The inside of the hoop has ridges on either side, so that it forms a track on which a ball can roll, as indicated in Figure $9-25$. If a ball of mass $2 M$ and radius $r=R / 4$ is released as shown, the system rocks back and forth until it comes to rest with the ball at the bottom of the hoop. When the ball comes to rest, what is the $x$ coordinate of its center?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 77

A $63-\mathrm{kg}$ canoeist stands in the middle of her $22-\mathrm{kg}$ canoe. The canoe is $3.0 \mathrm{m}$ long, and the end that is closest to land is $2.5 \mathrm{m}$ from the shore. The canoeist now walks toward the shore until she comes to the end of the canoe. (a) When the canoeist stops at the end of her canoe, is her distance from the shore equal to, greater than, or less than $2.5 \mathrm{m}$ ? Explain. (b) Verify your answer to part (a) by calculating the distance from the canoeist to shore.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 78

In the previous problem, suppose the canoeist is $3.4 \mathrm{m}$ from shore when she reaches the end of her canoe. What is the canoe's mass?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 79

Referring to Problem $56,$ find the reading on the scale (a) before and (b) after the string breaks, assuming the ball falls through the liquid with an acceleration equal to $0.250 g .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 80

A young hockey player stands at rest on the ice holding a $1.3-\mathrm{kg}$ helmet. The player tosses the helmet with a speed of $6.5 \mathrm{m} / \mathrm{s}$ in a direction $11^{\circ}$ above the horizontal, and recoils with a speed of $0.25 \mathrm{m} / \mathrm{s}$. Find the mass of the hockey player.

Nick Auwerda

Numerade Educator

Problem 81

Suppose the air carts in Example $9-9$ are both moving to the right initially. The cart to the left has a mass $m$ and an initial speed $v_{0} ;$ the cart to the right has an initial speed $v_{0} / 2 .$ If the center of mass of this system moves to the right with a speed $2 v_{0} / 3,$ what is the mass of the cart on the right?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 82

A long, uniform rope with a mass of 0.135 kg per meter lies on the ground. You grab one end of the rope and lift it at the constant rate of $1.13 \mathrm{m} / \mathrm{s}$. Calculate the upward force you must exert at the moment when the top end of the rope is $0.525 \mathrm{m}$ above the ground.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 83

The Center of Mass of Water Find the center of mass of a water molecule, referring to Figure $9-26$ for the relevant angles and distances. The mass of a hydrogen atom is $1.0 \mathrm{u}$, and the mass of an oxygen atom is $16 \mathrm{u}$, where $\mathrm{u}$ is the atomic mass unit (see Problem 40). Use the center of the oxygen atom as the origin of your coordinate system.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 84

The three air carts shown in Figure $9-27$ have masses, reading from left to right, of $m, 2 m,$ and $4 m,$ respectively. Initially, the cart on the right is at rest, whereas the other two carts are moving to the right with a speed $v_{0}$. All carts are equipped with putty bumpers that give completely inelastic collisions. (a) Find the final speed of the carts. (b) Calculate the ratio of the final kinetic energy of the system to the initial kinetic energy.

Nick Auwerda

Numerade Educator

Problem 85

A fireworks rocket is launched vertically into the night sky with an initial speed of $44.2 \mathrm{m} / \mathrm{s}$. The rocket coasts after being launched, then explodes and breaks into two pieces of equal mass 2.50 s later. (a) If each piece follows a trajectory that is initially at $45.0^{\circ}$ to the vertical, what was their speed immediately after the explosion? (b) What is the velocity of the rocket's center of mass before and after the explosion? (c) What is the acceleration of the rocket's center of mass before and after the explosion?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 86

The total momentum of two cars approaching an intersection is $\overrightarrow{\mathrm{p}}_{\text {total }}=(15,000 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}) \hat{\mathrm{x}}+(2100 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}) \hat{\mathrm{y}}$ (a) If the momentum of car 1 is $\overrightarrow{\mathrm{p}}_{1}=(11,000 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}) \hat{\mathrm{x}}+$ $(-370 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}) \hat{y},$ what is the momentum of car $2 ?$(b) Does your answer to part (a) depend on which car is closer to the
intersection? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 87

Four identical textbooks, each of length $L,$ are stacked near the edge of a table, as shown in Figure $9-28$. The books are stacked in such a way that the distance they overhang the edge of the table, $d$, is maximized. Find the maximum overhang distance $d$ in terms of $L .$ In particular, show that $d>L ;$ that is, the top book is completely to the right of the table edge. (In principle, the overhang distance $d$ can be made as large as desired simply by increasing the number of books in the stack.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 88

Consider a one-dimensional, head-on elastic collision. One object has a mass $m_{1}$ and an initial velocity $v_{1} ;$ the other has a mass $m_{2}$ and an initial velocity $v_{2}$. Use momentum conservation and energy conservation to show that the final velocities of the two masses are
$$\begin{array}{l}
v_{1, f}=\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right) v_{1}+\left(\frac{2 m_{2}}{m_{1}+m_{2}}\right) v_{2} \\
v_{2, f}=\left(\frac{2 m_{1}}{m_{1}+m_{2}}\right) v_{1}+\left(\frac{m_{2}-m_{1}}{m_{1}+m_{2}}\right) v_{2}
\end{array}$$

Nick Auwerda

Numerade Educator

Problem 89

Two air carts of mass $m_{1}=0.84 \mathrm{kg}$ and $m_{2}=0.42 \mathrm{kg}$ are placed on a frictionless track. Cart 1 is at rest initially, and has a spring bumper with a force constant of $690 \mathrm{N} / \mathrm{m}$. Cart 2 has a flat metal surface for a bumper, and moves toward the bumper of the stationary cart with an initial speed $v=0.68 \mathrm{m} / \mathrm{s}$. (a) What is the speed of the two carts at the moment when their speeds are equal? (b) How much energy is stored in the spring bumper when the carts have the same speed? (c) What is the final speed of the carts after the collision?

Nick Auwerda

Numerade Educator

Problem 90

Golden Earrings and the Golden Ratio A popular earring design features a circular piece of gold of diameter $D$ with a circular cutout of diameter $d$, as shown in Figure $9-29$. If this earring is to balance at the point $P$, show that the diameters must satisfy the condition $D=\phi d,$ where $\phi=(1+\sqrt{5}) / 2=1.61803 \ldots$ is the famous "golden ratio."

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 91

Two objects with masses $m_{1}$ and $m_{2}$ and initial velocities $v_{1, i}$ and $v_{2, i}$ move along a straight line and collide elastically. Assuming that the objects move along the same straight line after the collision, show that their relative velocities are unchanged; that is, show that $v_{1, d}-v_{2, i}=v_{2, f}-v_{1, f}$. (You can use the results given in Problem $88 .$ )

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 92

Amplified Rebound Height Two small rubber balls are dropped from rest at a height $h$ above a hard floor. When the balls are released, the lighter ball (with mass $m$ ) is directly above the heavier ball (with mass $M$ ). Assume the heavier ball reaches the floor first and bounces elastically; thus, when the balls collide, the ball of mass $M$ is moving upward with a speed $v$ and the ball of mass $m$ is moving downward with essentially the same speed. In terms of $h,$ find the height to which the ball of mass $m$ rises after the collision. (Use the results given in Problem $88,$ and assume the balls collide at ground level.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 93

On a cold winter morning, a child sits on a sled resting on smooth ice. When the 9.75 -kg sled is pulled with a horizontal force of $40.0 \mathrm{N},$ it begins to move with an acceleration of $2.32 \mathrm{m} / \mathrm{s}^{2} .$ The $21.0-\mathrm{kg}$ child accelerates too, but with a smaller acceleration than that of the sled. Thus, the child moves forward relative to the ice, but slides backward relative to the sled. Find the acceleration of the child relative to the ice.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 94

An object of mass $m$ undergoes an elastic collision with an identical object that is at rest. The collision is not head-on. Show that the angle between the velocities of the two objects after the collision is $90^{\circ}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 95

Weighing a Block on an Incline A wedge of mass $m_{1}$ is firmly attached to the top of a scale, as shown in Figure $9-30 .$ The inclined surface of the wedge makes an angle $\theta$ with the horizontal. Now, a block of mass $m_{2}$ is placed on the inclined surface of the wedge and allowed to accelerate without friction down the slope. (a) Show that the reading on the scale while the block slides is
$$\left(m_{1}+m_{2} \cos ^{2} \theta\right) g$$
(b) Explain why the reading on the scale is less than $\left(m_{1}+m_{2}\right) g .$
(c) Show that the expression in part (a) gives the expected results for $\theta=0$ and $\theta=90^{\circ}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 96

A uniform rope of length $L$ and mass $M$ rests on a table.
(a) If you lift one end of the rope upward with a constant speed, $v,$ show that the rope's center of mass moves upward with constant accelenation.
(b) Next, suppose you hold the rope suspended in air, with its lower end just touching the table. If you now lower the rope with a constant speed, $v$, onto the table, is the acceleration of the rope's center of mass upward or downward? Explain your answer. (c) Find the magnitude and direction of the acceleration of the rope's center of mass for the case described in part (b). Compare with part (a).

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 97

From the perspective of an observer on the planet, what is the spacecraft's speed of approach?
A. $v_{i}+u$
B. $v_{i}-u$
C. $u-v_{1}$
D. $v_{f}-u$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 98

From the perspective of an observer on the planet, what is the spacecraft's speed of departure?
A. $v_{f}+u$
B. $v_{f}-u$
C. $u-v_{f}$
D. $v_{1}-u$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 99

Set the speed of departure from Problem 98 equal to the speed of approach from Problem 97. Solving this relation for the final speed, $v_{\varphi}$ y ields:
A. $v_{i}=v_{i}+u$
B. $v_{t}=v_{1}-u$
C. $v_{\ell}=v_{\mathrm{i}}+2 u$
D. $v_{f}=v_{i}-2 u$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 100

Consider the special case in which $v_{i}=u$. By what factor does the kinetic energy of the spacecraft increase as a result of the encounter?
A. 4
B. 8
C. 9
D. 16

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 101

Referring to Example $9-5$ Suppose a bullet of mass $m=6.75 \mathrm{g}$ is fired into a ballistic pendulum whose bob has a mass of $M=0.675 \mathrm{kg}$. (a) If the bob rises to a height of $0.128 \mathrm{m},$ what was the initial speed of the bullet? (b) What was the speed of the bullet-bob combination immediately after the collision takes place?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 102

Referring to Example $9-5$ A bullet with a mass $m=8.10 \mathrm{g}$ and an initial speed $v_{0}=320 \mathrm{m} / \mathrm{s}$ is fired into a ballistic pendulum. What mass must the bob have if the bullet-bob combination is to rise to a maximum height of $0.125 \mathrm{m}$ after the collision?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 103

Referring to Example 9-9 Suppose that cart 1 has a mass of $3.00 \mathrm{kg}$ and an initial speed of $0.250 \mathrm{m} / \mathrm{s} .$ Cart 2 has a mass of 1.00 $\mathrm{kg}$ and is at rest initially. (a) What is the final speed of the carts? (b) How much kinetic energy is lost as a result of the collision?

Nick Auwerda

Numerade Educator

Problem 104

Referring to Example $9-9$ Suppose the two carts have equal masses and are both moving to the right before the collision. The initial speed of cart 1 (on the left) is $v_{0}$ and the initial speed of cart 2 (on the right) is $v_{0} / 2$. (a) What is the speed of the center of mass of this system? (b) What percentage of the initial kinetic energy is lost as a result of the collision? (c) Suppose the collision is elastic. What are the final speeds of the two carts in this case?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator