College Physics With an Integrated Approach to Forces and Kinematics

Alan Giambattista, Betty McCarthy Richardson , Robert C. Richardson

Chapter 7

Linear Momentum - all with Video Answers

Educators

Chapter Questions

Problem 1

Two cars, each of mass $1300 \mathrm{~kg}$, are approaching each other on a head-on collision course. Each speedometer reads $19 \mathrm{~m} / \mathrm{s}$. What is the magnitude of the total momentum of the system?

Mukesh Devi

Mukesh Devi

Numerade Educator

Problem 1

Two point charges, $+5.0 \mu \mathrm{C}$ and $-2.0 \mu \mathrm{C}$, are separated by $5.0 \mathrm{~m}$. What is the electric potential energy?

Mayukh Banik

Numerade Educator

Problem 2

What is the momentum of an automobile (weight = $9800 \mathrm{~N}$ ) when it is moving at $35 \mathrm{~m} / \mathrm{s}$ to the south?

Vysakh M

Numerade Educator

Problem 3

Verify that the SI unit of impulse is the same as the SI unit of momentum.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 4

A cue stick hits a cue ball with an average force of $24 \mathrm{~N}$ for a duration of $0.028 \mathrm{~s}$. If the mass of the ball is $0.16 \mathrm{~kg}$, how fast is it moving after being struck?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 5

A system consists of three particles with these masses and velocities: mass $3.0 \mathrm{~kg}$, moving north at $3.0 \mathrm{~m} / \mathrm{s}$; mass $4.0 \mathrm{~kg}$, moving south at $5.0 \mathrm{~m} / \mathrm{s}$; and $\mathrm{mass} 7.0 \mathrm{~kg}$, moving north at $2.0 \mathrm{~m} / \mathrm{s}$. What is the total momentum of the system?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 6

A sports car traveling along a straight line increases its speed from $20.0 \mathrm{mi} / \mathrm{h}$ to $60.0 \mathrm{mi} / \mathrm{h}$. (a) What is the ratio of the final to the initial magnitude of its momentum?
(b) What is the ratio of the final to the initial kinetic energy?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 7

A ball of mass $5.0 \mathrm{~kg}$ moving with a speed of $2.0 \mathrm{~m} / \mathrm{s}$ in the $+x$ -direction hits a wall and bounces back with the same speed in the $-x$ -direction. What is the change of momentum of the ball?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 8

An object of mass $3.0 \mathrm{~kg}$ is projected into the air at a $55^{\circ}$ angle. It hits the ground $3.4$ s later. What is its change in momentum while it is in the air? Ignore air resistance.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 9

An object of mass $3.0 \mathrm{~kg}$ is allowed to fall from rest under the force of gravity for $3.4 \mathrm{~s}$. What is the change in its momentum? Ignore air resistance.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 10

What average force is necessary to bring a $50.0-\mathrm{kg}$ sled from rest to a speed of $3.0 \mathrm{~m} / \mathrm{s}$ in a period of $20.0 \mathrm{~s} ?$ Assume frictionless ice.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 11

For a safe re-entry into the Earth's atmosphere, the pilots of a space capsule must reduce their speed from $2.6 \times 10^{4} \mathrm{~m} / \mathrm{s}$ to $1.1 \times 10^{4} \mathrm{~m} / \mathrm{s}$. The rocket engine pro-
duces a backward force on the capsule of $1.8 \times 10^{5} \mathrm{~N}$. The mass of the capsule is $3800 \mathrm{~kg}$. For how long must they fire their engine? [Hint: Ignore the change in mass of the capsule due to the expulsion of exhaust gases.]

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 12

A $0.15-\mathrm{kg}$ baseball traveling in a horizontal direction with a speed of $20 \mathrm{~m} / \mathrm{s}$ hits a bat and is popped straight up with a speed of $15 \mathrm{~m} / \mathrm{s}$. (a) What is the change in momentum (magnitude and direction) of the baseball?
(b) If the bat was in contact with the ball for $50 \mathrm{~ms}$, what was the average force of the bat on the ball?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 13

An automobile traveling at a speed of $30.0 \mathrm{~m} / \mathrm{s}$ applies its brakes and comes to a stop in $5.0 \mathrm{~s}$. If the automobile has a mass of $1.0 \times 10^{3} \mathrm{~kg}$, what is the average horizontal force exerted on it during braking? Assume the road is level.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 14

A $3.0-\mathrm{kg}$ body is initially moving northward at $15 \mathrm{~m} / \mathrm{s}$. Then a force of $15 \mathrm{~N}$, toward the east, acts on it for $4.0 \mathrm{~s}$. (a) At the end of the $4.0 \mathrm{~s}$, what is the body's final velocity? (b) What is the change in momentum during the $4.0 \mathrm{~s}$ ?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 15

A boy of mass $60.0 \mathrm{~kg}$ is rescued from a hotel fire by leaping into a firefighters' net. The window from which he leapt was $8.0 \mathrm{~m}$ above the net. The firefighters lower their arms as he lands in the net so that he is brought to a complete stop in a time of $0.40 \mathrm{~s}$. (a) What is his change in momentum during the $0.40$ -s interval?
(b) What is the impulse on the net due to the boy during the interval? [Hint: Do not ignore gravity.] (c) What is the average force on the net due to the boy during the interval?

Vishal Gupta

Numerade Educator

Problem 16

A $115-\mathrm{g}$ ball is traveling to the left with a speed of $30 \mathrm{~m} / \mathrm{s}$ when it is struck by a racket. The force on the ball, directed to the right and applied over $21 \mathrm{~ms}$ of contact time, is shown in the graph. What is the speed of the ball immediately after it leaves the racket? ( $\mathrm{W}$ tutorial: impulse)

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 17

A pole-vaulter of mass $60.0 \mathrm{~kg}$ vaults to a height of $6.0 \mathrm{~m}$ before dropping to thick padding placed below to cushion her fall. (a) Find the speed with which she lands. (b) If the padding brings her to a stop in a time of $0.50 \mathrm{~s}$, what is the average force on her body due to the padding during that time interval?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 18

A rifle has a mass of $4.5 \mathrm{~kg}$ and it fires a bullet of mass $10.0 \mathrm{~g}$ at a muzzle speed of $820 \mathrm{~m} / \mathrm{s}$. What is the recoil speed of the rifle as the bullet leaves the gun barrel?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 19

A $0.030-\mathrm{kg}$ bullet is fired vertically at $200 \mathrm{~m} / \mathrm{s}$ into a $0.15-\mathrm{kg}$ baseball that is initially at rest. The bullet lodges in the baseball and, after the collision, the baseball/ bullet rise to a height of $37 \mathrm{~m}$. (a) What was the speed of the baseball/bullet right after the collision? (b) What was the average force of air resistance while the baseball/bullet was rising?

Vishal Gupta

Numerade Educator

Problem 20

A submarine of mass $2.5 \times 10^{6} \mathrm{~kg}$ and initially at rest fires a torpedo of mass $250 \mathrm{~kg}$. The torpedo has an initial speed of $100.0 \mathrm{~m} / \mathrm{s}$. What is the initial recoil speed of the submarine? Neglect the drag force of the water.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 21

A uranium nucleus (mass $238 \mathrm{u}$ ), initially at rest, undergoes radioactive decay. After an alpha particle (mass $4.0 \mathrm{u}$ ) is emitted, the remaining nucleus is thorium (mass $234 \mathrm{u}$ ). If the alpha particle is moving at $0.050$ times the speed of light, what is the recoil speed of the thorium nucleus? (Note: "u" is a unit of mass; it is not necessary to convert it to $\mathrm{kg}$.)

Vishal Gupta

Numerade Educator

Problem 22

Dash is standing on his frictionless skateboard with three balls, each with a mass of $100 \mathrm{~g}$, in his hands. The combined mass of Dash and his skateboard is $60 \mathrm{~kg}$. How fast should dash throw the balls forward if he wants to move backward with a speed of $0.50 \mathrm{~m} / \mathrm{s}$ ? Do you think Dash can succeed? Explain.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 23

A $58-\mathrm{kg}$ astronaut is in space, far from any objects that would exert a significant gravitational force on him. He would like to move toward his spaceship, but his jet pack is not functioning. He throws a $720-\mathrm{g}$ socket wrench with a velocity of $5.0 \mathrm{~m} / \mathrm{s}$ in a direction away from the ship. After $0.50 \mathrm{~s}$, he throws a $800-\mathrm{g}$ spanner in the same direction with a speed of $8.0 \mathrm{~m} / \mathrm{s}$. After another $9.90 \mathrm{~s}$, he throws a mallet with a speed of $6.0 \mathrm{~m} / \mathrm{s}$ in the same direction. The mallet has a mass of $1200 \mathrm{~g}$. How fast is the astronaut moving after he throws the mallet?

Vishal Gupta

Numerade Educator

Problem 24

A man with a mass of $65 \mathrm{~kg}$ skis down a frictionless hill that is $5.0 \mathrm{~m}$ high. At the bottom of the hill the terrain levels out. As the man reaches the horizontal section, he grabs a $20-\mathrm{kg}$ backpack and skis off a $2.0-\mathrm{m}$ -high ledge. At what horizontal distance from the edge of the ledge does the man land?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 25

A cannon on a railroad car is facing in a direction parallel to the tracks. It fires a 98 -kg shell at a speed of $105 \mathrm{~m} / \mathrm{s}$ (relative to the ground) at an angle of $60.0^{\circ}$ above the horizontal. If the cannon plus car have a mass of $5.0 \times 10^{4} \mathrm{~kg}$, what is the recoil speed of the car if it was at rest before the cannon was fired? [Hint: A component of a system's momentum along an axis is conserved if the net external force acting on the system has no component along that axis.]

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 26

A marksman standing on a motionless railroad car fires a gun into the air at an angle of $30.0^{\circ}$ from the horizontal. The bullet has a speed of $173 \mathrm{~m} / \mathrm{s}$ (relative to the ground) and a mass of $0.010 \mathrm{~kg}$. The man and car move to the left at a speed of $1.0 \times 10^{-3} \mathrm{~m} / \mathrm{s}$ after he shoots. What is the mass of the man and car? (See the hint in Problem 25.)

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 27

Particle $\mathrm{A}$ is at the origin and has a mass of $30.0 \mathrm{~g}$. Particle $\mathrm{B}$ has a mass of $10.0 \mathrm{~g}$. Where must particle $\mathrm{B}$ be located if the coordinates of the $\mathrm{CM}$ are $(x, y)=(2.0 \mathrm{~cm}$, $5.0 \mathrm{~cm}) ?$

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 28

Particle A has a mass of $5.0 \mathrm{~g}$ and particle $\mathrm{B}$ has a mass of $1.0 \mathrm{~g} .$ Particle $\mathrm{A}$ is located at the origin and particle $\mathrm{B}$ is at the point $(x, y)=(25 \mathrm{~cm}, 0)$. What is the location of the CM? ( V) tutorial: center of mass)

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 29

The three bodies in the figure each have the same mass. If one of the bodies is moved $12 \mathrm{~cm}$ in the positive $x$ -direction, by how much does the CM move?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 30

The positions of three particles, written as $(x, y)$ coordinates, are: particle 1 (mass $4.0 \mathrm{~kg}$ ) at $(4.0 \mathrm{~m}, 0 \mathrm{~m})$; particle 2 (mass $6.0 \mathrm{~kg}$ ) at $(2.0 \mathrm{~m}, 4.0 \mathrm{~m})$; particle 3 (mass $3.0 \mathrm{~kg}$ ) at $(-1.0 \mathrm{~m},-2.0 \mathrm{~m}) .$ What is the location of the $\mathrm{CM}$ ?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 31

Belinda needs to find the cM of a sculpture she has made so that it will hang in a gallery correctly. The sculpture is all in one plane and consists of various shaped uniform objects with masses and sizes as shown. Where is the $\mathrm{CM}$ of this sculpture? Assume the thin rods connecting the larger pieces have no mass and place the reference frame origin at the top left corner of the sculpture.

Dading Chen

Numerade Educator

Problem 32

Jane is sitting on a chair with her lower leg at a $30.0^{\circ}$ angle with respect to the vertical, as shown. You need to develop a computer model of her leg to assist in some medical research. If you assume that her leg can be modeled as two uniform cylinders, one with mass $M=$ $20 \mathrm{~kg}$ and length $L=35 \mathrm{~cm}$ and one with mass $m=$ $10 \mathrm{~kg}$ and length $l=40 \mathrm{~cm}$, where is the $\mathrm{CM}$ of her leg?

Vishal Gupta

Numerade Educator

Problem 33

Find the $x$ -coordinate of the $\mathrm{CM}$ of the composite object shown in the figure. The sphere, cylinder, and rectangular solid all have a uniform composition. Their masses and dimensions are: sphere: $200 \mathrm{~g}$, diameter $=10 \mathrm{~cm}$; cylinder: $450 \mathrm{~g}$, length $=17 \mathrm{~cm}$, radius $=5.0 \mathrm{~cm}$; rectangular solid: $325 \mathrm{~g}$, length in $x$ -direction $=16 \mathrm{~cm}$, height $=10 \mathrm{~cm}$, depth $=12 \mathrm{~cm}$.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 34

Consider two falling bodies. Their masses are $3.0 \mathrm{~kg}$ and $4.0 \mathrm{~kg}$. At time $t=0$, the two are released from rest. What is the velocity of their $\mathrm{CM}$ at $t=10.0 \mathrm{~s}$ ? Ignore ain resistance.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 35

Body A of mass $3 \mathrm{~kg}$ is moving in the $+x$ -direction with a speed of $14 \mathrm{~m} / \mathrm{s}$. Body $\mathrm{B}$ of mass $4 \mathrm{~kg}$ is moving in the $-y$ -direction with a speed of $7 \mathrm{~m} / \mathrm{s}$. What are the $x$ - and $y$ -components of the velocity of the $\mathrm{CM}$ of the two bodies?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 36

If a particle of mass $5.0 \mathrm{~kg}$ is moving east at $10 \mathrm{~m} / \mathrm{s}$ and a particle of mass $15 \mathrm{~kg}$ is moving west at $10 \mathrm{~m} / \mathrm{s}$, what is the velocity of the $\mathrm{CM}$ of the pair?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 37

An object located at the origin and having mass $M$ explodes into three pieces having masses $M / 4, M / 3$, and $5 \mathrm{M} / 12 .$ The pieces scatter on a horizontal frictionless $x y$ -plane. The piece with mass $M / 4$ flies away with velocity $5.0 \mathrm{~m} / \mathrm{s}$ at $37^{\circ}$ above the $x$ -axis. The piece with mass $M / 3$ has velocity $4.0 \mathrm{~m} / \mathrm{s}$ directed at an angle of $45^{\circ}$ above the $-x$ -axis. (a) What are the velocity components of the third piece? (b) Describe the motion of the $\mathrm{CM}$ of the system after the explosion.

Paul A.

California State Polytechnic University, Pomona

Problem 38

Prove Eq. (7-13) $\Sigma \overrightarrow{\mathbf{F}}_{\text {ext }}=M \overrightarrow{\mathbf{a}}_{\mathrm{CM}}$. [Hint: Start with $\Sigma \overrightarrow{\mathbf{F}}_{\text {ext }}=\lim _{\Delta t \rightarrow 0}(\Delta \overrightarrow{\mathbf{p}} / \Delta t)$, where $\Sigma \overrightarrow{\mathbf{F}}_{\text {ext }}$ is the net external
force acting on a system and $\overrightarrow{\mathbf{p}}$ is the total momentum of the system.]

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 39

A helium atom (mass $4.00 \mathrm{u}$ ) moving at $618 \mathrm{~m} / \mathrm{s}$ to the right collides with an oxygen molecule (mass $32.0 \mathrm{u}$ ) moving in the same direction at $412 \mathrm{~m} / \mathrm{s}$. After the collision, the oxygen molecule moves at $456 \mathrm{~m} / \mathrm{s}$ to the right. What is the velocity of the helium atom after the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 40

A toy car with a mass of $120 \mathrm{~g}$ moves to the right with a speed of $0.75 \mathrm{~m} / \mathrm{s}$. A small child drops a $30.0-\mathrm{g}$ piece of clay onto the car. The clay sticks to the car and the car continues to the right. What is the change in speed of the car? Consider the frictional force between the car and the ground to be negligible.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 41

In the railroad freight yard, an empty freight car of mass $m$ rolls along a straight level track at $1.0 \mathrm{~m} / \mathrm{s}$ and collides with an initrally stationary, fully loaded boxcar of mass $4.0 \mathrm{~m}$. The two cars couple together on collision. (a) What is the speed of the two cars after the collision? (b) Suppose instead that the two cars are at rest after the collision. With what speed was the loaded boxcar moving before the collision if the empty one was moving at $1.0 \mathrm{~m} / \mathrm{s}$ ?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 42

A $0.020-\mathrm{kg}$ ballet traveling at $200.0 \mathrm{~m} / \mathrm{s}$ east hits a motionless $2.0-\mathrm{kg}$ block and bounces off it, retracing its original path with a velocity of $100.0 \mathrm{~m} / \mathrm{s}$ west. What is the final velocity of the block? Assume the block rests on a perfectly frictionless horizontal surface.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 43

A block of wood of mass $0.95 \mathrm{~kg}$ is initially at rest. $\mathrm{A}$ bullet of mass $0.050 \mathrm{~kg}$ traveling at $100.0 \mathrm{~m} / \mathrm{s}$ strikes the block and becomes embedded in it. With what speed do the block of wood and the ballet move just after the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 44

A $0.020-\mathrm{kg}$ bullet is shot horizontally and collides with a $2.00-\mathrm{kg}$ block of wood. The bullet embeds in the block and the block slides along a horizontal surface for $1.50 \mathrm{~m}$. If the coefficient of kinetic friction between the block and surface is $0.400$, what was the original speed of the bullet?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 45

A $2.0-\mathrm{kg}$ block is moving to the right at $1.0 \mathrm{~m} / \mathrm{s}$ just before it strikes and sticks to a $1.0-\mathrm{kg}$ block initially at rest. What is the total momentum of the two blocks after the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 46

A $75-\mathrm{kg}$ man is at rest on ice skates. A $0.20-\mathrm{kg}$ ball is thrown to him. The ball is moving horizontally at $25 \mathrm{~m} / \mathrm{s}$ just before the man catches it. How fast is the man mowing just after he catches the ball?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 47

A BMW of mass $2.0 \times 10^{3} \mathrm{~kg}$ is traveling at $42 \mathrm{~m} / \mathrm{s}$. It approaches a $1.0 \times 10^{3} \mathrm{~kg}$ Volkswagen going $25 \mathrm{~m} / \mathrm{s}$ in the same direction and strikes it in the rear. Neither driver applies the brakes. Neglect the relatively small frictional forces on the cars due to the road and due to air resistance. (a) If the collision slows the BMW down to $33 \mathrm{~m} / \mathrm{s}$, what is the speed of the VW after the collision? (b) During the collision, which car exerts a larger force on the other, or are the forces equal in magnitude? Explain.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 48

A 100 -g ball collides elastically with a 300 -g ball that is at rest. If the $100-\mathrm{g}$ ball was traveling in the positive $x$ direction at $5.00 \mathrm{~m} / \mathrm{s}$ before the collision, what are the welocities of the two balls after the collision? ( W) tutorial: elastic collision)

Vishal Gupta

Numerade Educator

Problem 49

A projectile of $1.0-\mathrm{kg}$ mass approaches a stationary body of $5.0 \mathrm{~kg}$ at $10.0 \mathrm{~m} / \mathrm{s}$ and, after colliding, rebounds in the reverse direction along the same line with a speed of $5.0 \mathrm{~m} / \mathrm{s}$. What is the speed of the $5.0-\mathrm{kg}$ body after the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 50

A $2.0-\mathrm{kg}$ object is at rest on a perfectly frictionless surface when it is hit by a $3.0-\mathrm{kg}$ object moving at $8.0 \mathrm{~m} / \mathrm{s}$. If the two objects are stuck together after the collision, what is the speed of the combination?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 51

A spring of negligible mass is compressed between two blocks, A and B, which are at rest on a frictionless horizontal surface at a distance of $1.0 \mathrm{~m}$ from a wall on the left and $3.0 \mathrm{~m}$ from a wall on the right. The sizes of the blocks and spring are small. When the spring is released, block A moves toward the left wall and strikes it at the same instant that block B strikes the right wall. The mass of $\mathrm{A}$ is $0.60 \mathrm{~kg}$. What is the mass of $\mathrm{B}$ ?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 52

Two identical gliders on an air track are held together by a piece of string, compressing a spring between the gliders. While they are moving to the right at a common speed of $0.50 \mathrm{~m} / \mathrm{s}$, someone holds a match under the string and burns it, letting the spring force the gliders apart. One glider is then observed to be moving to the right at $1.30 \mathrm{~m} / \mathrm{s}$. (a) What velocity does the other glider have? (b) Is the total kinetic energy of the two gliders after the collision greater than, less than, or equal to the total kinetic energy before the collision? If greater, where did the extra energy come from? If less, where did the "lost" energy go?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 53

A $0.010-\mathrm{kg}$ bullet traveling horizontally at $400.0 \mathrm{~m} / \mathrm{s}$ strikes a $4.0-\mathrm{kg}$ block of wood sitting at the edge of a table. The bullet is lodged into the wood. If the table height is $1.2 \mathrm{~m}$, how far from the table does the block. hit the floor?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 54

Two objects with masses $m_{1}$ and $m_{2}$ approach each other with equal and opposite momenta so that the total momentum is zero. Show that, if the collision is elastic, the final speed of each object must be the same as its initial speed. (The final velocity of each object is nor the same as its initial velocity, however.)

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 55

A $6.0$ -kg object is at rest on a perfectly frictionless surface when it is struck head-on by a $2.0-\mathrm{kg}$ object moving at $10 \mathrm{~m} / \mathrm{s}$. If the collision is perfectly elastic, what is the speed of the $6.0-\mathrm{kg}$ object after the collision? [Hint:
You will need two equations.]

Vishal Gupta

Numerade Educator

Problem 56

Use the result of Problem 54 to show that in $a n y$ elastic collision between two objects, the relative speed of the two is the same before and after the collision. [Hints: Look at the collision in its CM frame-the reference frame in which the cM is at rest. The relative speed of two objects is the same in any inertial reference frame.]

Vishal Gupta

Numerade Educator

Problem 57

A firecracker is tossed straight up into the air. It explodes into three pieces of equal mass just as it reaches the highest point. Two pieces mowe off at $120 \mathrm{~m} / \mathrm{s}$ at right angles to each other. How fast is the third piece moving?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 58

Body A of mass $M$ has an original velocity of $6.0 \mathrm{~m} / \mathrm{s}$ in the $+x$ -direction toward a stationary body (body B) of the same mass. After the collision, body A has velocity components of $1.0 \mathrm{~m} / \mathrm{s}$ in the $+x$ -direction and $2.0 \mathrm{~m} / \mathrm{s}$ in the $+y$ -direction. What is the magnitude of body $\underline{\text { B's }}$ velocity after the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 59

(a) With reference to Practice Problem $7.11$, find the momentum change of the ball of mass $m_{1}$ during the collision. Give your answer in $x$ - and $y$ -component form; express the components in terms of $m_{1}$ and $v_{i}$.
(b) Repeat for the ball of mass $m_{2}$. How are the momentum changes related?

Vishal Gupta

Numerade Educator

Problem 60

A hockey puck moving at $0.45 \mathrm{~m} / \mathrm{s}$ collides elastically with another puck that was at rest. The pucks have equal mass. The first puck is deflected $37^{\circ}$ to the right and moves off at $0.36 \mathrm{~m} / \mathrm{s}$. Find the speed and direction of the second puck after the collision.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 61

Puck 1 sliding along the $x$ -axis strikes stationary puck 2 of the same mass. After the elastic collision, puck 1 moves off at speed $v_{1 f}$ in the direction $60.0^{\circ}$ above the $x$ -axis; puck 2 moves off at speed $v_{2 f}$ in the direction $30.0^{\circ}$ below the $x$ -axis. Find $v_{2 f}$ in terms of $v_{1 f}$

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 62

Block A, with a mass of $220 \mathrm{~g}$, is traveling north on a frictionless surface with a speed of $5.0 \mathrm{~m} / \mathrm{s} .$ Block $\mathrm{B}$, with a mass of $300 \mathrm{~g}$ travels west on the same surface until it collides with A. After the collision, the blocks move off together with a velocity of $3.13 \mathrm{~m} / \mathrm{s}$ at an angle of $42.5^{\circ}$ to the north of west. What was B's speed just before the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 63

A projectile of mass $2.0 \mathrm{~kg}$ approaches a stationary target body at $5.0 \mathrm{~m} / \mathrm{s}$. The projectile is deflected through an angle of $60.0^{\circ}$ and its speed after the collision is $3.0 \mathrm{~m} / \mathrm{s}$. What is the magnitude of the momentum of the target body after the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 64

A $1500-\mathrm{kg}$ car moving east at $17 \mathrm{~m} / \mathrm{s}$ collides with a $1800-\mathrm{kg}$ car moving south at $15 \mathrm{~m} / \mathrm{s}$ and the two cars stick together. (a) What is the velocity of the cars right after the collision? (b) How much kinetic energy was converted to another form during the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 65

A car with a mass of $1700 \mathrm{~kg}$ is traveling directly northeast ( $45^{\circ}$ between north and east) at a speed of $14 \mathrm{~m} / \mathrm{s}$ (31 mph), and collides with a smaller car with a mass of $1300 \mathrm{~kg}$ that is traveling directly south at a speed of $18 \mathrm{~m} / \mathrm{s}$ (40 mph). The two cars stick together during the collision. With what speed and direction does the tangled mess of metal move right after the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 66

In a nuclear reactor, a neutron moving at speed $v_{\mathrm{i}}$ in the positive $x$ -direction strikes a deuteron, which is at rest. The neutron is deflected by $90.0^{\circ}$ and moves off with speed $v_{f} / \sqrt{3}$ in the positive $y$ -direction. Find the $x$ - and $y$ -components of the deuteron's velocity after the collision. (The mass of the deuteron is twice the mass of the neutron.)

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 67

Two identical pucks are on an air table. Puck $\mathrm{A}$ has an initial velocity of $2.0 \mathrm{~m} / \mathrm{s}$ in the $+x$ -direction. Puck $\mathrm{B}$ is at rest. Puck A collides elastically with puck B and A moves off at $1.0 \mathrm{~m} / \mathrm{s}$ at an angle of $60^{\circ}$ above the $x$ -axis. What is the speed and direction of puck $\mathrm{B}$ after the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 68

In a circus trapeze act, two acrobats actually fly through the air and grab on to one another, then together grab a swinging bar. One acrobat, with a mass of $60 \mathrm{~kg}$, is moving at $3.0 \mathrm{~m} / \mathrm{s}$ at an angle of $10^{\circ}$ above the horizontal and the other, with a mass of $80 \mathrm{~kg}$, is approaching her with a speed of $2.0 \mathrm{~m} / \mathrm{s}$ at an angle of $20^{\circ}$ above the horizontal. What is the direction and speed of the acrobats right after they grab on to each other? Let the positive $x$ -axis be in the horizontal direction and assume the first acrobat has positive velocity components in the positive $x$ - and $y$ -directions.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 69

Two African swallows fly toward one another, carrying coconuts. The first swallow is flying north horizontally with a speed of $20 \mathrm{~m} / \mathrm{s}$. The second swallow is flying at the same height as the first and in the opposite direction with a speed of $15 \mathrm{~m} / \mathrm{s} .$ The mass of the first swallow is $0.270 \mathrm{~kg}$ and the mass of his coconut is $0.80 \mathrm{~kg}$. The second swallow's mass is $0.220 \mathrm{~kg}$ and her coconut's mass is $0.70 \mathrm{~kg}$. The swallows collide and lose their coconuts. Immediately after the collision, the $0.80-\mathrm{kg}$ coconut travels $10^{\circ}$ west of south with a speed of $13 \mathrm{~m} / \mathrm{s}$ and the $0.70-\mathrm{kg}$ coconut moves $30^{\circ}$ east of north with a speed of $14 \mathrm{~m} / \mathrm{s}$. The two birds are tangled up with one another and stop flapping their wings as they travel off together. What is the velocity of the birds immediately after the collision?

Vishal Gupta

Numerade Educator

Problem 70

A sled of mass $5.0 \mathrm{~kg}$ is coasting along on a frictionless ice-covered lake at a constant speed of $1.0 \mathrm{~m} / \mathrm{s}$. A $1.0-\mathrm{kg}$ book is dropped vertically onto the sled. At what speed does the sled move once the book is on it?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 71

An automobile weighing $13.6 \mathrm{kN}$ is moving at $17.0 \mathrm{~m} / \mathrm{s}$ when it collides with a stopped car weighing $9.0 \mathrm{kN}$. If they lock bumpers and move off together, what is their speed just after the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 72

For a system of three particles moving along a line, an observer in a laboratory measures the following masses and velocities. What is the velocity of the $\mathrm{CM}$ of the system?
$$
\begin{array}{cc}
\hline \text { Mass }(\mathbf{k g}) & v_{x}(\mathbf{m} / \mathrm{s}) \\
\hline 3.0 & +290 \\
5.0 & -120 \\
2.0 & +52 \\
\hline
\end{array}
$$

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 73

An intergalactic spaceship is traveling through space far from any planets or stars, where no human has gone before. The ship carries a crew of 30 people (of total mass $2.0 \times 10^{3} \mathrm{~kg}$ ). If the speed of the spaceship is $1.0 \times 10^{5} \mathrm{~m} / \mathrm{s}$ and its mass (excluding the crew) is $4.8 \times 10^{4} \mathrm{~kg}$, what is the magnitude of the total momentum of the ship and the crew?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 74

A baseball player pitches a fastball toward home plate at a speed of $41 \mathrm{~m} / \mathrm{s}$. The batter swings, connects with the ball of mass $145 \mathrm{~g}$, and hits it so that the ball leaves the bat with a speed of $37 \mathrm{~m} / \mathrm{s}$. Assume that the ball is moving horizontally just before and just after the collision with the bat. (a) What is the magnitude of the change in momentum of the ball? (b) What is the impulse delivered to the ball by the bat? (c) If the bat and ball are in contact for $3.0 \mathrm{~ms}$, what is the magnitude of the average force exerted on the ball by the bat?

Narayan Hari

Numerade Educator

Problem 75

A tennis ball of mass $0.060 \mathrm{~kg}$ is served. It strikes the ground with a velocity of $54 \mathrm{~m} / \mathrm{s}(120 \mathrm{mi} / \mathrm{h})$ at an angle of $22^{\circ}$ below the horizontal. Just after the bounce it is moving at $53 \mathrm{~m} / \mathrm{s}$ at an angle of $18^{\circ}$ above the horizontal. If the interaction with the ground lasts $0.065 \mathrm{~s}$, what average force did the ground exert on the ball?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 76

A uniform rod of length $30.0 \mathrm{~cm}$ is bent into the shape of an inverted U. Each of the three sides is of length $10.0 \mathrm{~cm}$. Find the location, in $x$ - and $y$ -coordinates, of the $\mathrm{CM}$ as measured from the origin.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 77

A child places 12 wooden blocks together, as shown in the figure. If each block has the same mass and density, where is the $\mathrm{CM}$ of these blocks? Each block is a cube with sides of $1.0$ inch length. The origin of the coordinate system is at the center of the farthest block to the left.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 78

To contain some unruly demonstrators, the riot squad approaches with fire hoses. Suppose that the rate of flow of water through a fire hose is $24 \mathrm{~kg} / \mathrm{s}$ and the stream of water from the hose moves at $17 \mathrm{~m} / \mathrm{s}$. What force is exerted by such a stream on a person in the crowd? Assume that the water comes to a dead stop against the demonstrator's chest.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 79

An inexperienced catcher catches a $130 \mathrm{~km} / \mathrm{h}$ fastball of mass $140 \mathrm{~g}$ within $1 \mathrm{~ms}$, whereas an experienced catcher slightly retracts his hand during the catch, extending the stopping time to $10 \mathrm{~ms}$. What are the average forces imparted to the two gloved hands during the catches?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 80

A stationary $0.1-\mathrm{g}$ fly encounters the windshield of a $1000-\mathrm{kg}$ automobile traveling at $100 \mathrm{~km} / \mathrm{h}$. (a) What is the change in momentum of the car due to the fly? (b) What is the change of momentum of the fly due to the car?
(c) Approximately how many flies does it take to reduce the car's speed by $1 \mathrm{~km} / \mathrm{h}$ ?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 81

A $0.15-\mathrm{kg}$ baseball is pitched with a speed of $35 \mathrm{~m} / \mathrm{s}$ $(78 \mathrm{mph}) .$ When the ball hits the catcher's glove, the glove moves back by $5.0 \mathrm{~cm}(2$ in. $)$ as it stops the ball.
(a) What was the change in momentum of the baseball? (b) What impulse was applied to the baseball?
(c) Assuming a constant acceleration of the ball, what was the average force applied by the catcher's glove?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 82

A projectile of mass $2.0 \mathrm{~kg}$ approaches a stationary target body at $8.0 \mathrm{~m} / \mathrm{s}$. The projectile is deflected through an angle of $90.0^{\circ}$ and its speed after the collision is $6.0 \mathrm{~m} / \mathrm{s}$. What is the speed of the target body after the collision if the collision is perfectly elastic?

Vishal Gupta

Numerade Educator

Problem 83

A radioactive nucleus is at rest when it spontaneously decays by emitting an electron and neutrino. The momentum of the electron is $8.20 \times 10^{-19} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$ and it is directed at right angles to that of the neutrino. The neutrino's momentum has magnitude $5.00 \times 10^{-19} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$
(a) In what direction does the newly formed (daughter) nucleus recoil? (b) What is its momentum?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 84

A $60.0-\mathrm{kg}$ woman stands at one end of a $120-\mathrm{kg}$ raft that is $6.0 \mathrm{~m}$ long. The other end of the raft is $0.50 \mathrm{~m}$ from a pier. (a) The woman walks toward the pier until she gets to the other end of the raft and stops there. Now what is the distance between the raft and the pier? (b) In
(a), how far did the woman walk (relative to the pier)?

Paul A.

California State Polytechnic University, Pomona

Problem 85

A police officer is investigating the scene of an accident where two cars collided at an intersection. One car with a mass of $1100 \mathrm{~kg}$ moving west had collided with a $1300-\mathrm{kg}$ car moving north. The two cars, stuck together, skid at an angle of $30^{\circ}$ north of west for a distance of $17 \mathrm{~m}$. The coefficient of kinetic friction between the tires and the road is $0.80$. The speed limit for each car was $70 \mathrm{~km} / \mathrm{h}$. Was either car speeding?

Vishal Gupta

Numerade Educator

Problem 86

A jet plane is flying at $130 \mathrm{~m} / \mathrm{s}$ relative to the ground. There is no wind. The engines take in $81 \mathrm{~kg}$ of air per second. Hot gas (burned fuel and air) is expelled from the engines at high speed. The engines provide a forward force on the plane of magnitude $6.0 \times 10^{4} \mathrm{~N}$. At what speed relative to the ground is the gas being expelled? [Hint: Look at the momentum change of the air taken in by the engines during a time interval $\Delta t .]$ This calculation is approximate since we are ignoring the $3.0 \mathrm{~kg}$ of fuel consumed and expelled with the air each second.

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 87

Within cells, small organelles containing newly synthesized proteins are transported along microtubules by tiny molecular motors called kinesins. What force does a kinesin molecule need to deliver in order to accelerate an organelle with mass $0.01$ pg $\left(10^{-17} \mathrm{~kg}\right)$ from 0 to $1 \mu \mathrm{m} / \mathrm{s}$ within a time of $10 \mu \mathrm{s}$ ?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 88

The pendulum bobs in the figure are made of soft clay so that they stick together after impact. The mass of bob A is half that of bob B. Bob B is initially at rest. What is the ratio of the kinetic energy of the combined bobs, just after impact, to the kinetic energy of bob A just before impact?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 89

The pendulum bobs in the figure are made of soft clay so that they stick together after impact. The mass of bob A is half that of bob $B$. Bob $B$ is initially at rest. If bob A is released from a height $h$ above its lowest point, what is the maximum height attained by bobs $\mathrm{A}$ and $\mathrm{B}$ after the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 90

A flat, circular metal disk of uniform thickness has a radius of $3.0 \mathrm{~cm} .$ A hole is drilled in the disk that is $1.5 \mathrm{~cm}$ in radius. The hole is tangent to one side of the disk. Where is the $\mathrm{CM}$ of the disk now that the hole has been drilled? [Hint: The original disk (before the hole is drilled) can be thought of as having two pieces - the disk with the hole plus the smaller disk of metal drilled out. Write an equation that expresses $x_{\mathrm{CM}}$ of the original disk in terms of the $x_{\mathrm{CM}}$ 's of the two pieces. Since the thickness is uniform, the mass of any piece is proportional to its area.]

Vishal Gupta

Numerade Educator

Problem 91

Two pendulum bobs have equal masses and lengths $(5.1 \mathrm{~m}) .$ Bob $\mathrm{A}$ is initially held horizontally while bob $\mathrm{B}$ hangs vertically at rest. Bob A is released and collides elastically with bob B. How fast is bob B moving immediately after the collision?

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 92

Two identical gliders, each with elastic bumpers and mass $0.10 \mathrm{~kg}$, are on a horizontal air track. Friction is negligible. Glider 2 is stationary. Glider 1 moves toward glider 2 from the left with a speed of $0.20 \mathrm{~m} / \mathrm{s}$. They collide. After the collision, what are the velocities of glider 1 and glider $2 ?$

Joseph Petrullo

Joseph Petrullo

Numerade Educator

Problem 93

A radium nucleus (mass $226 \mathrm{u}$ ) at rest decays into a radon nucleus (symbol Rn, mass $222 \mathrm{u}$ ) and an alpha particle (symbol $\alpha$, mass $4 \mathrm{u}$ ). (a) Find the ratio of the speeds $v_{a} / v_{\mathrm{Rn}}$ after the decay. (b) Find the ratio of the magnitudes of the momenta $p_{w} / p_{\mathrm{Rn}}$. (c) Find the ratio of the kinetic energies $K_{\alpha} / K_{\mathrm{Rn}}$. (Note: "u" is a unit of mass; it is not necessary to convert it to $\mathrm{kg}$.)

Joseph Petrullo

Joseph Petrullo

Numerade Educator