Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 9

Linear Momentum and Collisions - all with Video Answers

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Chapter Questions

Problem 1

A $3.00-\mathrm{kg}$ particle has a velocity of $(3.00 \hat{\mathrm{i}}-4.00 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ . (a) Find its $x$ and $y$ components of momentum. the magnitude and direction of its momentum.

Sheh Lit Chang

Sheh Lit Chang

University of Washington

Problem 2

A 65.0-kg boy and his 40.0-kg sister, both wearing roller blades, face each other at rest. The girl pushes the boy hard, sending him backward with velocity 2.90 m/s toward the west. Ignore friction. (a) Describe the subsequent motion of the girl. (b) How much chemical energy is converted into mechanical energy in the girl’s muscles? (c) Is the momentum of the boy-girl system conserved in the pushing-apart process? How can it be, with large forces acting? How can it be, with no motion beforehand and plenty of motion afterward?

Sheh Lit Chang

University of Washington

Problem 3

How fast can you set the Earth moving? In particular, when you jump straight up as high as you can, what is the order of magnitude of the maximum recoil speed that you give to the Earth? Model the Earth as a perfectly solid object. In your solution, state the physical quantities you take as data and the values you measure or estimate for them.

Sheh Lit Chang

University of Washington

Problem 4

Two blocks of masses M and 3M are placed on a horizontal, frictionless surface. A light spring is attached to one of them, and the blocks are pushed together with the spring between them (Fig. P9.4). A cord initially holding the blocks together is burned; after that happens, the block of mass 3M moves to the right with a speed of 2.00 m/s. (a) What is the velocity of the block of mass M? (b) Find the system’s original elastic potential energy, taking M 0.350 kg. (c) Is the original energy in the spring or in the cord? Explain your answer. (d) Is momentum of the system conserved in the bursting-apart process? How can it be, with large forces acting? How can it be, with no motion beforehand and plenty of motion afterward?

Sheh Lit Chang

University of Washington

Problem 5

(a) A particle of mass $m$ moves with momentum of magnitude $p .$ Show that the kinetic energy of the particle is given by $K=p^{2} / 2 m$ . (b) Express the magnitude of the particle's momentum in terms of its kinetic energy and mass.

Sheh Lit Chang

University of Washington

Problem 6

A friend claims that as long as he has his seat belt on, he can hold on to a 12.0-kg child in a 60.0 mi/h head-on collision with a brick wall in which the car passenger compartment comes to a stop in 0.050 0 s. Is his claim true? Explain why he will experience a violent force during the collision, tearing the child from his arms. Evaluate the size of this force. (A child should always be in a toddler seat secured with a seat belt in the back seat of a car.)

Sheh Lit Chang

University of Washington

Problem 7

An estimated force–time curve for a baseball struck by a bat is shown in Figure P9.7. From this curve, determine (a) the impulse delivered to the ball, (b) the average force exerted on the ball, and (c) the peak force exerted on the ball.

Sheh Lit Chang

University of Washington

Problem 8

A ball of mass 0.150 kg is dropped from rest from a height of 1.25 m. It rebounds from the floor to reach a height of 0.960 m. What impulse was given to the ball by the floor?

Salamat Ali

Numerade Educator

Problem 9

A 3.00-kg steel ball strikes a wall with a speed of 10.0 $\mathrm{m} / \mathrm{s}$ at an angle of $60.0^{\circ}$ with the surface. It bounces off with the same speed and angle (Fig. P9.9). If the ball is in contact with the wall for 0.200 $\mathrm{s}$ , what is the average force exerted by the wall on the ball?

Sheh Lit Chang

University of Washington

Problem 10

A tennis player receives a shot with the ball (0.060 0 kg) traveling horizontally at 50.0 m/s and returns the shot with the ball traveling horizontally at 40.0 m/s in the opposite direction. (a) What is the impulse delivered to the ball by the tennis racquet? (b) What work does the racquet do on the ball?

Sheh Lit Chang

University of Washington

Problem 11

The magnitude of the net force exerted in the x direction on a 2.50-kg particle varies in time as shown in Figure P9.11. Find (a) the impulse of the force, (b) the final velocity the particle attains if it is originally at rest, (c) its final velocity if its original velocity is 2.00 m/s, and (d) the average force exerted on the particle for the time interval between 0 and 5.00 s.

Sheh Lit Chang

University of Washington

Problem 12

A force platform is a tool used to analyze the performance of athletes by measuring the vertical force that the athlete exerts on the ground as a function of time. Starting from rest, a 65.0-kg athlete jumps down onto the platform from a height of 0.600 m. While she is in contact with the platform during the time interval $0< t< 0.800 \mathrm{s}$ , the force she exerts on it is described by the function
$$F=(9200 \mathrm{N} / \mathrm{s}) t-\left(11500 \mathrm{N} / \mathrm{s}^{2}\right) t^{2}$$
(a) What impulse did the athlete receive from the platform? (b) With what speed did she reach the platform? (c) With what speed did she leave it? (d) To what height did she jump upon leaving the platform?

Sheh Lit Chang

University of Washington

Problem 13

A glider of mass m is free to slide along a horizontal air track. It is pushed against a launcher at one end of the track. Model the launcher as a light spring of force constant k compressed by a distance x. The glider is released from rest. (a) Show that the glider attains a speed of $v=$ $x(k / m)^{1 / 2}$ . (b) Does a glider of large or of small mass attain a greater speed? (c) Show that the impulse im- parted to the glider is given by the expression $x(k m)^{1 / 2}$ . (d) Is a greater impulse imparted to a large or a small mass? (e) Is more work done on a large or a small mass?

Sheh Lit Chang

University of Washington

Problem 14

Water falls without splashing at a rate of 0.250 L/s from a height of 2.60 m into a 0.750-kg bucket on a scale. If the bucket is originally empty, what does the scale read 3.00 s after water starts to accumulate in it?

Sheh Lit Chang

University of Washington

Problem 15

A 10.0-g bullet is fired into a stationary block of wood $(m=5.00 \mathrm{kg}) .$ The bullet imbeds into the block. The speed of the bullet-plus-wood combination immediately after the collision is 0.600 $\mathrm{m} / \mathrm{s}$ . What was the original speed of the bullet?

Robert Daine

Numerade Educator

Problem 16

A railroad car of mass $2.50 \times 10^{4} \mathrm{kg}$ is moving with a speed of 4.00 $\mathrm{m} / \mathrm{s}$ . It collides and couples with three other coupled railroad cars, each of the same mass as the single car and moving in the same direction with an initial speed of 2.00 $\mathrm{m} / \mathrm{s}$ . (a) What is the speed of the four cars immediately after the collision? (b) How much energy is transformed into internal energy in the collision?

Sheh Lit Chang

University of Washington

Problem 17

Four railroad cars, each of mass $2.50 \times 10^{4} \mathrm{kg},$ are coupled together and coasting along horizontal tracks at speed $v_{i}$ toward the south. A very strong movie actor, riding on the second car, uncouples the front car and gives it a big push, increasing its speed to 4.00 $\mathrm{m} / \mathrm{s}$ southward. The remaining three cars continue moving south, now at 2.00 $\mathrm{m} / \mathrm{s}$ (a) Find the initial speed of the cars. (b) How much work did the actor do? (c) State the relationship between the process described here and the process in Problem $16 .$

Sheh Lit Chang

University of Washington

Problem 18

As shown in Figure $\mathrm{P} 9.18$ (page $262 ),$ a bullet of mass $m$ and speed $v$ passes completely through a pendulum bob of mass $M$ . The bullet emerges with a speed of $v / 2 .$ The pendulum bob is suspended by a stiff rod of length $\ell$ and negligible mass. What is the minimum value of $v$ such that the pendulum bob will barely swing through a complete vertical circle?

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 19

Two blocks are free to slide along the frictionless wooden track $A B C$ shown in Figure $\mathrm{P} 9.19 .$ The block of mass $m_{1}=$ 5.00 $\mathrm{kg}$ is released from $A .$ Protruding from its front end is the north pole of a strong magnet, which is repelling the north pole of an identical magnet, embedded in the back end of the block of mass $m_{2}=10.0 \mathrm{kg}$ , initially at rest. The two blocks never touch. Calculate the maximum height to which $m_{1}$ rises after the elastic collision.

Sheh Lit Chang

University of Washington

Problem 20

A tennis ball of mass 57.0 g is held just above a basketball of mass 590 g. With their centers vertically aligned, both are released from rest at the same moment, to fall through a distance of 1.20 m, as shown in Figure P9.20. (a) Find the magnitude of the downward velocity with which the basketball reaches the ground. Assume an elastic collision with the ground instantaneously reverses the velocity of the basketball while the tennis ball is still moving down. Next, the two balls meet in an elastic collision. (b) To what height does the tennis ball rebound?

Vishal Gupta

Numerade Educator

Problem 21

A 45.0-kg girl is standing on a plank that has a mass of 150 kg. The plank, originally at rest, is free to slide on a frozen lake that constitutes a flat, frictionless supporting surface. The girl begins to walk along the plank at a constant velocity of 1.50 m/s relative to the plank. (a) What is her velocity relative to the ice surface? (b) What is the velocity of the plank relative to the ice surface?

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 22

A 7.00-g bullet, when fired from a gun into a 1.00-kg block of wood held in a vise, penetrates the block to a depth of 8.00 cm. This block of wood is placed on a frictionless horizontal surface, and a second 7.00-g bullet is fired from the gun into the block. To what depth does the bullet penetrate the block in this case?

Sheh Lit Chang

University of Washington

Problem 23

A neutron in a nuclear reactor makes an elastic head-on collision with the nucleus of a carbon atom initially at rest. (a) What fraction of the neutron’s kinetic energy is transferred to the carbon nucleus? (b) The initial kinetic energy of the neutron is $1.60 \times 10^{-13} \mathrm{J}$ . Find its final kinetic energy and the kinetic energy of the carbon nucleus after the collision. (The mass of the carbon nucleus is nearly 12.0 times the mass of the neutron.)

Prashant Bana

Numerade Educator

Problem 24

Three carts of masses 4.00 kg, 10.0 kg, and 3.00 kg move on a frictionless, horizontal track with speeds of 5.00 m/s, 3.00 m/s, and 4.00 m/s as shown in Figure P9.24. Velcro couplers make the carts stick together after colliding. Find the final velocity of the train of three carts. (b) What If? Does your answer require that all the carts collide and stick together at the same moment? What if they collide in a different order?

Sheh Lit Chang

University of Washington

Problem 25

A 12.0-g wad of sticky clay is hurled horizontally at a 100-g wooden block initially at rest on a horizontal surface. The clay sticks to the block. After impact, the block slides 7.50 m before coming to rest. If the coefficient of friction between the block and the surface is 0.650, what was the speed of the clay immediately before impact?

Vishal Gupta

Numerade Educator

Problem 26

In an American football game, a 90.0-kg fullback running east with a speed of 5.00 m/s is tackled by a 95.0-kg opponent running north with a speed of 3.00 m/s. (a) Explain why the successful tackle constitutes a perfectly inelastic collision. (b) Calculate the velocity of the players immediately after the tackle. (c) Determine the mechanical energy that disappears as a result of the collision. Account for the missing energy.

Sheh Lit Chang

University of Washington

Problem 27

A billiard ball moving at 5.00 $\mathrm{m} / \mathrm{s}$ strikes a stationary ball of the same mass. After the collision, the first ball moves, at 4.33 $\mathrm{m} / \mathrm{s}$ at an angle of $30.0^{\circ}$ with respect to the original line of motion. Assuming an elastic collision (and ignoring friction and rotational motion), find the struck ball's velocity after the collision.

Sanjeev Kumar

Numerade Educator

Problem 28

Two automobiles of equal mass approach an intersection. One vehicle is traveling with velocity 13.0 m/s toward the east, and the other is traveling north with speed $v_{2 i}$ . Neither driver sees the other. The vehicles collide in the intersection and stick together, leaving parallel skid marks at an angle of $55.0^{\circ}$ north of east. The speed limit for both roads is $35 \mathrm{mi} / \mathrm{h},$ and the driver of the northward-moving vehicle claims he was within the speed limit when the collision occurred. Is he telling the truth? Explain your reasoning.

Sheh Lit Chang

University of Washington

Problem 29

Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed of 5.00 m/s. After the collision, the orange disk moves along a direction that makes an angle of $37.0^{\circ}$ with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk.

Sheh Lit Chang

University of Washington

Problem 30

Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed $v_{i}$ . After the collision, the orange disk moves along a direction that makes an angle $\theta$ with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk.

Robert Daine

Numerade Educator

Problem 31

An object of mass 3.00 $\mathrm{kg}$ , moving with an initial velocity of $5.00 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s},$ collides with and sticks to an object of mass 2.00 $\mathrm{kg}$ with an initial velocity of $-3.00 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}$ . Find the final velocity of the composite object.

Sheh Lit Chang

University of Washington

Problem 32

Two particles with masses $m$ and 3$m$ are moving toward each other along the $x$ axis with the same initial speeds $v_{i}$. Particle $m$ is traveling to the left, and particle 3$m$ is traveling to the right. They undergo an elastic, glancing collision such that particle $m$ is moving downward after the collision at a right angle from its initial direction. (a) Find the final speeds of the two particles. (b) What is the angle $\theta$ at which the particle 3$m$ is scattered?

Sheh Lit Chang

University of Washington

Problem 33

An unstable atomic nucleus of mass $17.0 \times 10^{-27} \mathrm{kg}$ initially at rest disintegrates into three particles. One of the particles, of mass $5.00 \times 10^{-27} \mathrm{kg},$ moves in the $y$ direction with a speed of $6.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Another particle, of mass $8.40 \times 10^{-27} \mathrm{kg}$ , moves in the $x$ direction with a speed of $4.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Find $(\mathrm{a})$ the velocity of the third particle and $(\mathrm{b})$ the total kinetic energy increase in the process.

Robert Daine

Numerade Educator

Problem 34

The mass of the blue puck in Figure P9.34 is 20.0% greater than the mass of the green puck. Before colliding, the pucks approach each other with momenta of equal magnitudes and opposite directions, and the green puck has an initial speed of 10.0 m/s. Find the speed each puck has after the collision if half the kinetic energy of the system becomes internal energy during the collision.

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 35

Four objects are situated along the $y$ axis as follows: a 2.00 -kg object is located at $+3.00 \mathrm{m},$ a $3.00-\mathrm{kg}$ object is at $+2.50 \mathrm{m},$ a $2.50-\mathrm{kg}$ object is at the origin, and a $4.00-\mathrm{kg}$ object is at $-0.500 \mathrm{m}$ . Where is the center of mass of these objects?

Robert Daine

Numerade Educator

Problem 36

The mass of the Earth is $5.98 \times 10^{24} \mathrm{kg}$ , and the mass of the Moon is $7.36 \times 10^{22} \mathrm{kg}$ . The distance of separation, measured between their centers, is $3.84 \times 10^{8} \mathrm{m} .$ Locate the center of mass of the Earth-Moon system as measured from the center of the Earth.

Sheh Lit Chang

University of Washington

Problem 37

A uniform piece of sheet steel is shaped as shown in Figure P9.37. Compute the x and y coordinates of the center of mass of the piece.

Sheh Lit Chang

University of Washington

Problem 38

(a) Consider an extended object whose different portions have different elevations. Assume the free-fall acceleration is uniform over the object. Prove that the gravitational potential energy of the object-Earth system is given by $U_{g}=M g y_{\mathrm{CM}},$ where $M$ is the total mass of the object and $y_{\mathrm{CM}}$ is the elevation of its center of mass above the chosen reference level. (b) Calculate the gravitational potential energy associated with a ramp constructed on level ground with stone with density 3800 $\mathrm{kg} / \mathrm{m}^{3}$ and every- where 3.60 $\mathrm{m}$ wide. In a side view, the ramp appears as a right triangle with height 15.7 $\mathrm{m}$ at the top end and base 64.8 $\mathrm{m}$ (Fig. P9.38).

Sheh Lit Chang

University of Washington

Problem 39

A rod of length 30.0 $\mathrm{cm}$ has linear density (mass-per-length) given by
$$
\lambda=50.0 \mathrm{g} / \mathrm{m}+20.0 x \mathrm{g} / \mathrm{m}^{2}
$$
where $x$ is the distance from one end, measured in meters. (a) What is the mass of the rod? (b) How far from the $x=0$ end is its center of mass?

Mayukh Banik

Numerade Educator

Problem 40

In the 1968 Summer Olympic Games, University of Oregon high jumper Dick Fosbury introduced a new technique of high jumping called the “Fosbury flop.” It contributed to raising the world record by about 30 cm and is presently used by nearly every world-class jumper. In this technique, the jumper goes over the bar face up while arching his back as much as possible as shown in Figure P9.40a. This action places his center of mass outside his body, below his back. As his body goes over the bar, his center of mass passes below the bar. Because a given energy input implies a certain elevation for his center of mass, the action of arching his back means his body is higher than if his back were straight. As a model, consider the jumper as a thin, uniform rod of length $L .$ When the rod is straight, its center of mass is at its center. Now bend the rod in a circular arc so that it subter. Now bend $90.0^{\circ}$ at the center of the arc as shown in Figure $\mathrm{P} 9.40 \mathrm{b}$ . In this configuration, how far outside the rod is the center of mass?

Sheh Lit Chang

University of Washington

Problem 41

A $2.00-\mathrm{kg}$ particle has a velocity $(2.00 \hat{\mathbf{i}}-3.00 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s},$ and a $3.00-\mathrm{kg}$ particle has a velocity $(1.00 \hat{\mathrm{i}}+6.00 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ . Find $(\mathrm{a})$ the velocity of the center of mass and $(\mathrm{b})$ the total momentum of the system.

Sanjeev Kumar

Numerade Educator

Problem 42

The vector position of a 3.50 -g particle moving in the $x y$ plane varies in time according to $\overrightarrow{\mathbf{r}}_{1}=(3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}) t+2 \hat{\mathbf{j}} t^{2}$ . At the same time, the vector position of a 5.50 -g particle varies as $\overrightarrow{\mathbf{r}}_{2}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{i}} t^{2}-6 \hat{\mathbf{j}} t,$ where $t$ is in $s$ and $r$ is in $\mathrm{cm} .$ At $t=2.50$ s, determine (a) the vector position of the center of mass, (b) the linear momentum of the system, (c) the velocity of the center of mass, (d) the acceleration of the center of mass, and (e) the net force exerted on the two-particle system.

Efren Serra

Numerade Educator

Problem 43

Romeo (77.0 kg) entertains Juliet (55.0 kg) by playing his guitar from the rear of their boat at rest in still water, 2.70 m away from Juliet, who is in the front of the boat. After the serenade, Juliet carefully moves to the rear of the boat (away from shore) to plant a kiss on Romeo’s cheek. How far does the 80.0-kg boat move toward the shore it is facing?

Paul Gabriel

Numerade Educator

Problem 44

A ball of mass 0.200 $\mathrm{kg}$ has a velocity of $1.50 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s} ;$ a ball of mass 0.300 $\mathrm{kg}$ has a velocity of $-0.400 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s}$ . They meet in a head-on elastic collision. (a) Find their velocities after the collision. (b) Find the velocity of their center of mass before and after the collision.

Sheh Lit Chang

University of Washington

Problem 45

For a technology project, a student has built a vehicle, of total mass 6.00 kg, that moves itself. As shown in Figure P9.45, it runs on two light caterpillar tracks that pass around four light wheels. A reel is attached to one of the axles, and a cord originally wound on the reel passes over a pulley attached to the vehicle to support an elevated load. After the vehicle is released from rest, the load descends slowly, unwinding the cord to turn the axle and make the vehicle move forward. Friction is negligible in the pulley and axle bearings. The caterpillar tread does not slip on the wheels or the floor. The reel has a conical shape so that the load descends at a constant low speed while the vehicle moves horizontally across the floor with constant acceleration, reaching final velocity 3.00 m/s. (a) Does the floor impart impulse to the vehicle? If so, how much? (b) Does the floor do work on the vehicle? If so, how much? (c) Does it make sense to say that the final momentum of the vehicle came from the floor? If not, from where? (d) Does it make sense to say that the final kinetic energy of the vehicle came from the floor? If not, from where? (e) Can we say that one particular force causes the forward acceleration of the vehicle? What does cause it?

Sheh Lit Chang

University of Washington

Problem 46

A 60.0-kg person bends his knees and then jumps straight up. After his feet leave the floor his motion is unaffected by air resistance and his center of mass rises by a maximum of 15.0 cm. Model the floor as completely solid and motionless. (a) Does the floor impart impulse to the person? (b) Does the floor do work on the person? (c) With what momentum does the person leave the floor? (d) Does it make sense to say that this momentum came from the floor? Explain. (e) With what kinetic energy does the person leave the floor? (f) Does it make sense to say that this energy came from the floor? Explain.

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 47

A particle is suspended from a post on top of a cart by a light string of length $L$ as shown in Figure P9.47a. The cart and particle are initially moving to the right at constant speed $v_{i}$ , with the string vertical. The cart suddenly comes to rest when it runs into and sticks to a bumper as shown in Figure $\mathrm{P} 9.47 \mathrm{b}$ . The suspended particle swings through an angle $\theta .$ (a) Show that the original speed of the cart can be computed from $v_{i}=\sqrt{2 g L(1-\cos \theta)}$ . (b) Find the initial speed implied by $L=1.20 \mathrm{m}$ and $\theta=$ $35.0^{\circ} .$ (c) Is the bumper still exerting a horizontal force on the cart when the hanging particle is at its maximum angle from the vertical? At what moment in the observable motion does the bumper stop exerting a horizontal force on the cart?

Sheh Lit Chang

University of Washington

Problem 48

On a horizontal air track, a glider of mass m carries a shaped post. The post supports a small dense sphere, also of mass m, hanging just above the top of the glider on a cord of length L. The glider and sphere are initially at rest with the cord vertical. (Fig. P9.47a shows a cart and a sphere similarly connected.) A constant horizontal force of magnitude F is applied to the glider, moving it through
displacement x1; then the force is removed. During the time interval when the force is applied, the sphere moves through a displacement with horizontal component x2.] (a) Find the horizontal component of the velocity of the center of mass of the glider-sphere system when the force is removed. (b) After the force is removed, the glider continues to move on the track and the sphere swings back and forth, both without friction. Find an expression for the largest angle the cord makes with the vertical.

Sheh Lit Chang

University of Washington

Problem 49

Sand from a stationary hopper falls onto a moving conveyor belt at the rate of 5.00 $\mathrm{kg} / \mathrm{s}$ as shown in Figure $\mathrm{P} 9.49 .$ The conveyor belt is supported by frictionless rollers. It moves at a constant speed of 0.750 $\mathrm{m} / \mathrm{s}$ under the action of a constant horizontal external force $\overrightarrow{\mathbf{F}}_{\mathrm{ext}}$ supplied by the motor that drives the belt. Find (a) the sand's rate of change of momentum in the horizontal direction, (b) the force of friction exerted by the belt on the sand, (c) the external force $\overrightarrow{\mathbf{F}}_{\text { ext }},(\mathrm{d})$ the work done by $\overrightarrow{\mathbf{F}}_{\text { ext in }}$ 1 s, and $(\mathrm{e})$ the kinetic energy acquired by the falling sand each second due to the change in its horizontal motion. (f) Why are the answers to (d) and (e) different?

Sheh Lit Chang

University of Washington

Problem 50

Model rocket engines are sized by thrust, thrust duration, and total impulse, among other characteristics. A size C5 model rocket engine has an average thrust of 5.26 N, a fuel mass of 12.7 g, and an initial mass of 25.5 g. The duration of its burn is 1.90 s. (a) What is the average exhaust speed of the engine? (b) This engine is placed in a rocket body of mass 53.5 g. What is the final velocity of the rocket if it is fired in outer space? Assume the fuel burns at a constant rate.

Sheh Lit Chang

University of Washington

Problem 51

The first stage of a Saturn V space vehicle consumed fuel and oxidizer at the rate of $1.50 \times 10^{4} \mathrm{kg} / \mathrm{s}$ , with an exhaust speed of $2.60 \times 10^{3} \mathrm{m} / \mathrm{s}$ . (a) Calculate the thrust produced by this engine. (b) Find the acceleration the vehicle had just as it lifted off the launch pad on the Earth, taking the vehicle's initial mass as $3.00 \times 10^{6} \mathrm{kg} .$ Note: You must include the gravitational force to solve part (b).

Sheh Lit Chang

University of Washington

Problem 52

Rocket science. A rocket has total mass $M_{i}=360 \mathrm{kg}$ , including 330 $\mathrm{kg}$ of fuel and oxidizer. In interstellar space, it starts from rest at the position $x=0$ , turns on its engine at time $t=0,$ and puts out exhaust with relative speed $v_{e}=$ 1500 $\mathrm{m} / \mathrm{s}$ at the constant rate $k=2.50 \mathrm{kg} / \mathrm{s}$ . The fuel will last for an actual burn time of $330 \mathrm{kg} /(2.5 \mathrm{kg} / \mathrm{s})=132 \mathrm{s}$ but define a "projected depletion time" as $T_{p}=M_{i} / k=$ $360 \mathrm{kg} /(2.5 \mathrm{kg} / \mathrm{s})=144 \mathrm{s}$ (which would be the burn time if the rocket could use its payload and fuel tanks, and even the walls of the combustion chamber as fuel). (a) Show that during the burn the velocity of the rocket as a function of time is given by $$
v(t)=-v_{e} \ln \left(1-\frac{t}{T_{p}}\right)
$$
(b) Make a graph of the velocity of the rocket as a function of time for times running from 0 to 132 $\mathrm{s}$ . (c) Show that the acceleration of the rocket is $$a(t)=\frac{v_{e}}{T_{p}-t}$$(d) Graph the acceleration as a function of time. (e) Show
that the position of the rocket is
$$
x(t)=v_{e}\left(T_{p}-t\right) \ln \left(1-\frac{t}{T_{p}}\right)+v_{e} t
$$
(f) Graph the position during the burn as a function of time.

Sheh Lit Chang

University of Washington

Problem 53

A rocket for use in deep space is to be capable of boosting a total load (payload plus rocket frame and engine) of 3.00 metric tons to a speed of $10000 \mathrm{~m} / \mathrm{s}$.
(a) It has an engine and fuel designed to produce an exhaust speed of $2000 \mathrm{~m} / \mathrm{s}$. How much fuel plus oxidizer is required? (b) If a different fuel and engine design could give an exhaust speed of $5000 \mathrm{~m} / \mathrm{s},$ what amount of fuel and oxidizer would be required for the same task? This exhaust speed is 2.50 times higher than that in part
(a). Fxplain why the required fuel mass is 2.50 times smaller, or larger than that, or still smaller.

Mark J

Numerade Educator

Problem 54

Two gliders are set in motion on an air track. A spring of force constant $k$ is attached to the back end of the second glider. The first glider, of mass $m_{1},$ has velocity $\overrightarrow{\mathbf{v}}_{1},$ and the second glider, of mass $m_{2},$ moves more slowly, with velocity $\vec{v}_{2},$ as shown in Figure P9.54. When $m_{1}$ collides with the $\overrightarrow{\mathbf{v}} .$ In terms of $\overrightarrow{\mathbf{v}}_{1}, \overrightarrow{\mathbf{v}}_{2}, m_{1}, m_{2},$ and $k,$ find $(\mathrm{a})$ the velocity $\overrightarrow{\mathbf{v}}$ at maximum compression, $(\mathrm{b})$ the maximum compression $x_{\mathrm{max}},$ and $(\mathrm{c})$ the velocity of each glider after $m_{1}$ has lost contact with the spring.

Sheh Lit Chang

University of Washington

Problem 55

An 80.0-kg astronaut is taking a space walk to work on the engines of his ship, which is drifting through space with a constant velocity. The astronaut, wishing to get a better view of the Universe, pushes against the ship and much later finds himself 30.0 m behind the ship. Without a thruster or tether, the only way to return to the ship is to throw his 0.500-kg wrench directly away from the ship. If he throws the wrench with a speed of 20.0 m/s relative to the ship, after what time interval does the astronaut reach the ship?

Sheh Lit Chang

University of Washington

Problem 56

An aging Hollywood actor (mass 80.0 kg) has been cloned, but the genetic replica is far from perfect. The clone has a different mass m, his stage presence is poor, and he uses foul language. The clone, serving as the actor’s stunt double, stands on the brink of a cliff 36.0 m high, next to a sturdy tree. The actor stands on top of a Humvee, 1.80 m above the level ground, holding a taut rope tied to a tree branch directly above the clone. When the director calls “action,” the actor starts from rest and swings down on the rope without friction. The actor is momentarily hidden from the camera at the bottom of the arc, where he undergoes an elastic head-on collision with the clone, sending him over the cliff. Cursing vilely, the clone falls freely into the ocean below. The actor is prosecuted for making an obscene clone fall, and you are called as an expert witness at the sensational trial. (a) Find the horizontal component $R$ of the clone's displacement as it depends on $m$ . Evaluate $R$ (b) for $m=$ 79.0 $\mathrm{kg}$ and $(\mathrm{c})$ for $m=81.0 \mathrm{kg}$ (d) What value of $m$ gives a range of 30.0 $\mathrm{m}$ ? (e) What is the maximum possible value for $R,$ and $(f)$ to what value of $m$ does it correspond? What are $(g)$ the minimum values of $R$ and (h) the corresponding value of $m ?$ (i) For the actor-clone- Earth system, is mechanical energy conserved throughout the action sequence? Is this principle sufficient to solve the problem? Explain. (j) For the same system, is momentum conserved? Explain how this principle is used. (k) What If? Show that $R$ does not depend on the value of the gravitational acceleration. Is this result remarkable? State how one might make sense of it.

Sheh Lit Chang

University of Washington

Problem 57

A bullet of mass $m$ is fired into a block of mass $M$ initially at rest at the edge of a frictionless table of height $h$ (Fig. P9.57). The bullet remains in the block, and after impact the block lands a distance $d$ from the bottom of the table. Determine the initial speed of the bullet.

Sheh Lit Chang

University of Washington

Problem 58

A small block of mass $m_{1}=0.500 \mathrm{kg}$ is released from rest at the top of a curve-shaped, frictionless wedge of mass $m_{2}=3.00 \mathrm{kg},$ which sits on a frictionless horizontal surface as shown in Figure $\mathrm{P} 9.58$ a. When the block leaves the wedge, its velocity is measured to be 4.00 $\mathrm{m} / \mathrm{s}$ to the right as shown in the figure. (a) What is the velocity of the wedge after the block reaches the horizontal surface? (b) What is the height $h$ of the wedge?

Sheh Lit Chang

University of Washington

Problem 59

A 0.500 -kg sphere moving with a velocity given by $(2.00 \hat{\mathbf{i}}-3.00 \hat{\mathbf{j}}+1.00 \hat{\mathbf{k}}) \mathrm{m} / \mathrm{s}$ strikes another sphere of mass 1.50 $\mathrm{kg}$ that is moving with an initial velocity of $(-1.00 \hat{\mathbf{i}}+2.00 \hat{\mathbf{j}}-3.00 \hat{\mathbf{k}}) \mathrm{m} / \mathrm{s} .$ (a) The velocity of the $0.500-\mathrm{kg}$ sphere after the collision is given by $(-1.00 \hat{\mathrm{i}}+3.00 \hat{\mathrm{j}}-8.00 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s}$ . Find the final velocity of the $1.50-\mathrm{kg}$ sphere and identify the kind of collision (elastic, inelastic, or perfectly inelastic). (b) Now assume the velocity of the $0.500-\mathrm{kg}$ sphere after the collision is $(-0.250 \hat{\mathrm{i}}+0.750 \hat{\mathrm{j}}-2.00 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s}$ . Find the final velocity of the $1.50-\mathrm{kg}$ sphere and identify the kind of collision. (c) What If? Take the velocity of the $0.500-\mathrm{kg}$ sphere after the collision as $(-1.00 \hat{\mathrm{i}}+3.00 \hat{\mathrm{j}}+a \hat{\mathbf{k}}) \mathrm{m} / \mathrm{s}$ . Find the value of $a$ and the velocity of the $1.50-\mathrm{kg}$ sphere after an elastic collision.

Sheh Lit Chang

University of Washington

Problem 60

A 75.0-kg firefighter slides down a pole while a constant friction force of 300 N retards her motion. A horizontal 20.0-kg platform is supported by a spring at the bottom of the pole to cushion the fall. The firefighter starts from rest 4.00 m above the platform, and the spring constant is 4 000 N/m. Find (a) the firefighter’s speed immediately before she collides with the platform and (b) the maximum distance the spring is compressed. Assume the friction force acts during the entire motion.

Sheh Lit Chang

University of Washington

Problem 61

George of the Jungle, with mass m, swings on a light vine hanging from a stationary tree branch. A second vine of equal length hangs from the same point, and a gorilla of larger mass M swings in the opposite direction on it. Both vines are horizontal when the primates start from rest at the same moment. George and the gorilla meet at the lowest point of their swings. Each is afraid that one vine will break, so they grab each other and hang on. They swing upward together, reaching a point where the vines make an angle of $35.0^{\circ}$ with the vertical. (a) Find the value of the ratio $m / M .$ (b) What If? Try the following experiment at home. Tie a small magnet and a steel screw to opposite ends of a string. Hold the center of the string fixed to represent the tree branch, and reproduce a model of the motions of George and the gorilla. What changes in your analysis will make it apply to this situation? What If? Next assume the magnet is strong so that it noticeably attracts the screw over a distance of a few centimeters. Then the screw will be moving faster immediately before it sticks to the magnet. Does this extra magnet strength make a difference?

Sheh Lit Chang

University of Washington

Problem 62

A student performs a ballistic pendulum experiment using an apparatus similar to that shown in Figure 9.9b. She obtains the following average data: $h=8.68 \mathrm{cm}, m_{1}=$ $68.8 \mathrm{g},$ and $m_{2}=263 \mathrm{g} .$ The symbols refer to the quantities in Figure 9.9 $\mathrm{a}$ . (a) Determine the initial speed $v_{1 A}$ of the projectile. (b) The second part of her experiment is to obtain $v_{1 A}$ by firing the same projectile horizontally (with the pendulum removed from the path) and measuring its final horizontal position $x$ and distance of fall $y$ (Fig. P9.62). Show that the initial speed of the projectile is related to $x$ and $y$ by the equation
$$v_{1 A}=\frac{x}{\sqrt{2 y / g}}$$
What numerical value does she obtain for $v_{1 A}$ based on her measured values of $x=257 \mathrm{cm}$ and $y=85.3 \mathrm{cm} ?$ What factors might account for the difference in this value compared with that obtained in part (a)?

Sheh Lit Chang

University of Washington

Problem 63

Lazarus Carnot, an artillery general, managed the military draft for Napoleon. Carnot used a ballistic pendulum to measure the firing speeds of cannonballs. In the symbols defined in Example 9.6, he proved that the ratio of the kinetic energy immediately after the collision to the kinetic energy immediately before is $m_{1} /\left(m_{1}+m_{2}\right) .$ (a) Carry out the proof yourself. (b) If the cannonball has mass 9.60 $\mathrm{kg}$ and the block (a tree trunk) has mass 214 $\mathrm{kg}$ , what fraction of the original energy remains mechanical after the collision? (c) What is the ratio of the momentum immediately after the collision to the momentum immediately before? (d) A student believes that such a large loss of mechanical energy must be accompanied by at least a small loss of momentum. How would you convince this student of the truth? General Carnot’s son Sadi was the second most important engineer in the history of ideas; we will study his work in Chapter 22.

Sheh Lit Chang

University of Washington

Problem 64

Pursued by ferocious wolves, you are in a sleigh with no horses, gliding without friction across an ice-covered lake. You take an action described by these equations:
$$\begin{aligned}(270 \mathrm{kg})(7.50 \mathrm{m} / \mathrm{s}) \hat{\mathbf{i}} &=(15.0 \mathrm{kg})\left(-v_{11} \hat{\mathbf{i}}\right)+(255 \mathrm{kg})\left(v_{2 f} \hat{\mathbf{i}}\right) \\ v_{1 f}+v_{2 f} &=8.00 \mathrm{m} / \mathrm{s} \end{aligned}$$
(a) Complete the statement of the problem, giving the data and identifying the unknowns. (b) Find the values of $v_{1 f}$ and $v_{2 f}(c)$ Find the work you do.

Sheh Lit Chang

University of Washington

Problem 65

A light spring of force constant 3.85 N/m is compressed by 8.00 cm and held between a 0.250-kg block on the left and a 0.500-kg block on the right. Both blocks are at rest on a horizontal surface. The blocks are released simultaneously so that the spring tends to push them apart. Find the maximum velocity each block attains if the coefficient of kinetic friction between each block and the surface is (a) 0, (b) 0.100, and (c) 0.462. Assume the coefficient of static friction is greater than the coefficient of kinetic friction in every case.

Sheh Lit Chang

University of Washington

Problem 66

Consider as a system the Sun with the Earth in a circular orbit around it. Find the magnitude of the change in the velocity of the Sun relative to the center of mass of the system over a 6-month period. Ignore the influence of other celestial objects. You may obtain the necessary astronomical data from the endpapers of the book.

Sheh Lit Chang

University of Washington

Problem 67

A 5.00-g bullet moving with an initial speed of 400 m/s is fired into and passes through a 1.00-kg block as shown in Figure P9.67. The block, initially at rest on a frictionless, horizontal surface, is connected to a spring with force constant 900 N/m. The block moves 5.00 cm to the right after impact. Find (a) the speed at which the bullet emerges from the block and (b) the mechanical energy converted into internal energy in the collision.

Sheh Lit Chang

University of Washington

Problem 68

There are (one can say) three coequal theories of motion: Newton’s second law, stating that the total force on a particle causes its acceleration; the work–kinetic energy theorem, stating that the total work on a particle causes its change in kinetic energy; and the impulse–momentum theorem, stating that the total impulse on a particle causes its change in momentum. In this problem, you compare predictions of the three theories in one particular case. A 3.00-kg object has velocity 7.00$\hat{\mathrm{j}} \mathrm{m} / \mathrm{s}$ . Then, a total force 12.0$\hat{\mathrm{i}} \mathrm{N}$ acts on the object for 5.00 $\mathrm{s}$ . (a) Calculate the object's final velocity, using the impulse-momentum theorem. (b) Calculate its acceltion from $\overrightarrow{\mathbf{a}}=\Sigma \overrightarrow{\mathbf{F}} / m .$ (d) Find the object's vector displacement from $\Delta \overrightarrow{\mathbf{r}}=\overrightarrow{\mathbf{v}}_{i} t+\frac{1}{2} \overrightarrow{\mathbf{a}} t^{2} .$ (e) Find the work done on the object from $W=\mathbf{F} \cdot \Delta \overrightarrow{\mathbf{r}}$ . (f) Find the final kinetic energy from $\frac{1}{2} m v_{f}^{2}=\frac{1}{2} m \overrightarrow{\mathbf{v}}_{f} \cdot \overrightarrow{\mathbf{v}}_{f}$ (g) Find the final kinetic energy from $\frac{1}{2} m v_{i}^{2}+W .$ (h) State the result of comparing the answers to parts $\mathrm{b}$ and $\mathrm{c},$ and the answers to parts $\mathrm{f}$ and $\mathrm{g}$.

Sheh Lit Chang

University of Washington

Problem 69

A chain of length L and total mass M is released from rest with its lower end just touching the top of a table as shown in Figure P9.69a. Find the force exerted by the table on the chain after the chain has fallen through a distance x as shown in Figure P9.69b. (Assume each link comes to rest the instant it reaches the table.)

Sheh Lit Chang

University of Washington