College Physics

Raymond A. Serway, Chris Vuille, John Hughes

Chapter 6

Momentum, Impulse, and Collisions - all with Video Answers

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Chapter Questions

Problem 1

Calculate the magnitude of the linear momentum for the following cases: (a) a proton with mass equal to $1.67 \times$ $10^{-27} {kg},$ moving with a speed of $5.00 \times 10^{6} {m} / {s} ;({b})$ a 15.0 ${-g}$ bullet moving with a speed of $300 {m} / {s} ;({c})$ a 75.0 -kg sprinter running with a speed of $10.0 {m} / {s} ;$ (d) the Earth (mass $=$ $5.98 \times 10^{24} {kg}$ ) moving with an orbital speed equal to $2.98 \times$ $10^{4} {m} / {s} .$

Keshav Singh

Keshav Singh

Numerade Educator

Problem 2

A high-speed photograph of a club hitting a golf ball is shown in Figure 6.3. The club was in contact with a ball, initially at rest, for about 0.0020 s. If the ball has a mass of 55 ${g}$ and leaves the head of the club with a speed of $2,0 \times 10^{2} {ft} / {s},$ find the average force exerted on the ball by the club.

Averell Hause

Carnegie Mellon University

Problem 3

A pitcher claims he can throw a 0.145-kg baseball with as much momentum as a 3.00-g bullet moving with a speed of $1.50 \times 10^{3} {m} / {s}$ . (a) What must the baseball's speed be if the
pitcher's claim is valid? (b) Which has greater kinetic energy, the ball or the bullet?

Keshav Singh

Numerade Educator

Problem 4

A 0.280-kg volleyball approaches a player horizontally with a speed of 15.0 m/s. The player strikes the ball with her fist and causes the ball to move in the opposite direction with a speed of 22.0 m/s. (a) What impulse is delivered to the ball by the player? (b) If the player’s fist is in contact with the ball for
0.060 0 s, find the magnitude of the average force exerted on the player’s fist.

Averell Hause

Carnegie Mellon University

Problem 5

Q C Drops of rain fall perpendicular to the roof of a parked car during a rainstorm. The drops strike the roof with a speed of 12 m/s, and the mass of rain per second striking the roof is 0.035 kg/s. (a) Assuming the drops come to rest after striking the roof, find the average force exerted by the rain on the
roof. (b) If hailstones having the same mass as the raindrops fall on the roof at the same rate and with the same speed, how would the average force on the roof compare to that found in part (a)?

Linda Winkler

Numerade Educator

Problem 6

Show that the kinetic energy of a particle of mass m is related to the magnitude of the momentum p of that particle by $K E=p^{2} / 2 m .$ (Note: This expression is invalid for particles traveling at speeds near that of light.)

Averell Hause

Carnegie Mellon University

Problem 7

An object has a kinetic energy of 275 J and a momentum of magnitude 25.0 kg ? m/s. Find the (a) speed and (b) mass of the object.

MS

Michael Sorola

Numerade Educator

Problem 8

An estimated force vs. time curve for a baseball struck by a bat is shown in Figure P6.8. From this curve, deter- mine (a) the impulse delivered to the ball and (b) the average force exerted on the ball.

Averell Hause

Carnegie Mellon University

Problem 9

A soccer player takes a corner kick, lofting a stationary ball 35.0° above the horizon at 22.5 m/s. If the soccer ball has
a mass of 0.425 kg and the player’s foot is in contact with it for $5.00 \times 10^{-2} {s}$ , find (a) the $x-$ and $y$ -components of the soccer ball's change in momentum and (b) the magnitude of the average force exerted by the player's foot on the ball.

Keshav Singh

Numerade Educator

Problem 10

A man claims he can safely hold on to a 12.0-kg child in a head-on collision with a relative speed of 120-mi/h lasting for 0.10 s as long as he has his seat belt on. (a) Find the magnitude of the average force needed to hold onto the child. (b) Based on the result to part (a), is the man’s claim valid? (c) What does the answer to this problem say about laws requiring the use of proper safety devices such as seat belts
and special toddler seats?

Averell Hause

Carnegie Mellon University

Problem 11

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?

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 12

A tennis player receives a shot with the ball $(0.0600 {kg})$ 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 racket? (b) What work does the racket do on the ball?

Averell Hause

Carnegie Mellon University

Problem 13

A car is stopped for a traffic signal. When the light turns green, the car accelerates, increasing its speed from 0 to 5.20 m/s in 0.832 s. What are the magnitudes of (a) the linear impulse and (b) the average total force experienced by a 70.0-kg passenger in the car during the time the car accelerates?

Keshav Singh

Numerade Educator

Problem 14

A 65.0-kg basketball player jumps vertically and leaves the floor with a velocity of 1.80 m/s upward. (a) What impulse does the player experience? (b) What force does the floor exert on the player before the jump? (c) What is the total average force exerted by the floor on the player if the player is in
contact with the floor for 0.450 s during the jump?

Averell Hause

Carnegie Mellon University

Problem 15

The force shown in the force vs. time diagram in Figure P6.15 acts on a 1.5-kg object. Find (a) the impulse of the force, (b) the final velocity of the object if it is initially at rest, and (c) the final velocity of the object if it is initially moving along the x - axis with a velocity of 22.0 m/s.

Surendra Kumar

Numerade Educator

Problem 16

A force of magnitude $F_{x}$ acting in the $x$ -direction on a 2.00 -kg particle varies in time as shown in Figure P6.16. Find (a) the impulse of the force, (b) the final velocity of the particle if it is initially at rest, and (c) the final velocity of the particle if it is initially moving along the x - axis with a velocity of 22.00 m/s.

Suzanne W.

Numerade Educator

Problem 17

The forces shown in the force vs. time diagram in Figure P6.17 act on a 1.5 - kg particle. Find (a) the impulse for the interval from t 5 0 to t 5 3.0 s and (b) the impulse for the interval from t 5 0 to t 5 5.0 s. If the forces act on a 1.5 - kg particle that is initially at rest, find the particle’s speed (c) at t 5 3.0 s and (d) at t 5 5.0 s.

Sheh Lit Chang

University of Washington

Problem 18

V A 3.00-kg steel ball strikes a massive wall at 10.0 m/s at an angle of u 5 60.0° with the plane of the wall. It bounces off the wall with the same speed and angle (Fig. P6.18). If the ball is in contact with the wall for 0.200 s, what is the average force exerted by the wall on the ball?

Averell Hause

Carnegie Mellon University

Problem 19

T The front 1.20 m of a 1 400-kg car is designed as a “crumple zone” that collapses to absorb the shock of a collision. If a car traveling 25.0 m/s stops uniformly in 1.20 m, (a) how long does the collision last, (b) what is the magnitude of the average force on the car, and (c) what is the acceleration of the car?Express the acceleration as a multiple of the acceleration of gravity.

Sheh Lit Chang

University of Washington

Problem 20

Q C A pitcher throws a 0.14-kg baseball toward the batter so that it crosses home plate horizontally and has a speed of 42 m/s just before it makes contact with the bat. The batter then hits the ball straight back at the pitcher with a speed of 48 m/s. Assume the ball travels along the same line leaving the bat as it followed before contacting the bat. (a) What is the magnitude of the impulse delivered by the bat to the baseball? (b) If the ball is in contact with the bat for 0.005 0 s, what is the magnitude of the average force exerted by the bat on the ball? (c) How does your answer to part (b) compare to the weight of the ball?

Averell Hause

Carnegie Mellon University

Problem 21

V High-speed stroboscopic photographs show that the head of a $2.00 \times 10^{2}$ -g golf club is traveling at 55.0 $\mathrm{m} / \mathrm{s}$ just before it strikes a 46.0 - g golf ball at rest on a tee. After the collision, the club head travels (in the same direction) at 40.0 m/s. Find the speed of the golf ball just after impact.

Keshav Singh

Numerade Educator

Problem 22

A rifle with a weight of 30.0 ${N}$ fires a $5.00-{g}$ bullet with a speed of $3.00 \times 10^{2} {m} / {s}$ . (a) Find the recoil speed of the rifle. (b) If ${a} 7.00 \times 10^{2}-{N}$ man holds the rifle firmly against his shoulder, find the recoil speed of the man and rifle.

Averell Hause

Carnegie Mellon University

Problem 23

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

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 24

This is a symbolic version of Problem 23. A girl of mass mG is standing on a plank of mass mP. Both are originally at rest on a frozen lake that constitutes a frictionless, flat surface. The girl begins to walk along the plank at a constant velocity vGP to the right relative to the plank. (The subscript GP denotes the girl
relative to plank.) (a) What is the velocity vPI of the plank relative to the surface of the ice? (b) What is the girl’s velocity vGI relative to the ice surface?

Averell Hause

Carnegie Mellon University

Problem 25

Squids are the fastest marine invertebrates, using a powerful set of muscles to take in and then eject water in a form of jet propulsion that can propel them to speeds of over 11.5 m/s. What speed would a stationary 1.50 - kg squid achieve by ejecting 0.100 kg of water (not included in the squid’s mass) at 3.25 m/s?

Sheh Lit Chang

University of Washington

Problem 26

A 75 -kg fisherman in a $125-{kg}$ boat throws a package of mass $m=15 {kg}$ horizontally toward the right with a speed of $v_{i}=$
4.5 ${m} / {s}$ as in Figure ${P} 6.26 .$ Neglecting water resistance, and assuming the boat is at rest before the package is thrown, find the velocity of the boat after the package is thrown.

Suzanne W.

Numerade Educator

Problem 27

A 65.0 - kg person throws a 0.045 0 - kg snowball forward with a ground speed of 30.0 m/s. A second person, with a mass of 60.0 kg, catches the snowball. Both people are on skates. The first person is initially moving forward with a speed of 2.50 m/s, and the second person is initially at rest. What are
the velocities of the two people after the snowball is exchanged? Disregard friction between the skates and the ice.

Keshav Singh

Numerade Educator

Problem 28

Two objects of masses $m_{1}=$ 0.56 ${kg}$ and $m_{2}=0.88 {kg}$ are placed on a horizontal frictionless surface and a compressed spring of force constant k 5 280 N/m is placed between them as in Figure P6.28a. Neglect the mass of the spring. The spring is not attached to either object and is compressed a distance of 9.8 cm. If the objects are released from rest, find the final velocity of each object as shown in Figure P6.28b.

Averell Hause

Carnegie Mellon University

Problem 29

An astronaut in her space suit has a total mass of 87.0 kg, including suit and oxygen tank. Her tether line lose its attachment to her spacecraft while she’s on a spacewalk. Initially at rest with respect to her spacecraft, she throws her 12.0 - kg oxygen tank away from her spacecraft with a speed of 8.00 m/s to propel herself back toward it (Fig. P6.29). (a) Determine the maximum distance she can be from the craft and still return within 2.00 min (the amount of time the air in her helmet remains breathable). (b) Explain in terms of Newton’s laws of motion why this strategy works.

Surendra Kumar

Numerade Educator

Problem 30

Three ice skaters meet at the center of a rink and each stands at rest facing the center, within arm's reach of the other two. On a signal, each skater pushes himself away from the other two across the frictionless ice. After the push, skater A
with mass $m_{{A}}=80.0 {kg}$ moves in the negative $y$ -direction at 3.50 ${m} / {s}$ and skater ${B}$ with mass $m_{{B}}=75.0 {kg}$ moves in the negative $x$ -direction at 4.00 ${m} / {s}$ . Find the $x$ - and $y-{components of the 90.0 -kg skater C's) velocity after the push.

Averell Hause

Carnegie Mellon University

Problem 31

A man of mass $m_{1}=70.0 {kg}$ is skating at $v_{1}=8.00 {m} / {s}$ behind his wife of mass $m_{2}=50.0 {kg},$ who is skating at $v_{2}=4.00 {m} / {s}$ . Instead of passing her, he inadvertently collides with her. He grabs her around the waist, and they maintain their balance. (a) Sketch the problem with before-and-after diagrams, representing the skaters as blocks. (b) Is the collision best described as elastic, inelastic, or perfectly inelastic? Why? (c) Write the general equation for conservation of momentum in terms of $m_{1}, v_{1}, m_{2}, v_{2},$ and final velocity $v_{j},$ (d) Solve the momentum equation for $v_{f},({c})$ Substitute valucs, obtaining the numerical value for $v_{f},$ their speed after the collision.

Keshav Singh

Numerade Educator

Problem 32

An archer shoots an arrow toward a $3.00 \times 10^{2}-{g}$ target that is sliding in her direction at a speed of 2.50 ${m} / {s}$ on a smooth, slippery surface. The $22.5-{g}$ arrow is shot with a speed of 35.0 ${m} / {s}$ and passes through the target, which is stopped by the impact. What is the speed of the arrow after passing through the target?

Averell Hause

Carnegie Mellon University

Problem 33

Gayle runs at a speed of 4.00 m/s and dives on a sled, initially at rest on the top of a frictionless, snow-covered hill. After she has descended a vertical distance of 5.00 m, her brother, who is initially at rest, hops on her back, and they continue down the hill together. What is their speed at the bottom of the hill
if the total vertical drop is 15.0 m? Gayle’s mass is 50.0 kg, the sled has a mass of 5.00 kg, and her brother has a mass of 30.0 kg.

Keshav Singh

Numerade Educator

Problem 34

A 75.0-kg ice skater moving at 10.0 m/s crashes into a stationary skater of equal mass. After the collision, the two skaters move as a unit at 5.00 m/s. Suppose the average force a skater can experience without breaking a bone is 4 500 N. If the impact time is 0.100 s, does a bone break?

Averell Hause

Carnegie Mellon University

Problem 35

A railroad car of mass 2.00 3 104 kg moving at 3.00 m/s co lides and couples with two coupled railroad cars, each of the same mass as the single car and moving in the same direction at 1.20 m/s. (a) What is the speed of the three coupled cars after the collision? (b) How much kinetic energy is lost in the
collision?

Keshav Singh

Numerade Educator

Problem 36

This is a symbolic version of Problem $35 .$ A railroad car of mass $M$ moving at a speed $v_{1}$
coupled railroad cars, each of the same mass $M$ and moving in the same direction at a speed $v_{2} .$ (a) What is the speed $v_{f}$ of the three coupled cars after the collision in terms of $v_{1}$ and $v_{2}$ ? (b) How much kinetic energy is lost in the collision? Answer in terms of $M, v_{1},$ and $v_{2}$

Averell Hause

Carnegie Mellon University

Problem 37

Consider the ballistic pendulum device discussed in Example 6.5 and illustrated in Figure 6.13. (a) Determine the ratio of the momentum immediately after the collision to the momentum immediately before the collision. (b) Show that the ratio of the kinetic energy immediately after the collision to the kinetic energy immediately before the collision is $m_{1} /\left(m_{1}+m_{2}\right)$

Keshav Singh

Numerade Educator

Problem 38

A cue ball traveling at 4.00 m/s makes a glancing, elastic collision with a target ball of equal mass that is initially at rest. The target ball deflects the cue ball so that its subsequent motion makes an angle of 30.0° with respect to its original direction of travel. Find (a) the angle between the velocity vectors of
the two balls after the collision and (b) the speed of each ball after the collision.

BS

Bradley Sciacca

Numerade Educator

Problem 39

In a Broadway performance, an 80.0-kg actor swings from a 3.75-m-long cable that is horizontal when he starts. At the bottom of his arc, he picks up his 55.0-kg costar in an inelastic collision. What maximum height do they reach after thei upward swing?

Keshav Singh

Numerade Educator

Problem 40

Two shuffleboard disks of equal mass, one orange and the other green, are involved in a perfectly elastic glancing collision. The green disk is initially at rest and is struck by the orange disk moving initially to the right at 5.00 m/s as in Figure P6.40a. After the collision, the orange disk moves in a direction that
makes an angle of 37.0° with the horizontal axis while the green disk makes an angle of 53.0° with this axis as in Figure P6.40b. Determine the speed of each disk after the collision.

Averell Hause

Carnegie Mellon University

Problem 41

A 0.030-kg bullet is fired vertically at 200 m/s into a 0.15-kg baseball that is initially at rest. How high does the combined bullet and baseball rise after the collision, assuming the bullet embeds itself in the ball?

Keshav Singh

Numerade Educator

Problem 42

T An bullet of mass m 5 8.00 g is fired into a block of mass M 5 250 g that is initially at rest at the edge of a table of height h 5 1.00 m (Fig. P6.42). The bullet remains in the block, and after the impact the block lands d 5 2.00 m from the bottom of the table. Determine the initial speed of the bullet.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 43

A 12.0-g bullet is fired horizontally into a 100-g wooden block that is initially at rest on a frictionless horizontal surface and connected to a spring having spring constant 150 N/m. The bullet becomes embedded in the block. If the bullet– block system compresses the spring by a maximum of 80.0 cm,
what was the speed of the bullet at impact with the block?

Keshav Singh

Numerade Educator

Problem 44

A 1200-kg car traveling initially with a speed of 25.0 m/s in an easterly direction crashes into the rear end of a 9 000-kg truck moving in the same direction at 20.0 m/s (Fig. P6.44). The velocity of the car right after the collision is 18.0 m/s to the east. (a) What is the velocity of the truck right after the
collision? (b) How much mechanical energy is lost in the collision? Account for this loss in energy.

Averell Hause

Carnegie Mellon University

Problem 45

A tennis ball of mass 57.0 g is held just above a basketball of mass 590 g. With their centers vertically aligned, both balls are released from rest at the same time, falling through a distance of 1.20 m, as shown in Figure P6.45. (a) Find the magnitude of the basketball’s velocity the instant before the basketball reaches the ground. (b) Assume that an elastic collision with the ground instantaneously reverses the velocity of the basketball so that it collides with the tennis ball just above it. To what height does the tennis ball rebound?

Sanu Kumar

Numerade Educator

Problem 46

A space probe, initially at rest, undergoes an internal mechanical malfunction and breaks into three pieces. One piece of mass m 1 5 48.0 kg travels in the positive x - direction at 12.0 m/s, and a second piece of mass m 2 5 62.0 kg travels in the xy-plane at an angle of 105° at 15.0 m/s. The third piece has mass m 3 5 112 kg. (a) Sketch a diagram of the situation, labeling the different masses and their velocities. (b) Write the general expression for conservation of momentum in the
x- and y-directions in terms of m 1, m 2, m 3, v 1, v 2, and v 3 and the sines and cosines of the angles, taking u to be the unknown angle. (c) Calculate the final x-components of the momenta of m 1 and m 2. (d) Calculate the final y-components of the momenta of m 1 and m 2. (e) Substitute the known momentum components into the general equations of momentum for the x- and y-directions, along with the known mass m 3. (f) Solve the two momentum equations for v 3 cos u and v 3 sin u, respectively, and use the identity cos2 u 1 sin2 u 5 1 to obtain v 3. (g) Divide the equation for v 3 sin u by that for v 3 cos u to obtain tan u, then obtain the angle by taking the inverse tangent of both sides. (h) In general, would three such pieces necessarily have to move in the same plane? Why?

Keshav Singh

Numerade Educator

Problem 47

A 25.0-g object moving to the right at 20.0 cm/s overtakes and collides elastically with a 10.0-g object moving in the same direction at 15.0 cm/s. Find the velocity of each object after the collision.

Vishal Gupta

Numerade Educator

Problem 48

A billiard ball rolling across a table at 1.50 m/s makes a head- on elastic collision with an identical ball. Find the speed of each ball after the collision (a) when the second ball is initially at rest, (b) when the second ball is moving toward the first at a speed of 1.00 m/s, and (c) when the second ball is
moving away from the first at a speed of 1.00 m/s.

Averell Hause

Carnegie Mellon University

Problem 49

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) Why does the tackle constitute a perfectly inelastic collision? (b) Calculate the velocity of the players immediately after the tackle and (c) determine the mechanical energy that is lost as a result of the collision. (d) Where did the lost energy go?

Keshav Singh

Numerade Educator

Problem 50

Identical twins, each with mass 55.0 kg, are on ice skates and at rest on a frozen lake, which may be taken as frictionless. Twin A is carrying a backpack of mass 12.0 kg. She throws it horizontally at 3.00 m/s to Twin B. Neglecting any gravity effects, what are the subsequent speeds of Twin A and Twin B?

Averell Hause

Carnegie Mellon University

Problem 51

A $2.00 \times 10^{3}$ -kg car moving east at 10.0 ${m} / {s}$ collides with a $3.00 \times 10^{3}-{kg}$ car moving north. The cars stick together and move as a unit after the collision, at an angle of $40.0^{\circ}$ north of east and a speed of 5.22 ${m} / {s}$ . Find the speed and direction of the $3.00 \times 10^{3}-{kg}$ car before the collision.

Keshav Singh

Numerade Educator

Problem 52

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 velocity $v_{2 i}$ 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 ${mi} / {h}$ , and the driver of the northward-moving vehicle claims he was within the limit when the collision occurred. Is he telling the
truth?

Averell Hause

Carnegie Mellon University

Problem 53

A billiard ball moving at 5.00 m/s strikes a stationary ball of the same mass. After the collision, the first ball moves at 4.33 m/s at an angle of 30.0° with respect to the original line of motion. (a) Find the velocity (magnitude and direction) of the second ball after collision. (b) Was the collision inelastic
or elastic?

Sheh Lit Chang

University of Washington

Problem 54

The Merlin rocket engines developed by SpaceX produce $8.01 \times 10^{5} {N}$ of instantaneous thrust with an exhaust speed of $3.05 \times 10^{3} {m} / {s}$ in vacuum. What mass of fuel does the engine bum each second?

Averell Hause

Carnegie Mellon University

Problem 55

One of the first ion engines on a commercial satellite used Xenon as a propellant and could eject the ionized gas at a rate of $3.03 \times 10^{-6} {kg} / {s}$ with an exhaust speed of $3.04 \times 10^{4} {m} / {s}$ . What instantaneous thrust could the engine provide?

Keshav Singh

Numerade Educator

Problem 56

NASA's Saturn V rockets that launched astronauts to the moon were powered by the strongest rocket engine ever developed, providing $6.77 \times 10^{6} {N}$ of thrust while burning fuel at a rate of $2.63 \times 10^{3} {kg} / {s}$ . Calculate the engine's exhaust speed.

Averell Hause

Carnegie Mellon University

Problem 57

A typical person begins to lose consciousness if subjected to accelerations greater than about 5g $\left(49.0 {m} / {s}^{2}\right)$ for more than a few seconds. Suppose a $3.00 \times 10^{4}$ -kg manned spaceship's engine has an exhaust speed of $2.50 \times 10^{3} {m} / {s}$ .
What maximum burn rate $|\Delta M / \Delta t|$ could the engine reach before the ship's acceleration exceeded 5$g$ and its human occupants began to lose consciousness?

Keshav Singh

Numerade Educator

Problem 58

A spaceship at rest relative to a nearby star in interplanetary space has a total mass of $2.50 \times 10^{4} {kg}$ . Its engines fire at $t=0$ , steadily burning fuel at 76.7 ${kg} / {s}$ with an exhaust speed of

Averell Hause

Carnegie Mellon University

Problem 59

A spaceship's orbital maneuver requires a speed increase of $1.20 \times 10^{3} {m} / {s}$ . If its engine has an exhaust speed of $2.50 \times$ $10^{3} {m} / {s}$ , determine the required ratio $M_{i} / M_{j}$ of its initial mass to its final mass. (The difference $M_{i}-M_{f}$ equals the mass of the ejected fuel.)

Salamat Ali

Numerade Educator

Problem 60

BIO In research in cardiology and exercise physiology, it is often important to know the mass of blood pumped by a person’s heart in one stroke. This information can be obtained by means of a ballistocardiograph. The instrument works as follows: The subject lies on a horizontal pallet floating on a film of air. Friction on the pallet is negligible. Initially, the momentum of the system is zero. When the heart beats, it expels a mass m of blood into the aorta with speed v, and the body and platform move in the opposite direction with speed V. The speed of the blood can be determined independently (e.g., by observing an ultrasound Doppler shift). Assume that the blood’s speed is
50.0 cm/s in one typical trial. The mass of the subject plus the pallet is 54.0 kg. The pallet moves at a speed of 6.00 3 1025m in 0.160 s after one heartbeat. Calculate the mass of blood that leaves the heart. Assume that the mass of blood is negligible compared with the total mass of the person. This simplified example illustrates the principle of ballistocardiography, but in practice a more sophisticated model of heart function is used.

Averell Hause

Carnegie Mellon University

Problem 61

Most of us know intuitively that in a head-on collision between a large dump truck and a subcompact car, you are better off being in the truck than in the car. Why is this? Many people imagine that the collision force exerted on the car is much greater than that exerted on the truck. To substantiate this view, they point out that the car is crushed, whereas the truck is only dented. This idea of unequal forces, of course, is false; Newton’s third law tells us that both objects are acted
upon by forces of the same magnitude. The truck suffers less damage because it is made of stronger metal. But what about the two drivers? Do they experience the same forces? To answer this question, suppose that each vehicle is initially moving at 8.00 m/s and that they undergo a perfectly inelas- tic head-on collision. Each driver has mass 80.0 kg. Including the masses of the drivers, the total masses of the vehicles are 800 kg for the car and $4.00 \times 10^{3} {kg}$ for the truck. If the col- lision time is 0.120 s, what force does the seat belt exert on each driver?

Keshav Singh

Numerade Educator

Problem 62

Consider a trictionless track as shown in Figure $P 6.62 .$ A block of mass $m_{1}=5.00 {kg}$ is released from $@$ . It makes a head-on elastic collision at $\Theta$ with a block of mass $m_{2}=10.0 {kg}$ that is initially at rest. Calculate the maximum height to which $m_{1}$
rises after the collision.

Averell Hause

Carnegie Mellon University

Problem 63

A 2.0-g particle moving at 8.0 m/s makes a perfectly elastic head-on collision with a resting 1.0-g object. (a) Find the speed of each particle after the collision. (b) Find the speed of each particle after the collision if the stationary particle has a mass of 10 g. (c) Find the final kinetic energy of the incident 2.0-g particle in the situations described in parts (a) and (b). In which case does the incident particle lose more kinetic energy?

Averell Hause

Carnegie Mellon University

Problem 64

A bullet of mass m and speed v passes completely through a pendulum bob of mass M as shown in
Figure P6.64. 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 bob will barely swing through a complete vertical circle?

Averell Hause

Carnegie Mellon University

Problem 65

An amateur skater of mass M is trapped in the middle of an ice rink and is unable to return to the side
where there is no ice. Every motion she makes causes her to slip on the ice and remain in the same spot. She decides to try to return to safety by removing her gloves of mass m and throwing them in the direction opposite the safe side. (a) She throws the gloves as hard as she can, and they leave her hand
with a velocity $\overrightarrow{{v}}_{\text { gloves. Explain whether or not she moves. If }}$
If she does move, calculate her velocity v girl relative to the Earth after she throws the gloves. (b) Discuss her motion from the point of view of the forces acting on her.

Keshav Singh

Numerade Educator

Problem 66

A $0.400-\mathrm{kg}$ blue bead slides on a frictionless, curved wire, starting from rest at point $@$ in Figure $\mathrm{P} 6.66$ where $h=1.50 \mathrm{m} .$ At point $(\widehat{B}),$ the bead collides
elastically with a $0.600-k g$ green bead at rest. Find the maximum height the green bead rises as it moves up the wire.

Averell Hause

Carnegie Mellon University

Problem 67

A 730-N man stands in the middle of a frozen pond of radius 5.0 m. He is unable to get to the other side because of a lack of friction between his shoes and the ice. To overcome this difficulty, he throws his 1.2-kg physics textbook horizontally toward the north shore at a speed of 5.0 m/s. How long does
it take him to reach the south shore?

Sheh Lit Chang

University of Washington

Problem 68

An unstable nucleus of mass $1.7 \times 10^{-26} {kg},$ initially at rest at the origin of a coordinate system, disintegrates into three particles. One particle, having a mass of $m_{1}=5.0 \times$ $10^{-27} {kg}$ , moves in the positive $y$ direction with speed $v_{1}=$ $6.0 \times 10^{6} {m} / {s}$ . Another particle, of mass $m_{2}=8.4 \times 10^{-27} {kg},$ moves in the positive $x$ -direction with speed $v_{2}=4.0 \times 10^{6}$ $v_{2}=4.0 \times 10^{6}$ ${m} / {s}$ . Find the magnitude and direction of the velocity of the third particle.

Averell Hause

Carnegie Mellon University

Problem 69

Two blocks of masses $m_{1}$ and $m_{2}$ approach each other on a horizontal table with the same constant speed, $v_{0},$ as measured by a laboratory observer. The blocks undergo a perfectly elastic collision, and it is observed that $m_{1}$ stops but $m_{2}$ moves opposite its original motion with some constant speed, $v$ (a) Determine the ratio of the two masses, $m_{1} / m_{2}$ . (b) What is
the ratio of their speeds, $v / v_{0} ?$

Keshav Singh

Numerade Educator

Problem 70

Two blocks of masses $m_{1}=2.00 {kg}$ and $m_{2}=4.00 {kg}$ are each released from rest at a height of $h=5.00 {m}$ on a frictionless track, as shown in Figure ${P} 6.70,$ and undergo an elastic head- on collision. (a) Determine the velocity of each block just before the collision. (b) Determine the velocity of each block immediately after the collision. (c) Determine the maximum heights to which $m_{1}$ and $m_{2}$ rise after the collision.

Averell Hause

Carnegie Mellon University

Problem 71

A block with mass m 1 5 0.500 kg is released from rest on a frictionless track at a distance h1 5 2.50 m above the top of a table. It then collides elastically with an object having mass m 2 5 1.00 kg that is initially at rest on the table, as shown in Figure P6.71. (a) Determine the velocities of the two objects just after the collision. (b) How high up the track does the 0.500-kg object travel back after the collision? (c) How far away from the bottom of the table does the 1.00 -kg object land, given that the height of the table is $h_{2}=2.00 {m} ?$ (d) How far away from the bottom of the table does the $0.500-{kg}$ object
eventually land?

Keshav Singh

Numerade Educator

Problem 72

Two objects of masses m and 3m are moving toward each other along the x-axis with the same initial speed$v_{0}$ .The object with mass m is traveling to the left, and the object with mass 3m is traveling to the right. They undergo an elastic glancing collision such that m is moving downward after the collision at right angles from its initial direction. (a) Find the final speeds of the two objects. (b) What is the angle u at which the object with mass 3m is scattered?

Averell Hause

Carnegie Mellon University

Problem 73

A small block of mass $m_{1}=0.500 {kg}$ is released from rest at the top of a curved wedge of mass $m_{2}=3.00 {kg},$ which sits on a frictionless horizontal surface as in Figure P6.73a. When the block leaves the wedge, its velocity is measured to be 4.00 $\mathrm{m} / {s}$ to the right, as in Figure ${P} 6.73 {b}$ . (a) What is the velocity of the wedge after the block reaches the horizontal surface? (b) What is the height $h$ of the wedge?

Keshav Singh

Numerade Educator

Problem 74

A car of mass $m$ moving at a speed $v_{1}$ collides and couples with the back of a truck of mass 2$m$ moving initially in the same direction as the car at a lower speed $v_{2},(\text { a })$ What is the speed $v_{f}$ of the two vehicles immediately after the collision? (b) What is the change in kinetic energy of the car-truck system in the collision?

Averell Hause

Carnegie Mellon University

Problem 75

A cannon is rigidly attached to a carriage, which can move along horizontal rails, but is connected to a post by a large spring, initially unstretched and with force constant $k=$ $2.00 \times 10^{4} {N} / {m},$ as in Figure ${P} 6.75$ . The cannon fires a $200 \times 10^{2}-{kg}$ projectile at a velocity of 125 ${m} / {s}$ directed $45.0^{\circ}$ above the horizontal. cannon and its carriage is $5.00 \times 10^{3}-{kg}$ , find the recoil speed of the cannon. (b) Determine the maximum extension of the spring. (c) Find the maximum force the spring exerts on the carriage. (d) Consider the system consisting of the cannon, the carriage, and the shell. Is the momentum of this system conserved during the firing? Why or why not?

Keshav Singh

Numerade Educator

Problem 76

Two blocks collide on a frictionless surface. After the collision, the blocks stick together. Block A has a mass M and is initially moving to the right at speed v. Block B has a mass 2M and is initially at rest. System C is composed of both blocks. (a) Draw a force diagram for each block at an instant during the collision. (b) Rank the magnitudes of the horizontal forces in your diagram. Explain your reasoning. (c) Calculate the change in momentum of block A, block B, and system C. (d) Is kinetic energy conserved in this collision? Explain your answer. (This problem is courtesy of Edward F. Redish. For more such problems, visit http://www.physics.umd.edu/perg.)

Averell Hause

Carnegie Mellon University

Problem 77

(a) A car traveling due east strikes a car traveling due north at an intersection, and the two move together as a unit. A property owner on the southeast corner of the intersection claims that his fence was torn down in the collision. Should he be awarded damages by the insurance company? Defend your answer. (b) Let the eastward-moving car have a mass of 1.30 3 $10^{3}-{kg}$ and a speed of 30.0 ${km} / {h}$ and the northward-moving car a mass of $1.10 \times 10^{3} {kg}$ and a speed of 20.0 ${km} / {h}$ . Find the velocity after the collision. Are the results consistent with your answer to part (a)?

Keshav Singh

Numerade Educator

Problem 78

A 60-kg soccer player jumps vertically upwards and heads the 0.45-kg ball as it is descending vertically with a speed of 25 m/s. (a) If the player was moving upward with a speed of 4.0 m/s just before impact, what will be the speed of the ball immediately after the collision if the ball rebounds vertically upwards and the collision is elastic? (b) If the ball is in contact with the player’s head for 20 ms, what is the average acceleration of the ball? (Note that the force of gravity may be ignored during the brief collision time.)

Averell Hause

Carnegie Mellon University

Problem 79

A boy of mass $m_{b}$ and his girlfriend of mass $m_{F}$ , both wearing ice skates, face each other at rest while standing on a frictionless ice rink. The boy pushes the girl, giving her a velocity $v_{g}$ toward the east. Assume that $m_{b} > m_{K}$ (a) Describe the subsequent motion of the boy. (b) Find expressions for the final kinetic energy of the girl and the final kinetic energy of the boy, and show that the girl has greater kinetic energy than the boy. (c) The boy and girl had zero kinetic energy before the
boy pushed the girl, but ended up with kinetic energy after the event. How do you account for the appearance of mechanical energy?

Keshav Singh

Numerade Educator

Problem 80

A 20.0-kg toboggan with 70.0-kg driver is sliding down a frictionless chute directed 30.0° below the horizontal at 8.00 m/s when a 55.0-kg woman drops from a tree limb straight down behind the driver. If she drops through a vertical displacement of 2.00 m, what is the subsequent velocity of the toboggan immediately after impact?

Averell Hause

Carnegie Mellon University

Problem 81

Measuring the speed of a bullet. A bullet of mass m is fired horizontally into a wooden block of mass M lying on a table. The bullet remains in the block after the collision. The coefficient of friction between the block and table is m, and the block slides a distance d before stopping. Find the initial speed $v_{0}$ of the bullet in terms of $M, m, \mu, g,$ and $d$

Keshav Singh

Numerade Educator

Problem 82

A flying squid (family Ommastrephidae) is able to “jump” off the surface of the sea by taking water into its body cavity and then ejecting the water vertically downward. A 0.850-kg squid is able to eject 0.300 kg of water with a speed of 20.0 m/s. (a) What will be the speed of the squid immediately after ejecting
the water? (b) How high in the air will the squid rise?

Averell Hause

Carnegie Mellon University

Problem 83

A 0.30-kg puck, initially at rest on a frictionless horizontal surface, is struck by a 0.20-kg puck that is initially moving along the x-axis with a velocity of 2.0 m/s. After the collision, the 0.20-kg puck has a speed of 1.0 m/s at an angle of u 5 53° to the positive x-axis. (a) Determine the velocity of the 0.30-kg
puck after the collision. (b) Find the fraction of kinetic energy lost in the collision.

Sheh Lit Chang

University of Washington

Problem 84

A wooden block of mass M rests on a table over a large hole as in Figure P6.84. A bullet of mass m with an initial velocity $v_{i}$ is fired upward into the bottom of the block and remains in the block after the collision. The block and bullet rise to a maximum height of $h .$ (a) Describe how you would find the initial velocity of the bullet using ideas you have learned in this topic. (b) Find an expression for the initial velocity of the bullet.

Averell Hause

Carnegie Mellon University

Problem 85

A 1.25-kg wooden block rests on a table over a large hole as in Figure P6.84. A 5.00-g bullet with an initial velocity $v_{i}$ is fired upward into the bottom of the block and remains in the block after the collision. The block and bullet rise to a maximum height of 22.0 ${cm} .$ (a) Describe how you would find
the initial velocity of the bullet using ideas you have learned in this topic. (b) Calculate the initial velocity of the bullet from the information provided.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator