Physics for Scientists and Engineers with Modern Physics

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

Chapter 4

Motion in Two Dimensions - all with Video Answers

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Chapter Questions

Problem 1

A motorist drives south at 20.0 $\mathrm{m} / \mathrm{s}$ for 3.00 $\mathrm{min}$ , then turns west and travels at 25.0 $\mathrm{m} / \mathrm{s}$ for 2.00 $\mathrm{min}$ , and finally travels northwest at 30.0 $\mathrm{m} / \mathrm{s}$ for $1.00 \mathrm{min} .$ For this 6.00 -min trip, find $(\mathrm{a})$ the total vector displacement, (b) the average speed, and $(\mathrm{c})$ the average velocity. Let the positive $x$ axis point east.

Anand Jangid

Anand Jangid

Numerade Educator

Problem 2

When the Sun is directly overhead, a hawk dives toward the ground with a constant velocity of 5.00 $\mathrm{m} / \mathrm{s}$ at $60.0^{\circ}$ below the horizontal. Calculate the speed of its shadow on the level ground.

Jacob Paiste

Numerade Educator

Problem 3

A golf ball is hit off a tee at the edge of a cliff. Its $x$ and $y$ coordinates as functions of time are given by $x=18,0 t$ and $y=4.00 t-4.90 t^{2},$ where $x$ and $y$ are in meters and $t$ is in seconds. (a) Write a vector expression for the ball's position as a function of time, using the unit vectors i and $\hat{\mathbf{j}}$ . tBy taking derivatives, obtain expressions for (b) the velocity vector $\overrightarrow{\mathbf{v}}$ as a function of time and (c) the acceleration vector $\overrightarrow{\mathbf{a}}$ as a function of time. (d) Next use unit-vector notation to write expressions for the position, the velocity, and the acceleration of the golf ball at $t=3.00 \mathrm{s}$ .

Lauren Madey

Numerade Educator

Problem 4

The coordinates of an object moving in the $x y$ plane vary with time according to the equations $x=-5.00 \sin \omega t$ and $y=4.00-5.00 \cos \omega t,$ where $\omega$ is a constant, $x$ and $y$ are in meters, and $t$ is in seconds. (a) Determine the components of velocity of the object at $t=0 .$ (b) Determine the components of acceleration of the object at $t=0 .$ (c) Write expressions for the position vector, the velocity vector, and the acceleration vector of the object at any time $l>0$ . (d) Describe the path of the object in an $x y$ plot.

Keshav Singh

Numerade Educator

Problem 5

The vector position of a particle varies in time according to the expression $\overrightarrow{\mathbf{r}}=3.00 \hat{\mathbf{i}}-6.00 t^{2} \hat{\mathbf{j}},$ where $\overrightarrow{\mathbf{r}}$ is in meters and $t$ is in seconds. (a) Find an expression for the velocity of the particle as a function of time. (b) Determine the acceleration of the particle as a function of time. (c) Calculate the particle's position and velocity at $t=1.00$ s.

Lauren Madey

Numerade Educator

Problem 6

A particle initially located at the origin has an acceleration of $\overrightarrow{\mathbf{a}}=3.00 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}^{2}$ and an initial velocity of $\overrightarrow{\mathbf{v}}_{i}=5.00 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s}$. Find (a) the vector position of the particle at any time $\iota$ (b) the velocity of the particle at any time $t,(c)$ the coordinates of the particle at $t=2.00 \mathrm{s},$ and $(\mathrm{d})$ the speed of the particle at $t=2.00 \mathrm{s}$

Jacob Paiste

Numerade Educator

Problem 7

A fish swimming in a horizontal plane has velocity $\overrightarrow{\mathbf{v}}_{i}=(4.00 \hat{\mathbf{i}}+1.00 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}$ at a point in the ocean where the position relative to a certain rock is $\overrightarrow{\mathbf{r}}_{i}=(10.0 \hat{\mathbf{i}}-4.00 \hat{\mathbf{j}}) \mathrm{m}$ After the fish swims with constant acceleration for 20.0 $\mathrm{s}$ its velocity is $\overrightarrow{\mathbf{v}}=(20.0 \hat{\mathbf{i}}-5.00 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}$ (a) What are the components of the acceleration of the fish? (b) What is the direction of its acceleration with respect to unit vector $\hat{\mathbf{i}}$ ? (c) If the fish maintains constant acceleration, where is it at $t=25.0 \mathrm{s}$ and in what direction is it moving?

Anand Jangid

Numerade Educator

Problem 8

Review. A snowmobile is originally at the point with position vector 29.0 $\mathrm{m}$ at $95.0^{\circ}$ counterclockwise from the $x$ axis, moving with velocity 4.50 $\mathrm{m} / \mathrm{s}$ at $40.0^{\circ} .$ It moves with constant acceleration 1.90 $\mathrm{m} / \mathrm{s}^{2}$ at $200^{\circ} .$ After 5.00 s have elapsed, find (a) its velocity and (b) its position vector.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 9

In a local bar, a customer slides an empty beer mug down the counter for a refill. The height of the counter is 1.22 m. The mug slides off the counter and strikes the floor 1.40 m from the base of the counter. (a) With what velocity did the mug leave the counter? (b) What was the direction of the mug’s velocity just before it hit the floor?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 10

In a local bar, a customer slides an empty beer mug down the counter for a refill. The height of the counter is $h$. The mug slides off the counter and strikes the floor at distance d from the base of the counter. (a) With what velocity did the mug leave the counter? (b) What was the direction of the mug’s velocity just before it hit the floor?

Jacob Paiste

Numerade Educator

Problem 11

A projectile is fired in such a way that its horizontal range is equal to three times its maximum height. What is the angle of projection?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 12

To start an avalanche on a mountain slope, an artillery shell is fired with an initial velocity of 300 $\mathrm{m} / \mathrm{s}$ at $55.0^{\circ}$ above the horizontal. It explodes on the mountainside 42.0 $\mathrm{s}$ after firing. What are the $x$ and $y$ coordinates of the shell where it explodes, relative to its firing point?

Jacob Paiste

Numerade Educator

Problem 13

Chinook salmon are able to move through water especially fast by jumping out of the water periodically. This behavior is called porpoising. Suppose a salmon swimming in still water jumps out of the water with velocity 6.26 $\mathrm{m} / \mathrm{s}$ at $45.0^{\circ}$ above the horizontal, sails through the air a distance $L$ before returning to the water, and then swims the same distance $L$ underwater in a straight, horizontal line with velocity 3.58 $\mathrm{m} / \mathrm{s}$ before jumping out again. (a) Determine the average velocity of the fish for the entire process of jumping and swimming underwater. (b) Consider the time interval required to travel the entire distance of $2 L .$ By what percentage is this time interval reduced by the jumping/swimming process compared with simply swimming underwater at 3.58 $\mathrm{m} / \mathrm{s}$ ?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 14

A rock is thrown upward from level ground in such a way that the maximum height of its flight is equal to its horizontal range $R$ (a) At what angle $\theta$ is the rock thrown? (b) In terms of its original range $R,$ what is the range $R_{\max }$ the rock can attain if it is launched at the same speed but at the optimal angle for maximum range? (c) What If? Would your answer to part (a) be different if the rock is thrown with the same speed on a different planet? Explain.

Donald Albin

Numerade Educator

Problem 15

A firefighter, a distance $d$ from a burning building, directs a stream of water from a fire hose at angle $\theta_{i}$ above the horizontal as shown in Figure $\mathrm{P} 4.15 .$ If the initial speed of the stream is $v_{i}$ at what height $h$ does the water strike the building?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 16

A ball is tossed from an upper-story window of a building. The ball is given an initial velocity of 8.00 $\mathrm{m} / \mathrm{s}$ at an angle of $20.0^{\circ}$ below the horizontal. It strikes the ground 3.00 s later. (a) How far horizontally from the base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown. (c) How long does it take the ball to reach a point 10.0 $\mathrm{m}$ below the level of launching?

Jacob Paiste

Numerade Educator

Problem 17

A placekicker must kick a football from a point 36.0 $\mathrm{m}$ (about 40 yards) from the goal. Half the crowd hopes the ball will clear the crossbar, which is 3.05 $\mathrm{m}$ high. When kicked, the ball leaves the ground with a speed of 20.0 $\mathrm{m} / \mathrm{s}$ at an angle of $53.0^{\circ}$ to the horizontal. (a) By how much does the ball clear or fall short of clearing the crossbar? (b) Does the ball approach the crossbar while still rising or while falling?

Anand Jangid

Numerade Educator

Problem 18

A landscape architect is planning an artificial waterfall in a city park. Water flowing at 1.70 $\mathrm{m} / \mathrm{s}$ will leave the end of a horizontal channel at the top of a vertical wall $h=2.35 \mathrm{m}$ high, and from there it will fall into a pool (Fig. P4.18). (a) Will the space behind the waterfall be wide enough for a pedestrian walkway? (b) To sell her plan to the city council, the architect wants to build a model to standard scale, which is one-twelfth actual size. How fast should the water flow in the channel in the model?

Surendra Kumar

Numerade Educator

Problem 19

A playground is on the flat roof of a city school, 6.00 $\mathrm{m}$ above the street below (Fig. P4.19). The vertical wall of the building is $h=7.00 \mathrm{m}$ high, forming a 1 -m-high railing around the playground. A ball has fallen to the street below, and a passerby returns it by launching it an angle of $\theta=53.0^{\circ}$ above the horizontal at a point $d=24.0 \mathrm{m}$ from the base of the building wall. The ball takes 2.20 $\mathrm{s}$ to reach a point vertically above the wall. (a) Find the speed at which the ball was launched. (b) Find the vertical distance by which the ball clears the wall. (c) Find the horizontal distance from the wall to the point on the roof where the ball lands.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 20

A basketball star covers 2.80 $\mathrm{m}$ horizontally in a jump to dunk the ball (Fig. P4.20a). His motion through space can be modeled precisely as that of a particle at his center of mass, which we will define in Chapter $9 .$ His center of mass is at elevation 1.02 $\mathrm{m}$ when he leaves the floor. It reaches a maximum height of 1.85 $\mathrm{m}$ above the floor and is at elevation 0.900 $\mathrm{m}$ when he touches down again. Determine (a) his time of flight (his "hang time"), (b) his horizontal and (c) vertical velocity components at the instant of take-off, and (d) his takeoff angle. (e) For comparison, determine the hang time of a whitetail deer making a jump (Fig. P4.20b) with center-of-mass elevations $y_{i}=1.20 \mathrm{m}, y_{\max }=$ $2.50 \mathrm{m},$ and $y_{f}=0.700 \mathrm{m} .$

Jacob Paiste

Numerade Educator

Problem 21

A soccer player kicks a rock horizontally off a 40.0 -m-high cliff into a pool of water. If the player hears the sound of the splash 3.00 s later, what was the initial speed given to the rock? Assume the speed of sound in air is 343 $\mathrm{m} / \mathrm{s}$ .

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 22

The motion of a human body through space can be modeled as the motion of a particle at the body's center of mass as we will study in Chapter 9. The components of the displacement of an athlete's center of mass from the beginning to the end of a certain jump are described by the equations
$$x_{f}=0+(11.2 \mathrm{m} / \mathrm{s})\left(\cos 18.5^{\circ}\right) t$$
$$0.360 \mathrm{m}=0.840 \mathrm{m}+(11.2 \mathrm{m} / \mathrm{s})\left(\sin 18.5^{\circ}\right) t-\frac{1}{2}\left(9.80 \mathrm{m} / \mathrm{s}^{2}\right) t^{2}$$
where $t$ is in seconds and is the time at which the athlete ends the jump. Identify (a) the athlete's position and (b) his vector velocity at the takeoff point. (c) How far did he jump?

Jacob Paiste

Numerade Educator

Problem 23

A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of $v_{i}=18.0 \mathrm{m} / \mathrm{s}$ . The cliff is $h=50.0 \mathrm{m}$ above a body of water as shown in Figure P4.23. ( a) What are the coordinates of the initial position of the stone? (b) What are the components of the initial velocity of the stone? (c) What is the appropriate analysis model-for the vertical motion of the stone? (d) What is the appropriate analysis model for the horizontal motion of the stone? (e) Write symbolic equations for the x and y components of the velocity of the stone as a function of time. (f) Write symbolic equations for the position of the stone as a function of time. (g) How long after being released does the stone strike the water below the cliff? (h) With what speed and angle of impact does the stone land?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 24

The record distance in the sport of throwing cowpats is 81.1 $\mathrm{m}$ . This record toss was set by Steve Uren of the United States in $1981 .$ Assuming the initial launch angle was $45^{\circ}$ and neglecting air resistance, determine (a) the initial speed of the projectile and (b) the total time interval the projectile was in flight. (c) How would the answers change if the range were the same but the launch angle were greater than $45^{\circ} ?$ Explain.

Jacob Paiste

Numerade Educator

Problem 25

A boy stands on a diving board and tosses a stone into a swimming pool. The stone is thrown from a height of 2.50 $\mathrm{m}$ above the water surface with a velocity of 4.00 $\mathrm{m} / \mathrm{s}$ at an angle of $60.0^{\circ}$ above the horizontal. As the stone strikes the water surface, it immediately slown to exactly half the speed it had when it struck the water and maintains that speed while in the water. After the stone enters the water, it moves in a straight line in the direction of the velocity it had when it struck the water. If the pool is 3.00 $\mathrm{m}$ deep, how much time elapses between when the stone is thrown and when it strikes the bottom of the pool?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 26

A projectile is fired from the top of a cliff of height $h$ above the ocean below. The projectile is fired at an angle $\theta$ above the horizontal and with an initial speed $v_{i}$ . (a) Find a symbolic expression in terms of the variables $v_{i}, g,$ and $\theta$ for the time at which the projectile reaches its maximum height. (b) Using the result of part (a), find an expression for the maximum height $h_{\max }$ above the ocean attained by the projectile in terms of $h, v_{i}, g,$ and $\theta .$

Jacob Paiste

Numerade Educator

Problem 27

The athlete shown in Figure P4.27 rotates a 1.00-kg discus along a circular path of radius 1.06 m. The maximum speed of the discus is 20.0 $\mathrm{m} / \mathrm{s}$ . Determine the magnitude of the maximum radial acceleration of the discus.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 28

In Example $4.6,$ we found the centripetal acceleration of the Earth as it revolves around the Sun. From information on the endpapers of this book, compute the centripetal acceleration of a point on the surface of the Earth at the equator caused by the rotation of the Earth about its axis.

Jacob Paiste

Numerade Educator

Problem 29

Review. The 20 -g centrifuge at NASA's Ames Research Center in Mountain View, California, is a horizontal, cylindrical tube 58 ft long and is represented in Figure $\mathrm{P} 4.29$ Assume an astronaut in training sits in a seat at one end, facing the axis of rotation 29.0 $\mathrm{ft}$ away. Determine the rotation rate, in revolutions per second, required to give the astronaut a centripetal acceleration of 20.0$g$ .

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 30

An athlete swings a ball, connected to the end of a chain, in a horizontal circle. The athlete is able to rotate the ball at the rate of 8.00 $\mathrm{rev} / \mathrm{s}$ when the length of the chain is $0.600 \mathrm{m} .$ When he increases the length to $0.900 \mathrm{m},$ he is able to rotate the ball only 6.00 rev/s. (a) Which rate of rotation gives the greater speed for the ball? (b) What is the centripetal acceleration of the ball at 8.00 $\mathrm{rev} / \mathrm{s}$ ? (c) What is the centripetal acceleration at 6.00 $\mathrm{rev} / \mathrm{s}$ ?

Jacob Paiste

Numerade Educator

Problem 31

A train slows down as it rounds a sharp horizontal turn, going from 90.0 $\mathrm{km} / \mathrm{h}$ to 50.0 $\mathrm{km} / \mathrm{h}$ in the 15.0 $\mathrm{s}$ it takes to round the bend. The radius of the curve is $150 \mathrm{m} .$ Compute the acceleration at the moment the train speed reaches 50.0 $\mathrm{km} / \mathrm{h}$ . Assume the train continues to slow down at this time at the same rate.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 32

Figure $\mathrm{P} 4.32$ represents the total acceleration of a particle moving clockwise in a circle of radius 2.50 $\mathrm{m}$ at a certain instant of time. For that instant, find (a) the radial acceleration of the particle, (b) the speed of the particle, and (c) its tangential acceleration.

Jacob Paiste

Numerade Educator

Problem 33

(a) Can a particle moving with instantaneous speed 3.00 $\mathrm{m} / \mathrm{s}$ on a path with radius of curvature 2.00 $\mathrm{m}$ have an acceleration of magnitude 6.00 $\mathrm{m} / \mathrm{s}^{2} ?$ (b) Can it have an acceleration of magnitude 4.00 $\mathrm{m} / \mathrm{s}^{2} ?$ In each case, if the answer is yes, explain how it can happen; if the answer is no, explain why not.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 34

A ball swings in a vertical circle at the end of a rope 1.50 $\mathrm{m}$ long. When the ball is $36.9^{\circ}$ past the lowest point on its way up, its total acceleration is $(-22.5 \hat{\mathbf{i}}+20.2 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^{2} .$ For that instant, (a) sketch a vector diagram showing the components of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the ball.

Jacob Paiste

Numerade Educator

Problem 35

A police car traveling at 95.0 $\mathrm{km} / \mathrm{h}$ is traveling west, chasing a motorist traveling at 80.0 $\mathrm{km} / \mathrm{h}$ . ( a) What is the velocity of the motorist relative to the police car? (b) What is the velocity of the police car relative to the motorist? (c) If they are originally 250 $\mathrm{m}$ apart, in what time interval will the police car overtake the motorist?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 36

If A moving beltway at an airport has a speed $v_{1}$ and a length $L .$ A woman stands on the beltway as it moves from one end to the other, while a man in a hurry to reach his flight walks on the beltway with a speed of $v_{2}$ relative to the moving beltway. (a) What time interval is required for the woman to travel the distance L? (b) What time interval is required for the man to travel this distance? (c) A second beltway is located next to the first one. It is identical to the first one but moves in the opposite direction at speed $v_{1} \cdot$ Just as the man steps onto the beginning of the beltway and begins to walk at speed $v_{2}$ relative to his beltway, a child steps on the other end of the adjacent beltway. The child stands at rest relative to this second beltway. How long after stepping on the beltway does the man pass the child?

Jacob Paiste

Numerade Educator

Problem 37

An airplane maintains a speed of 630 $\mathrm{km} / \mathrm{h}$ relative to the air it is flying through as it makes a trip to a city 750 $\mathrm{km}$ away to the north. ( a) What time interval is required for the trip if the plane flies through a headwind blowing at 35.0 $\mathrm{km} / \mathrm{h}$ toward the south? (b) What time interval is required if there is a tailwind with the same speed? (c) What time interval is required if there is a crosswind blowing at 35.0 $\mathrm{km} / \mathrm{h}$ to the east relative to the ground?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 38

A car travels due east with a speed of 50.0 $\mathrm{km} / \mathrm{h}$ . Raindrops are falling at a constant speed vertically with respect to the Earth. The traces of the rain on the side windows of the car make an angle of $60.0^{\circ}$ with the vertical. Find the velocity of the rain with respect to (a) the car and (b) the Earth.

Jacob Paiste

Numerade Educator

Problem 39

A bolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 $\mathrm{m} / \mathrm{s}^{2}$ . (a) What is the acceleration of the bolt relative to the train car? (b) What is the acceleration of the bolt relative to the Earth? (c) Describe the trajectory of the bolt as seen by an observer inside the train car. (d) Describe the trajectory of the bolt as seen by an observer fixed on the Earth.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 40

A river has a steady speed of 0.500 $\mathrm{m} / \mathrm{s}$ . A student swims upstream a distance of 1.00 $\mathrm{km}$ and swims back to the starting point. ( a) If the student can swim at a speed of 1.20 $\mathrm{m} / \mathrm{s}$ in still water, how long does the trip take? (b) How $\mathrm{m}$ much time is required in still water for the same length swim? (c) Intuitively, why does the swim take longer when there is a current?

Jacob Paiste

Numerade Educator

Problem 41

A river flows with a steady speed $v$ . A student swims upstream a distance $d$ and then back to the starting point. The student can swim at speed $c$ in still water. (a) In terms of $d, v,$ and $c$ , what time interval is required for the round trip? (b) What time interval would be required if the water were still? (c) Which time interval is larger? Explain whether it is always larger.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 42

A farm truck moves due east with a constant velocity of 9.50 $\mathrm{m} / \mathrm{s}$ on a limitless, horizontal stretch of road. A boy riding on the back of the truck throws a can of soda upward (Fig. P4.42) and catches the projectile at the same location on the truck bed, but 16.0 $\mathrm{m}$ farther down the road. (a) In the frame of reference of the truck, at what angle to the vertical does the boy throw the can? (b) What is the initial speed of the can relative to the truck? (c) What is the shape of the can's trajectory as seen by the boy? An observer on the ground watches the boy throw the can and catch it. In this observer's frame of reference, (d) describe the shape of the can's path and (e) determine the initial velocity of the can.

Jacob Paiste

Numerade Educator

Problem 43

A science student is riding on a flatcar of a train traveling along a straight, horizontal track at a constant speed of 10.0 $\mathrm{m} / \mathrm{s}$ . The student throws a ball into the air along a path that he judges to make an initial angle of $60.0^{\circ}$ with the horizontal and to be in line with the track. The student's professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does she see the ball rise?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 44

A Coast Guard cutter detects an unidentified ship at a distance of 20.0 $\mathrm{km}$ in the direction $15.0^{\circ}$ east of north. The ship is traveling at 26.0 $\mathrm{km} / \mathrm{h}$ on a course at $40.0^{\circ}$ east of north. The Coast Guard wishes to send a speedboat to intercept and investigate the vessel. If the speedboat travels at 50.0 $\mathrm{km} / \mathrm{h}$ , in what direction should it head? Express the direction as a compass bearing with respect to due north.

Jacob Paiste

Numerade Educator

Problem 45

Why is the following situation impossible? A normally proportioned adult walks briskly along a straight line in the $+x$ direction, standing straight up and holding his right arm vertical and next to his body so that the arm does not swing. His right hand holds a ball at his side a distance $h$ above the floor. When the ball passes above a point marked as $x=0$ on the horizontal floor, he opens his fingers to release the ball from rest relative to his hand. The ball strikes the ground for the first time at position $x=7.00 h .$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 46

A particle starts from the origin with velocity 5$\hat{i} \mathrm{m} / \mathrm{s}$ at $t=0$ and moves in the $x y$ plane with a varying acceleration given by $\overrightarrow{\mathbf{a}}=(6 \sqrt{t} \hat{\mathbf{j}}),$ where $\overrightarrow{\mathbf{a}}$ is in meters per second squared and $t$ is in seconds. (a) Determine the velocity of the particle as a function of time. (b) Determine the position of the particle as a function of time.

Jacob Paiste

Numerade Educator

Problem 47

The "Vomit Comet." In microgravity astronaut training and equipment testing, NASA flies a KC135A aircraft along a parabolic flight path. As shown in Figure $\mathrm{P} 4.47,$ the aircraft climbs from $24000 \mathrm{ft}$ to $31000 \mathrm{ft},$ where it enters a parabolic path with a velocity of $143 \mathrm{~m} / \mathrm{s}$ nose high at $45.0^{\circ}$ and exits with velocity $143 \mathrm{~m} / \mathrm{s}$ at $45.0^{\circ}$ nose low. During this portion of the flight, the aircraft and objects inside its padded cabin are in free fall; astronauts and equipment float freely as if there were no gravity. What are the aircraft's
(a) speed and (b) altitude at the top of the maneuver?
(c) What is the time interval spent in microgravity?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 48

A boy throws a stone horizontally from the top of a cliff of height $h$ toward the ocean below. The stone strikes the ocean at distance $d$ from the base of the cliff. In terms of $h, d,$ and $g,$ find expressions for (a) the time $t$ at which the stone lands in the ocean, (b) the initial speed of the stone, (c) the speed of the stone immediately before it reaches the ocean, and (d) the direction of the stone's velocity immediately before it reaches the ocean.

Jacob Paiste

Numerade Educator

Problem 49

Lisa in her Lamborghini accelerates at the rate of $(3.00 \hat{\mathrm{i}}-2.00 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}^{2},$ while Jill in her Jaguar accelerates at $(1.00 \hat{\mathrm{i}}+3.00 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}^{2}$ . They both start from rest at the origin of an $x y$ coordinate system. After $5.00 \mathrm{s},$ (a) what is Lisa's speed with respect to Jill, (b) how far apart are they, and $(\mathrm{c})$ what is Lisa's acceleration relative to Jill?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 50

A basketball player is standing on the floor 10.0 $\mathrm{m}$ from the basket as in Figure $\mathrm{P} 4.50 .$ The height of the basket is $3.05 \mathrm{m},$ and he shoots the ball at a $40.0^{\circ}$ angle with the horizontal from a height of 2.00 $\mathrm{m}$ . (a) What is the acceleration of the basketball at the highest point in its trajectory? (b) At what speed must the player throw the basketball so that the ball goes through the hoop without striking the backboard?

Jacob Paiste

Numerade Educator

Problem 51

A flea is at point on a horizontal turntable, 10.0 cm from the center. The turntable is rotating at 33.3 rev/min in the clockwise direction. The flea jumps straight up to a height of 5.00 cm. At takeoff, it gives itself no horizontal velocity relative to the turntable. The flea lands on the turntable at point . Choose the origin of coordinates to be at the center of the turntable and the positive x axis passing through at the moment of takeoff. Then the original position of the flea is 10.0$\hat{\mathbf{i}} \mathrm{cm}$. (a) Find the position of point when the flea lands. (b) Find the position of point when the flea lands.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 52

As some molten metal splashes, one droplet flies off to the east with initial velocity $v_{i}$ at angle $\theta_{i}$ above the horizontal, and another droplet flies off to the west with the same speed at the same angle above the horizontal as shown in Figure $\mathrm{P} 4.52$ . In terms of $v_{i}$ and $\theta_{i},$ find the distance between the two droplets as a function of time.

Jacob Paiste

Numerade Educator

Problem 53

Why is the following situation impossible? Manny Ramírez hits a home run so that the baseball just clears the top row of bleachers, 24.0 $\mathrm{m}$ high, located 130 $\mathrm{m}$ from home plate. The ball is hit at 41.7 $\mathrm{m} / \mathrm{s}$ at an angle of $35.0^{\circ}$ to the horizontal, and air resistance is negligible.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 54

A pendulum with a cord of length $r=1.00 \mathrm{m}$ swings in a vertical plane (Fig. P4.54). When the pendulum is in the two horizontal positions $\theta=$ $90.0^{\circ}$ and $\theta=270^{\circ}$ , its speed is 5.00 $\mathrm{m} / \mathrm{s}$ . Find the magnitude of (a) the radial acceleration and (b) the tangential acceleration for these positions. (c) Draw vector diagrams to determine the direction of the total acceleration for these two positions. (d) Calculate the magnitude and direction of the total acceleration at these two positions.

Ummatul Choudary

Ummatul Choudary

Numerade Educator

Problem 55

Two blocks of masses $m_{1}$ and $m_{2}$ are placed on a table in contact with each other as discussed in Example 5.7 and shown in Active Figure $5.12 \mathrm{a}$. The coefficient of kinetic friction between the block of mass $m_{1}$ and the table is $\mu_{1}$, and that between the block of mass $m_{2}$ and the table is $\mu_{2}$. A horizontal force of magnitude $F$ is applied to the block of mass $m_{1} .$ We wish to find $P,$ the magnitude of the contact force between the blocks. (a) Draw diagrams showing the forces for each block. (b) What is the net force on the system of two blocks? (c) What is the net force acting on $m_{1}$ ? (d) What is the net force acting on $m_{2} ?$ (e) Write Newton's second law in the $x$ direction for each block. (f) Solve the two equations in two unknowns for the acceleration $a$ of the blocks in terms of the masses, the applied force $F$, the coefficients of friction, and $g .(\mathrm{g})$ Find the magnitude $P$ of the contact force between the blocks in terms of the same quantities.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 56

A projectile is launched from the point $(x=0, y=0)$, with velocity $(12.0 \mathrm{i}+49.0 \mathrm{j}) \mathrm{m} / \mathrm{s},$ at $t=0 .$ (a) Make a table listing the projectile's distance $|\overrightarrow{\mathrm{r}}|$ from the origin at the end of each second thereafter, for $0 \leq t \leq 10 \mathrm{s}$ . Tabulating the $x$ and $y$ coordinates and the components of velocity $v_{x}$ and $v_{y}$ will also be useful. (b) Notice that the projectile's distance from its starting point increases with time, goes through a maximum, and starts to decrease. Prove that the distance is a maximum when the position vector is perpendicular to the velocity. Suggestion: Argue that if $\vec{v}$ is not perpendicular to $\overrightarrow{\mathbf{r}},$ then $|\overrightarrow{\mathbf{r}}|$ must be increasing or decreasing. (c) Determine the magnitude of the maximum displacement. (d) Explain your method for solving part (c).

Jacob Paiste

Numerade Educator

Problem 57

A spring cannon is located at the edge of a table that is 1.20 $\mathrm{m}$ above the floor. A steel ball is launched from the cannon with speed $v_{i}$ at $35.0^{\circ}$ above the horizontal. (a) Find the horizontal position of the ball as a function of $v_{i}$ at the instant it lands on the floor. We write this function as $x\left(v_{i}\right)$ Evaluate $x$ for $\left(\text { b ) } v_{i}=0.100 \mathrm{m} / \mathrm{s} \text { and for } v_{i}=100 \mathrm{m} / \mathrm{s}\right.$ (d) Assume $v_{i}$ is close to but not equal to zero. Show that one term in the answer to part (a) dominates so that the function $x\left(v_{i}\right)$ reduces to a simpler form. (e) If $v_{i}$ is very large, what is the approximate form of $x\left(v_{i}\right) ?(\text { ( ) Describe }$ the overall shape of the graph of the function $x\left(v_{i}\right)$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 58

An outfielder throws a baseball to his catcher in an attempt to throw out a runner at home plate. The ball bounces once before reaching the catcher. Assume the angle at which the bounced ball leaves the ground is the same as the angle at which the outfielder threw it as shown in Figure $\mathrm{P} 4.58$ , but that the ball's speed after the bounce is one-half of what it was before the bounce. (a) Assume the ball is always thrown with the same initial speed and ignore air resistance. At what angle $\theta$ should the fielder throw the ball to make it go the same distance $D$ with one bounce (blue path) as a ball thrown upward at $45.0^{\circ}$ with no bounce (green path)? (b) Determine the ratio of the time interval for the one- bounce throw to the flight time for the no-bounce throw.

Abhishek Kumar

Numerade Educator

Problem 59

A World War II bomber flies horizontally over level terrain with a speed of 275 $\mathrm{m} / \mathrm{s}$ relative to the ground and at an altitude of 3.00 $\mathrm{km}$ . The bombardier releases one bomb. (a) How far does the bomb travel horizontally between its release and its impact on the ground? Ignore the effects of air resistance. (b) The pilot maintains the plane's original course, altitude, and speed through a storm of flak. Where is the plane when the bomb hits the ground? (c) The bomb hits the target seen in the telescopic bombsight at the moment of the bomb's release. At what angle from the vertical was the bombsight set?

Keshav Singh

Numerade Educator

Problem 60

A truck loaded with cannonball watermelons stops suddenly to avoid running over the edge of a washed-out bridge (Fig. P4.60). The quick stop causes a number of melons to fly off the truck. One melon leaves the hood of the truck with an initial speed $v_{i}=10.0 \mathrm{m} / \mathrm{s}$ in the horizontal direction. A cross section of the bank has the shape of the bottom half of a parabola, with its vertex at the initial location of the projected watermelon and with the equation $y^{2}=16 x,$ where $x$ and $y$ are measured in meters. What are the $x$ and $y$ coordinates of the melon when it splatters on the bank?

Donald Albin

Numerade Educator

Problem 61

A hawk is flying horizontally at 10.0 $\mathrm{m} / \mathrm{s}$ in a straight line, 200 $\mathrm{m}$ above the ground. A mouse it has been carrying struggles free from its talons. The hawk continues on its path at the same speed for 2.00 s before attempting to retrieve its prey. To accomplish the retrieval, it dives in a straight line at constant speed and recaptures the mouse 3.00 $\mathrm{m}$ above the ground. (a) Assuming no air resistance acts on the mouse, find the diving speed of the hawk. (b) What angle did the hawk make with the horizontal during its descent? (c) For what time interval did the mouse experience free fall?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 62

An aging coyote cannot run fast enough to catch a roadrunner. He purchases on eBay a set of jet-powered roller skates, which provide a constant horizontal acceleration of 15.0 $\mathrm{m} / \mathrm{s}^{2}$ (Fig. P4.62). The coyote starts at rest 70.0 m from the edge of a cliff at the instant the road-runner zips past in the direction of the cliff. (a) Determine the minimum constant speed the roadrunner must have to reach the cliff before the coyote. At the edge of the cliff, the roadrunner escapes by making a sudden turn, while the coyote continues straight ahead. The coyote's skates remain horizontal and continue to operate while he is in flight, so his acceleration while in the air is $(15.0 \hat{\mathbf{i}}-9.80 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^{2}$ . (b) The cliff is 100 $\mathrm{m}$ above the flat floor of the desert. Determine how far from the base of the vertical cliff the coyote lands. (c) Determine the components of the coyote's impact velocity.

Jacob Paiste

Numerade Educator

Problem 63

A car is parked on a steep incline, making an angle of $37.0^{\circ}$ below the horizontal and overlooking the ocean, when its brakes fail and it begins to roll. Starting from rest at $t=0,$ the car rolls down the incline with a constant acceleration of 4.00 $\mathrm{m} / \mathrm{s}^{2}$ , traveling 50.0 $\mathrm{m}$ to the edge of a vertical cliff. The cliff is 30.0 $\mathrm{m}$ above the ocean. Find (a) the speed of the car when it reaches the edge of the cliff, (b) the time interval elapsed when it arrives there, (c) the velocity of the car when it lands in the ocean, (d) the total time interval the car is in motion, and (e) the position of the car when it lands in the ocean, relative to the base of the cliff.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 64

Do not hurt yourself; do not strike your hand against anything. Within these limitations, describe what you do to give your hand a large acceleration. Compute an order-of-magnitude estimate of this acceleration, stating the quantities you measure or estimate and their values.

Jacob Paiste

Numerade Educator

Problem 65

A fisherman sets out upstream on a river. His small boat, powered by an outboard motor, travels at a constant speed $v$ in still water. The water flows at a lower constant speed $v_{w}$ . The fisherman has traveled upstream for 2.00 $\mathrm{km}$ when his ice chest falls out of the boat. He notices that the chest is missing only after he has gone upstream for another 15.0 $\mathrm{min}$ . At that point, he turns around and heads back downstream, all the time traveling at the same speed relative to the water. He catches up with the floating ice chest just as he returns to his starting point. How fast is the river flowing? Solve this problem in two ways. (a) First, use the Earth as a reference frame. With respect to the Earth, the boat travels upstream at speed $v-v_{w}$ and downstream at $v+v_{w^{*}}$ (b) $A$ second much simpler and more elegant solution is obtained by using the water as the reference frame. This approach has important applications in many more complicated problems; examples are calculating the motion of rockets and satellites and analyzing the scattering of subatomic particles from massive targets.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 66

Two swimmers, Chris and Sarah, start together at the same point on the bank of a wide stream that flows with a speed $v .$ Both move at the same speed $c$ (where $c > v )$ relative to the water. Chris swims downstream a distance $L$ and then upstream the same distance. Sarah swims so that her motion relative to the Earth is perpendicular to the banks of the stream. She swims the distance $L$ and then back the same distance, with both swimmers returning to the starting point. In terms of $L, c,$ and $v,$ find the time intervals required (a) for Chris's round trip and (b) for Sarah's round trip. (c) Explain which swimmer returns first.

Surendra Kumar

Numerade Educator

Problem 67

A skier leaves the ramp of a ski jump with a velocity of $v=10.0 \mathrm{m} / \mathrm{s}$ at $\theta=15.0^{\circ}$ above the horizontal as shown in Figure $\mathrm{P} 4.67 .$ The slope where she will land is inclined downward at $\phi=50.0^{\circ},$ and air resistance is negligible. Find (a) the distance from the end of the ramp to where the jumper lands and (b) her velocity components just before the landing. (c) Explain how you think the results might be affected if air resistance were included.

Keshav Singh

Numerade Educator

Problem 68

A person standing at the top of a hemispherical rock of radius $R$ kicks a ball (initially at rest on the top of the rock) to give it horizontal velocity $\vec{v}_{i}$ as shown in Figure $\mathrm{P} 4.68$ . (a) What must be its minimum initial speed if the ball is never to hit the rock after it is kicked? (b) With this initial speed, how far from the base of the rock does the ball hit the ground?

Jacob Paiste

Numerade Educator

Problem 69

A dive-bomber has a velocity of 280 $\mathrm{m} / \mathrm{s}$ at an angle $\theta$ below the horizontal. When the altitude of the aircraft is $2.15 \mathrm{km},$ it releases a bomb, which subsequently hits a target on the ground. The magnitude of the displacement from the point of release of the bomb to the target is $3.25 \mathrm{km} .$ Find the angle $\theta .$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 70

A fireworks rocket explodes at height $h,$ the peak of its vertical trajectory. It throws out burning fragments in all directions, but all at the same speed $v$ . Pellets of solidified metal fall to the ground without air resistance. Find the smallest angle that the final velocity of an impacting fragment makes with the horizontal.

Jacob Paiste

Numerade Educator

Problem 71

An enemy ship is on the east side of a mountain island as shown in Figure $\mathrm{P} 4.71$ . The enemy ship has maneuvered to within 2500 $\mathrm{m}$ of the 1800 -m-high mountain peak and can shoot projectiles with an initial speed of 250 $\mathrm{m} / \mathrm{s}$ . If the western shoreline is horizontally 300 $\mathrm{m}$ from the peak, what are the distances from the western shore at which a ship can be safe from the bombardment of the enemy ship?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 72

In the What If? section of Example 4.5 , it was claimed that the maximum range of a ski jumper occurs for a launch angle $\theta$ given by
$$\theta=45^{\circ}-\frac{\phi}{2}$$
where $\phi$ is the angle the hill makes with the horizontal in Figure 4.14 . Prove this claim by deriving the equation above.

Jacob Paiste

Numerade Educator