Chapter 4

A suspicious-looking man runs as fast as he can along a moving sidewalk from one end to the other, taking \(2.50 \mathrm{~s}\). Then security agents appear, and the man runs as fast as he can back along the sidewalk to his starting point, taking \(10.0 \mathrm{~s}\). What is the ratio of the man's running speed to the sidewalk's speed?

A rugby player runs with the ball directly toward his opponent's goal, along the positive direction of an \(x\) axis. He can legally pass the ball to a teammate as long as the ball's velocity relative to the field does not have a positive \(x\) component. Suppose the player runs at speed \(4.0 \mathrm{~m} / \mathrm{s}\) relative to the field while he passes the ball with velocity \(\vec{v}_{B P}\) relative to himself. If \(\vec{v}_{B P}\) has magnitude \(6.0 \mathrm{~m} / \mathrm{s}\), what is the smallest angle it can have for the pass to be legal?

After flying for 15 min in a wind blowing \(42 \mathrm{~km} / \mathrm{h}\) at an angle of \(20^{\circ}\) south of east, an airplane pilot is over a town that is \(55 \mathrm{~km}\) due north of the starting point. What is the speed of the airplane relative to the air?

A train travels due south at \(30 \mathrm{~m} / \mathrm{s}\) (relative to the ground) in a rain that is blown toward the south by the wind. The path of each raindrop makes an angle of \(70^{\circ}\) with the vertical, as measured by an observer stationary on the ground. An observer on the train, however, sees the drops fall perfectly vertically. Determine the speed of the raindrops relative to the ground.

A light plane attains an airspeed of \(500 \mathrm{~km} / \mathrm{h}\). The pilot sets out for a destination \(800 \mathrm{~km}\) due north but discovers that the plane must be headed \(20.0^{\circ}\) east of due north to fly there directly. The plane arrives in \(2.00 \mathrm{~h}\). What were the (a) magnitude and (b) direction of the wind velocity?

Snow is falling vertically at a constant speed of \(8.0 \mathrm{~m} / \mathrm{s}\). At what angle from the vertical do the snowflakes appear to be falling as viewed by the driver of a car traveling on a straight, level road with a speed of \(50 \mathrm{~km} / \mathrm{h} ?\)

Two ships, \(A\) and \(B\), leave port at the same time. Ship \(A\) travels northwest at 24 knots, and ship \(B\) travels at 28 knots in a direction \(40^{\circ}\) west of south. \((1 \mathrm{knot}=1\) nautical mile per hour; see Appendix D.) What are the (a) magnitude and (b) direction of the velocity of ship \(A\) relative to \(B ?\) (c) After what time will the ships be 160 nautical miles apart? (d) What will be the bearing of \(B\) (the direction of \(B\) 's position) relative to \(A\) at that time?

A \(200-\mathrm{m}\) -wide river flows due east at a uniform speed of \(2.0 \mathrm{~m} / \mathrm{s} .\) A boat with a speed of \(8.0 \mathrm{~m} / \mathrm{s}\) relative to the water leaves the south bank pointed in a direction \(30^{\circ}\) west of north. What are the (a) magnitude and (b) direction of the boat's velocity relative to the ground? (c) How long does the boat take to cross the river?

\(A\) is located \(4.0 \mathrm{~km}\) north and \(2.5 \mathrm{~km}\) east of ship \(B\). Ship \(A\) has a velocity of \(22 \mathrm{~km} / \mathrm{h}\) toward the south, and ship \(B\) has a velocity of \(40 \mathrm{~km} / \mathrm{h}\) in a direction \(37^{\circ}\) north of east. (a) What is the velocity of \(A\) relative to \(B\) in unit-vector notation with \(\hat{\mathrm{i}}\) toward the east? (b) Write an expression (in terms of \(\hat{\mathrm{i}}\) and \(\hat{\mathrm{j}}\) ) for the position of \(A\) relative to \(B\) as a function of \(t\), where \(t=0\) when the ships are in the positions described above. (c) At what time is the separation between the ships least? (d) What is that least separation?

A woman who can row a boat at \(6.4 \mathrm{~km} / \mathrm{h}\) in still water faces a long, straight river with a width of \(6.4 \mathrm{~km}\) and a current of \(3.2 \mathrm{~km} / \mathrm{h}\). Let \(\hat{\text { i }}\) point directly across the river and \(\hat{\mathrm{j}}\) point directly downstream. If she rows in a straight line to a point directly opposite her starting position, (a) at what angle to î must she point the boat and (b) how long will she take? (c) How long will she take if, instead, she rows \(3.2 \mathrm{~km}\) down the river and then back to her starting point? (d) How long if she rows \(3.2 \mathrm{~km} u p\) the river and then back to her starting point? (e) At what angle to \(\hat{\mathbf{i}}\) should she point the boat if she wants to cross the river in the shortest possible time? (f) How long is that shortest time?

You are kidnapped by political-science majors (who are upset because you told them political science is not a real science). Although blindfolded, you can tell the speed of their car (by the whine of the engine), the time of travel (by mentally counting off seconds), and the direction of travel (by turns along the rectangular street system). From these clues, you know that you are taken along the following course: \(50 \mathrm{~km} / \mathrm{h}\) for \(2.0 \mathrm{~min}\), turn \(90^{\circ}\) to the right, \(20 \mathrm{~km} / \mathrm{h}\) for \(4.0 \mathrm{~min}\), turn \(90^{\circ}\) to the right, \(20 \mathrm{~km} / \mathrm{h}\) for \(60 \mathrm{~s}\), turn \(90^{\circ}\) to the left, \(50 \mathrm{~km} / \mathrm{h}\) for \(60 \mathrm{~s}\), turn \(90^{\circ}\) to the right, \(20 \mathrm{~km} / \mathrm{h}\) for \(2.0 \mathrm{~min}\), turn \(90^{\circ}\) to the left, \(50 \mathrm{~km} / \mathrm{h}\) for \(30 \mathrm{~s}\). At that point, (a) how far are you from your starting point, and (b) in what direction relative to your initial direction of travel are you?

A radar station detects an airplane approaching directly from the east. At first observation, the airplane is at distance \(d_{1}=360 \mathrm{~m}\) from the station and at angle \(\theta_{1}=40^{\circ}\) above the horizon (Fig. \(4-49\) ). The airplane is tracked through an angular change \(\Delta \theta=123^{\circ}\) in the vertical east-west plane; its distance is then \(d_{2}=790 \mathrm{~m}\). Find the (a) magnitude and (b) direction of the airplane's displacement during this period.

A baseball is hit at ground level. The ball reaches its maximum height above ground level \(3.0 \mathrm{~s}\) after being hit. Then \(2.5 \mathrm{~s}\) after reaching its maximum height, the ball barely clears a fence that is \(97.5 \mathrm{~m}\) from where it was hit. Assume the ground is level. (a) What maximum height above ground level is reached by the ball? (b) How high is the fence? (c) How far beyond the fence does the ball strike the ground?

Long flights at midlatitudes in the Northern Hemisphere encounter the jet stream, an eastward airflow that can affect a plane's speed relative to Earth's surface. If a pilot maintains a certain speed relative to the air (the plane's airspeed), the speed relative to the surface (the plane's ground speed) is more when the flight is in the direction of the jet stream and less when the flight is opposite the jet stream. Suppose a round-trip flight is scheduled between two cities separated by \(4000 \mathrm{~km}\), with the outgoing flight in the direction of the jet stream and the return flight opposite it. The airline computer advises an airspeed of \(1000 \mathrm{~km} / \mathrm{h}\), for which the difference in flight times for the outgoing and return flights is \(70.0 \mathrm{~min}\). What jet-stream speed is the computer using?

A particle starts from the origin at \(t=0\) with a velocity of \(8.0 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}\) and moves in the \(x y\) plane with constant acceleration \((4.0 \hat{\mathrm{i}}+2.0 \mathrm{j}) \mathrm{m} / \mathrm{s}^{2} .\) When the particle's \(x\) coordinate is \(29 \mathrm{~m}\), what are its (a) \(y\) coordinate and (b) speed?

An astronaut is rotated in a horizontal centrifuge at a radius of \(5.0 \mathrm{~m}\). (a) What is the astronaut's speed if the centripetal acceleration has a magnitude of \(7.0 \mathrm{~g} ?\) (b) How many revolutions per minute are required to produce this acceleration? (c) What is the period of the motion?

\(A\) is \(90 \mathrm{~km}\) due west of oasis \(B\). A desert camel leaves \(A\) and takes \(50 \mathrm{~h}\) to walk \(75 \mathrm{~km}\) at \(37^{\circ}\) north of due east. Next it takes \(35 \mathrm{~h}\) to walk \(65 \mathrm{~km}\) due south. Then it rests for \(5.0 \mathrm{~h}\). What are the (a) magnitude and (b) direction of the camel's displacement relative to \(A\) at the resting point? From the time the camel leaves \(A\) until the end of the rest period, what are the (c) magnitude and (d) direction of its average velocity and (e) its average speed? The camel's last drink was at \(A\); it must be at \(B\) no more than \(120 \mathrm{~h}\) later for its next drink. If it is to reach \(B\) just in time, what must be the (f) magnitude and (g) direction of its average velocity after the rest period?

For women's volleyball the top of the net is \(2.24 \mathrm{~m}\) above the floor and the court measures \(9.0 \mathrm{~m}\) by \(9.0 \mathrm{~m}\) on each side of the net. Using a jump serve, a player strikes the ball at a point that is \(3.0 \mathrm{~m}\) above the floor and a horizontal distance of \(8.0 \mathrm{~m}\) from the net. If the initial velocity of the ball is horizontal, (a) what minimum magnitude must it have if the ball is to clear the net and (b) what maximum magnitude can it have if the ball is to strike the floor inside the back line on the other side of the net?

A rifle is aimed horizontally at a target \(30 \mathrm{~m}\) away. The bullet hits the target \(1.9 \mathrm{~cm}\) below the aiming point. What are (a) the bullet's time of flight and (b) its speed as it emerges from the rifle?

A particle is in uniform circular motion about the origin of an \(x y\) coordinate system, moving clockwise with a period of \(7.00 \mathrm{~s}\). At one instant, its position vector (measured from the origin) is \(\vec{r}=(2.00 \mathrm{~m}) \hat{\mathrm{i}}-(3.00 \mathrm{~m}) \hat{\mathrm{j}}\). At that instant, what is its velocity in unit-vector notation?

An iceboat sails across the surface of a frozen lake with constant acceleration produced by the wind. At a certain instant the boat's velocity is \((6.30 \hat{\mathrm{i}}-8.42 \mathrm{j}) \mathrm{m} / \mathrm{s}\). Three seconds later, because of a wind shift, the boat is instantaneously at rest. What is its average acceleration for this \(3.00 \mathrm{~s}\) interval?

A magnetic field forces an electron to move in a circle with radial acceleration \(3.0 \times 10^{14} \mathrm{~m} / \mathrm{s}^{2} .\) (a) What is the speed of the electron if the radius of its circular path is \(15 \mathrm{~cm} ?\) (b) What is the period of the motion?

In \(3.50 \mathrm{~h}\), a balloon drifts \(21.5 \mathrm{~km}\) north, \(9.70 \mathrm{~km}\) east, and \(2.88 \mathrm{~km}\) upward from its release point on the ground. Find (a) the magnitude of its average velocity and (b) the angle its average velocity makes with the horizontal.

A ball is thrown horizontally from a height of \(20 \mathrm{~m}\) and hits the ground with a speed that is three times its initial speed. What is the initial speed?

A projectile is launched with an initial speed of \(30 \mathrm{~m} / \mathrm{s}\) at an angle of \(60^{\circ}\) above the horizontal. What are the (a) magnitude and (b) angle of its velocity \(2.0 \mathrm{~s}\) after launch, and \((\mathrm{c})\) is the angle above or below the horizontal? What are the (d) magnitude and (e) angle of its velocity \(5.0 \mathrm{~s}\) after launch, and \((\mathrm{f})\) is the angle above or below the horizontal?

The position vector for a proton is initially \(\vec{r}=\) \(5.0 \hat{\mathrm{i}}-6.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}\) and then later is \(\vec{r}=-2.0 \hat{\mathrm{i}}+6.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}\), all in meters. (a) What is the proton's displacement vector, and (b) to what plane is that vector parallel?

A particle \(P\) travels with constant speed on a circle of radius \(r=\) \(3.00 \mathrm{~m}\) (Fig. \(4-56)\) and completes one revolution in \(20.0 \mathrm{~s}\). The particle passes through \(O\) at time \(t=0 .\) State the following vectors in magnitudeangle notation (angle relative to the positive direction of \(x\) ). With respect to \(O\), find the particle's position vector at the times \(t\) of (a) \(5.00 \mathrm{~s}\), (b) \(7.50 \mathrm{~s}\), and \((\mathrm{c}) 10.0 \mathrm{~s}\). (d) For the \(5.00 \mathrm{~s}\) interval from the end of the fifth second to the end of the tenth second, find the particle's displacement. For that interval, find (e) its average velocity and its velocity at the (f) beginning and (g) end. Next, find the acceleration at the (h) beginning and (i) end of that interval.

The fast French train known as the TGV (Train à Grande Vitesse) has a scheduled average speed of \(216 \mathrm{~km} / \mathrm{h} .\) (a) If the train goes around a curve at that speed and the magnitude of the acceleration experienced by the passengers is to be limited to \(0.050 \mathrm{~g}\), what is the smallest radius of curvature for the track that can be tolerated? (b) At what speed must the train go around a curve with a \(1.00 \mathrm{~km}\) radius to be at the acceleration limit?

A person walks up a stalled 15 -m-long escalator in \(90 \mathrm{~s}\). When standing on the same escalator, now moving, the person is carried up in \(60 \mathrm{~s}\). How much time would it take that person to walk up the moving escalator? Does the answer depend on the length of the escalator?

(a) What is the magnitude of the centripetal acceleration of an object on Earth's equator due to the rotation of Earth? (b) What would Earth's rotation period have to be for objects on the equator to have a centripetal acceleration of magnitude \(9.8 \mathrm{~m} / \mathrm{s}^{2} ?\)

The range of a projectile depends not only on \(v_{0}\) and \(\theta_{0}\) but also on the value \(g\) of the free-fall acceleration, which varies from place to place. In 1936 , Jesse Owens established a world's running broad jump record of \(8.09 \mathrm{~m}\) at the Olympic Games at Berlin (where \(g=9.8128 \mathrm{~m} / \mathrm{s}^{2}\) ). Assuming the same values of \(v_{0}\) and \(\theta_{0}\), by how much would his record have differed if he had competed instead in 1956 at Melbourne (where \(\left.g=9.7999 \mathrm{~m} / \mathrm{s}^{2}\right)\) ?

The position vector \(\vec{r}\) of a particle moving in the \(x y\) plane is \(\vec{r}=2 \hat{i}+2 \sin [(\pi / 4 \mathrm{rad} / \mathrm{s}) t] \hat{\mathrm{j}}, \quad\) with \(\vec{r}\) in meters and \(t\) in seconds. (a) Calculate the \(x\) and \(y\) components of the particle's position at \(t=0,1.0,2.0,3.0\), and \(4.0 \mathrm{~s}\) and sketch the particle's path in the \(x y\) plane for the interval \(0 \leq t \leq\) \(4.0 \mathrm{~s}\). (b) Calculate the components of the particle's velocity at \(t=1.0,2.0\), and \(3.0 \mathrm{~s}\). Show that the velocity is tangent to the path of the particle and in the direction the particle is moving at each time by drawing the velocity vectors on the plot of the particle's path in part (a). (c) Calculate the components of the particle's acceleration at \(t=1.0,2.0\), and \(3.0 \mathrm{~s}\).

An electron having an initial horizontal velocity of magnitude \(1.00 \times 10^{9} \mathrm{~cm} / \mathrm{s}\) travels into the region between two horizontal metal plates that are electrically charged. In that region, the electron travels a horizontal distance of \(2.00 \mathrm{~cm}\) and has a constant downward acceleration of magnitude \(1.00 \times 10^{17} \mathrm{~cm} / \mathrm{s}^{2}\) due to the charged plates. Find (a) the time the electron takes to travel the \(2.00 \mathrm{~cm},(\mathrm{~b})\) the vertical distance it travels during that time, and the magnitudes of its (c) horizontal and (d) vertical velocity components as it emerges from the region.

An elevator without a ceiling is ascending with a constant speed of \(10 \mathrm{~m} / \mathrm{s}\). A boy on the elevator shoots a ball directly upward, from a height of \(2.0 \mathrm{~m}\) above the elevator floor, just as the elevator floor is \(28 \mathrm{~m}\) above the ground. The initial speed of the ball with respect to the elevator is \(20 \mathrm{~m} / \mathrm{s}\). (a) What maximum height above the ground does the ball reach? (b) How long does the ball take to return to the elevator floor?

A football player punts the football so that it will have a "hang time" (time of flight) of \(4.5 \mathrm{~s}\) and land \(46 \mathrm{~m}\) away. If the ball leaves the player's foot \(150 \mathrm{~cm}\) above the ground, what must be the (a) magnitude and (b) angle (relative to the horizontal) of the ball's initial velocity?

An airport terminal has a moving sidewalk to speed passengers through a long corridor. Larry does not use the moving sidewalk; he takes \(150 \mathrm{~s}\) to walk through the corridor. Curly, who simply stands on the moving sidewalk, covers the same distance in \(70 \mathrm{~s}\). Moe boards the sidewalk and walks along it. How long does Moe take to move through the corridor? Assume that Larry and Moe walk at the same speed.

A wooden boxcar is moving along a stra?ght ralroad track at speed \(v_{1}\). A sniper fires a bullet (initial speed \(v_{2}\) ) at it from a high-powered rifle. The bullet passes through both lengthwise walls of the car, its entrance and exit holes being exactly opposite each other as viewed from within the car. From what direction, relative to the track, is the bullet fired? Assume that the bullet is not deflected upon entering the car, but that its speed decreases by \(20 \%\). Take \(v_{1}=85 \mathrm{~km} / \mathrm{h}\) and \(v_{2}=650 \mathrm{~m} / \mathrm{s}\). (Why don't you need to know the width of the boxcar?)

A sprinter running on a circular track has a velocity of constant magnitude \(9.20 \mathrm{~m} / \mathrm{s}\) and a centripetal acceleration of magnitude \(3.80 \mathrm{~m} / \mathrm{s}^{2}\). What are (a) the track radius and (b) the period of the circular motion?

Suppose that a space probe can withstand the stresses of a \(20 g\) acceleration. (a) What is the minimum turning radius of such a craft moving at a speed of one-tenth the speed of light? (b) How long would it take to complete a \(90^{\circ}\) turn at this speed?

You are to throw a ball with a speed of \(12.0 \mathrm{~m} / \mathrm{s}\) at a target that is height \(h=5.00 \mathrm{~m}\) above the level at which you release the ball (Fig. 4-58). You want the ball's velocity to be horizontal at the instant it reaches the target. (a) At what angle \(\theta\) above the horizontal must you throw the ball? (b) What is the horizontal distance from the release point to the target? (c) What is the speed of the ball just as it reaches the target?

A graphing surprise. At time \(t=0\), a burrito is launched from level ground, with an initial speed of \(16.0 \mathrm{~m} / \mathrm{s}\) and launch angle \(\theta_{0}\). Imagine a position vector \(\vec{r}\) continuously directed from the launching point to the burrito during the flight. Graph the magnitude \(r\) of the position vector for (a) \(\theta_{0}=40.0^{\circ}\) and (b) \(\theta_{0}=80.0^{\circ}\). For \(\theta_{0}=40.0^{\circ}\), (c) when does \(r\) reach its maximum value, (d) what is that value, and how far (e) horizontally and (f) vertically is the burrito from the launch point? For \(\theta_{0}=80.0^{\circ},(\mathrm{g})\) when does \(r\) reach its maximum value, (h) what is that value, and how far (i) horizontally and (j) vertically is the burrito from the launch point?

A frightened rabbit moving at \(6.00 \mathrm{~m} / \mathrm{s}\) due east runs onto a large area of level ice of negligible friction. As the rabbit slides across the ice, the force of the wind causes it to have a constant acceleration of \(1.40 \mathrm{~m} / \mathrm{s}^{2}\), due north. Choose a coordinate system with the origin at the rabbit's initial position on the ice and the positive \(x\) axis directed toward the east. In unit-vector notation, what are the rabbit's (a) velocity and (b) position when it has slid for \(3.00 \mathrm{~s}\) ?

The pilot of an aircraft flies due east relative to the ground in a wind blowing \(20.0 \mathrm{~km} / \mathrm{h}\) toward the south. If the speed of the aircraft in the absence of wind is \(70.0 \mathrm{~km} / \mathrm{h}\), what is the speed of the aircraft relative to the ground?

A helicopter is flying in a straight line over a level field at a constant speed of \(6.20 \mathrm{~m} / \mathrm{s}\) and at a constant altitude of \(9.50 \mathrm{~m}\). A package is ejected horizontally from the helicopter with an initial velocity of \(12.0 \mathrm{~m} / \mathrm{s}\) relative to the helicopter and in a direction opposite the helicopter's motion. (a) Find the initial speed of the package relative to the ground. (b) What is the horizontal distance between the helicopter and the package at the instant the package strikes the ground? (c) What angle does the velocity vector of the package make with the ground at the instant before impact, as seen from the ground?

A car travels around a flat circle on the ground, at a constant speed of \(12.0 \mathrm{~m} / \mathrm{s}\). At a certain instant the car has an acceleration of \(3.00 \mathrm{~m} / \mathrm{s}^{2}\) toward the east. What are its distance and direction from the center of the circle at that instant if it is traveling (a) clockwise around the circle and (b) counterclockwise around the circle?

A baseball is hit at Fenway Park in Boston at a point \(0.762 \mathrm{~m}\) above home plate with an initial velocity of \(33.53 \mathrm{~m} / \mathrm{s}\) directed \(55.0^{\circ}\) above the horizontal. The ball is observed to clear the \(11.28\) -m-high wall in left field (known as the "green monster") \(5.00 \mathrm{~s}\) after it is hit, at a point just inside the left-field foulline pole. Find (a) the horizontal distance down the left-field foul line from home plate to the wall; (b) the vertical distance by which the ball clears the wall; (c) the horizontal and vertical displacements of the ball with respect to home plate \(0.500 \mathrm{~s}\) before it clears the wall.

A woman can row a boat at \(6.40 \mathrm{~km} / \mathrm{h}\) in still water. (a) If she is crossing a river where the current is \(3.20 \mathrm{~km} / \mathrm{h}\), in what direction must her boat be headed if she wants to reach a point directly opposite her starting point? (b) If the river is \(6.40 \mathrm{~km}\) wide, how long will she take to cross the river? (c) Suppose that instead of crossing the river she rows \(3.20 \mathrm{~km}\) down the river and then back to her starting point. How long will she take? (d) How long will she take to row \(3.20 \mathrm{~km} u p\) the river and then back to her starting point? (e) In what direction should she head the boat if she wants to cross in the shortest possible time, and what is that time?