Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

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 (a) the total vector displacement,
(b) the average speed, and (c) the average velocity. Use a coordinate system in which east is the positive $x$ axis.

Thomas Thompson

Thomas Thompson

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Problem 2

Suppose that the position vector for a particle is given as $\mathbf{r}=x \mathbf{i}+y \mathbf{j}$, with $x=a t+b$ and $y=c t^{2}+d$, where $a=1.00 \mathrm{~m} / \mathrm{s}, b=1.00 \mathrm{~m}, c=0.125 \mathrm{~m} / \mathrm{s}^{2}$, and $d=1.00$
m. (a) Calculate the average velocity during the time interval from $t=2.00 \mathrm{~s}$ to $t=4.00 \mathrm{~s}$. (b) Determine the velocity and the speed at $t=2.00 \mathrm{~s}$.

Shelby Mohamed

Numerade Educator

Problem 3

A golf ball is hit off a tee at the edge of a cliff. Its $x$ and $y$ coordinates versus time are given by the following expressions:
$$
x=(18.0 \mathrm{~m} / \mathrm{s}) t
$$
and
$$
y=(4.00 \mathrm{~m} / \mathrm{s}) t-\left(4.90 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}
$$
(a) Write a vector expression for the ball's position as a function of time, using the unit vectors $\mathbf{i}$ and $\mathbf{j}$. By taking derivatives of your results, write expressions for (b) the velocity vector as a function of time and (c) the acceleration vector as a function of time. Now use unit vector notation to write expressions for (d) the position, (e) the velocity, and (f) the acceleration of the ball, all at $t=3.00 \mathrm{~s}$

Shelby Mohamed

Numerade Educator

Problem 4

The coordinates of an object moving in the $x y$ plane vary with time according to the equations
$$
x=-(5.00 \mathrm{~m}) \sin \omega t
$$
and
$$
y=(4.00 \mathrm{~m})-(5.00 \mathrm{~m}) \cos \omega t
$$
where $t$ is in seconds and $\omega$ has units of seconds $^{-1}$.
(a) Determine the components of velocity and components of acceleration at $t=0 .$ (b) Write expressions for the position vector, the velocity vector, and the acceleration vector at any time $t>0 .$ (c) Describe the path of the object on an $x y$ graph.

Shelby Mohamed

Numerade Educator

Problem 5

At $t=0$, a particle moving in the $x y$ plane with constant acceleration has a velocity of $\mathbf{v}_{i}=(3.00 \mathbf{i}-2.00 \mathbf{j}) \mathrm{m} / \mathrm{s}$
when it is at the origin. At $t=3.00 \mathrm{~s}$, the particle's velocity is $\mathbf{v}=(9.00 \mathbf{i}+7.00 \mathbf{j}) \mathrm{m} / \mathrm{s}$. Find (a) the acceleration of the particle and (b) its coordinates at any time $t$.

Shelby Mohamed

Numerade Educator

Problem 6

The vector position of a particle varies in time aceording to the expression $\mathbf{r}=\left(3.00 \mathbf{i}-6.00 t^{2} \mathbf{j}\right) \mathrm{m} .$ (a) Find expressions for the velocity and acceleration as functions of time. (b) Determine the particle's position and velocity at $t=1.00 \mathrm{~s}$

Shelby Mohamed

Numerade Educator

Problem 7

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

Shelby Mohamed

Numerade Educator

Problem 8

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

Shelby Mohamed

Numerade Educator

Problem 9

In a local bar, a customer slides an empty beer mug down the counter for a refill. The bartender is momentarily distracted and does not see the mug, which slides off the counter and strikes the floor $1.40 \mathrm{~m}$ from the base of the counter. If the height of the counter is $0.860 \mathrm{~m}$, (a) with what velocity did the mug leave the counter and (b) what was the direction of the mug's velocity just before it hit the floor?

Aaron Miller

Numerade Educator

Problem 10

In a local bar, a customer slides an empty beer mug down the counter for a refill. The bartender is momentarily distracted and does not see the mug, which slides off the counter and strikes the floor at distance $d$ from the base of the counter. If the height of the counter is $h$,
(a) with what velocity did the mug leave the counter and (b) what was the direction of the mug's velocity just before it hit the floor?

Aaron Miller

Numerade Educator

Problem 11

One strategy in a snowball fight is to throw a first snowball at a high angle over level ground. While your opponent is watching the first one, you throw a second one at a low angle and timed to arrive at your opponent before or at the same time as the first one. Assume both snowballs are thrown with a speed of $25.0 \mathrm{~m} / \mathrm{s}$. The first one is thrown at an angle of $70.0^{\circ}$ with respect to the horizontal. (a) At what angle should the second (lowangle) snowball be thrown if it is to land at the same point as the first? (b) How many seconds later should the second snowball be thrown if it is to land at the same time as the first?

Shelby Mohamed

Numerade Educator

Problem 12

A tennis player standing $12.6 \mathrm{~m}$ from the net hits the ball at $3.00^{\circ}$ above the horizontal. To clear the net, the ball must rise at least $0.330 \mathrm{~m}$. If the ball just clears the net at the apex of its trajectory, how fast was the ball moving when it left the racket?

Shelby Mohamed

Numerade Educator

Problem 13

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 a 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?

Shelby Mohamed

Numerade Educator

Problem 14

An astronaut on a strange planet finds that she can jump a maximum horizontal distance of $15.0 \mathrm{~m}$ if her initial speed is $8.00 \mathrm{~m} / \mathrm{s}$. What is the free-fall acceleration on the planet?

Shelby Mohamed

Numerade Educator

Problem 15

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? Give your answer to three significant figures.

Prabhu Ramji

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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 $3.00 \mathrm{~s}$ 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?

Chasen Shaw

Numerade Educator

Problem 17

A cannon with a muzzle speed of $1000 \mathrm{~m} / \mathrm{s}$ is used to start an avalanche on a mountain slope. The target is $2000 \mathrm{~m}$ from the cannon horizontally and $800 \mathrm{~m}$ above the cannon. At what angle, above the horizontal, should the cannon be fired?

Shelby Mohamed

Numerade Educator

Problem 18

Consider a projectile that is launched from the origin of an $x y$ coordinate system with speed $v_{i}$ at initial angle $\theta_{i}$ above the horizontal. Note that at the apex of its trajectory the projectile is moving horizontally, so that the slope of its path is zero. Use the expression for the trajectory given in Equation $4.12$ to find the $x$ coordinate that corresponds to the maximum height. Use this $x$ coordinate and the symmetry of the trajectory to determine the horizontal range of the projectile.

Shelby Mohamed

Numerade Educator

Problem 19

A placekicker must kick a football from a point $36.0 \mathrm{~m}$ (about 40 yards) from the goal, and 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 $58.0^{\circ}$ 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?

Shelby Mohamed

Numerade Educator

Problem 20

A firefighter $50.0 \mathrm{~m}$ away from a burning building directs a stream of water from a fire hose at an angle of $30.0^{\circ}$ above the horizontal, as in Figure $\mathrm{P} 4.20$. If the speed of the stream is $40.0 \mathrm{~m} / \mathrm{s}$, at what height will the water strike the building?

Shelby Mohamed

Numerade Educator

Problem 21

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 in Figure $\mathrm{P} 4.20 .$ 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 22

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

Shelby Mohamed

Numerade Educator

Problem 23

A basketball star covers $2.80 \mathrm{~m}$ horizontally in a jump to dunk the ball (Fig. P4.23). His motion through space can be modeled as that of a particle at a point called his center of mass (which we shall 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 takeoff, and (d) his takeoff angle. (e) For comparison, determine the hang time of a whitetail deer making a jump with center-of-mass elevations $y_{i}=1.20 \mathrm{~m}, y_{\max }=2.50 \mathrm{~m}, y_{f}=0.700 \mathrm{~m}$.

Shelby Mohamed

Numerade Educator

Problem 24

The orbit of the Moon about the Earth is approximately circular, with a mean radius of $3.84 \times 10^{8} \mathrm{~m}$. It takes $27.3$ days for the Moon to complete one revolution about the Earth. Find (a) the mean orbital speed of the Moon and (b) its centripetal acceleration.

Shelby Mohamed

Numerade Educator

Problem 25

The athlete shown in Figure $\mathrm{P} 4.25$ rotates a $1.00-\mathrm{kg}$ discus along a circular path of radius $1.06 \mathrm{~m}$. 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 26

From information on the endsheets of this book, compute, for a point located on the surface of the Earth at the equator, the radial acceleration due to the rotation of the Earth about its axis.

Shelby Mohamed

Numerade Educator

Problem 27

A tire $0.500 \mathrm{~m}$ in radius rotates at a constant rate of 200 rev $/ \mathrm{min}$. Find the speed and acceleration of a small stone lodged in the tread of the tire (on its outer edge). (Hint: In one revolution, the stone travels a distance equal to the circumference of its path, $2 \pi r .$ )

Shelby Mohamed

Numerade Educator

Problem 28

During liftoff, Space Shuttle astronauts typically feel accelerations up to $1.4 g$, where $g=9.80 \mathrm{~m} / \mathrm{s}^{2}$. In their training, astronauts ride in a device where they experience such an acceleration as a centripetal acceleration. Specifically, the astronaut is fastened securely at the end of a mechanical arm that then turns at constant speed in a horizontal circle. Determine the rotation rate, in revolutions per second, required to give an astronaut a centripetal acceleration of $1.40 g$ while the astronaut moves in a circle of radius $10.0 \mathrm{~m}$.

Nishant Kumar

Numerade Educator

Problem 29

Young David who slew Goliath experimented with slings before tackling the giant. He found that he could revolve a sling of length $0.600 \mathrm{~m}$ at the rate of $8.00 \mathrm{rev} / \mathrm{s}$. If he increased the length to $0.900 \mathrm{~m}$, he could revolve the sling only $6.00$ times per second. (a) Which rate of rotation gives the greater speed for the stone at the end of the sling? (b) What is the centripetal acceleration of the stone at $8.00 \mathrm{rev} / \mathrm{s} ?(\mathrm{c})$ What is the centripetal acceleration at $6.00 \mathrm{rev} / \mathrm{s} ?$

Mayukh Banik

Numerade Educator

Problem 29

Young David who slew Goliath experimented with slings before tackling the giant. He found that he could revolve a sling of length $0.600 \mathrm{~m}$ at the rate of $8.00 \mathrm{rev} / \mathrm{s}$. If he increased the length to $0.900 \mathrm{~m}$, he could revolve the sling only $6.00$ times per second. (a) Which rate of rotation gives the greater speed for the stone at the end of the sling? (b) What is the centripetal acceleration of the stone at $8.00 \mathrm{rev} / \mathrm{s} ?(\mathrm{c})$ What is the centripetal acceleration at $6.00 \mathrm{rev} / \mathrm{s} ?$

Mayukh Banik

Numerade Educator

Problem 30

The astronaut orbiting the Earth in Figure $\mathrm{P} 4.30$ is preparing to dock with a Westar VI satellite. The satellite is in a circular orbit $600 \mathrm{~km}$ above the Earth's surface. where the free-fall acceleration is $8.21 \mathrm{~m} / \mathrm{s}^{2}$. The radius of the Earth is $6400 \mathrm{~km}$. Determine the speed of the satellite and the time required to complete one orbit around the Earth.

Shelby Mohamed

Numerade Educator

Problem 31

A train slows down as it rounds a sharp horizontal curve, slowing from $90.0 \mathrm{~km} / \mathrm{h}$ to $50.0 \mathrm{~km} / \mathrm{h}$ in the $15.0 \mathrm{~s}$ that it takes to round the curve. 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 that the train slows down at a uniform rate during the $15.0-\mathrm{s}$ interval.

Shelby Mohamed

Numerade Educator

Problem 32

An automobile whose speed is increasing at a rate of $0.600 \mathrm{~m} / \mathrm{s}^{2}$ travels along a circular road of radius $20.0 \mathrm{~m} .$ When the instantaneous speed of the automobile is $4.00$ $\mathrm{m} / \mathrm{s}$, find (a) the tangential acceleration component,
(b) the radial acceleration component, and (c) the magnitude and direction of the total acceleration.

Shelby Mohamed

Numerade Educator

Problem 33

Figure $\mathrm{P} 4.38$ shows the total acceleration and velocity of a particle moving clockwise in a circle of radius $2.50 \mathrm{~m}$

Shelby Mohamed

Numerade Educator

Problem 33

Figure $\mathrm{P} 4.38$ shows the total acceleration and velocity of a particle moving clockwise in a circle of radius $2.50 \mathrm{~m}$ at a given instant of time. At this instant, find (a) the radial acceleration, (b) the speed of the particle, and
(c) its tangential acceleration.

Shelby Mohamed

Numerade Educator

Problem 34

A student attaches a ball to the end of a string $0.600 \mathrm{~m}$ in length and then swings the ball in a vertical circle. The speed of the ball is $4.30 \mathrm{~m} / \mathrm{s}$ at its highest point and $6.50 \mathrm{~m} / \mathrm{s}$ at its lowest point. Find the acceleration of the ball when the string is vertical and the ball is at
(a) its highest point and (b) its lowest point.

Shelby Mohamed

Numerade Educator

Problem 35

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 and on its way up, its total acceleration is $(-22.5 \mathbf{i}+$ $20.2 \mathrm{j}) \mathrm{m} / \mathrm{s}^{2}$. At that instant, (a) sketch a vector diagram showing the components of this acceleration, (b) determine the magnitude of its radial acceleration, and
(c) determine the speed and velocity of the ball.

Shelby Mohamed

Numerade Educator

Problem 36

Heather in her Corvette accelerates at the rate of $(8.00 \mathbf{i}-2.00 \mathrm{j}) \mathrm{m} / \mathrm{s}^{2}$, while Jill in her Jaguar accelerates at $(1.00 \mathbf{i}+3.00 \mathbf{j}) \mathrm{m} / \mathrm{s}^{2}$. They both start from rest at the origin of an $x y$ coordinate system. After $5.00 \mathrm{~s}$, (a) what is Heather's speed with respect to Jill, (b) how far apart are they, and (c) what is Heather's acceleration relative to Jill?

Shelby Mohamed

Numerade Educator

Problem 37

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. If the student can swim at a speed of $1.20 \mathrm{~m} / \mathrm{s}$ in still water, how long does the trip take? Compare this with the time the trip would take if the water were still.

Mayukh Banik

Numerade Educator

Problem 38

How long does it take an automobile traveling in the left lane at $60.0 \mathrm{~km} / \mathrm{h}$ to pull alongside a car traveling in the right lane at $40.0 \mathrm{~km} / \mathrm{h}$ if the cars' front bumpers are initially $100 \mathrm{~m}$ apart?

Shelby Mohamed

Numerade Educator

Problem 39

The pilot of an airplane notes that the compass indicates a heading due west. The airplane's speed relative to the air is $150 \mathrm{~km} / \mathrm{h}$. If there is a wind of $90.0 \mathrm{~km} / \mathrm{h}$ toward the north, find the velocity of the airplane relative to the ground.

Shelby Mohamed

Numerade Educator

Problem 40

Two swimmers, Alan and Beth, start at the same point in a stream that flows with a speed $v$. Both move at the same speed $c(c>v)$ relative to the stream. Alan swims downstream a distance $L$ and then upstream the same distance. Beth swims such that her motion relative to the ground is perpendicular to the banks of the stream. She swims a distance $L$ in this direction and then back. The result of the motions of Alan and Beth is that they both return to the starting point. Which swimmer returns first? (Note: First guess at the answer.)

Shelby Mohamed

Numerade Educator

Problem 41

A child in danger of drowning in a river is being carried downstream by a current that has a speed of $2.50 \mathrm{~km} / \mathrm{h}$. The child is $0.600 \mathrm{~km}$ from shore and $0.800 \mathrm{~km}$ upstream of a boat landing when a rescue boat sets out.
(a) If the boat proceeds at its maximum speed of $20.0 \mathrm{~km} / \mathrm{h}$ relative to the water, what heading relative to the shore should the pilot take? (b) What angle does the boat velocity make with the shore? (c) How long does it take the boat to reach the child?

Supratim Pal

Numerade Educator

Problem 42

A bolt drops from the ceiling of a train car that is accelerating northward at a rate of $2.50 \mathrm{~m} / \mathrm{s}^{2}$. What is the $\mathrm{ac}$ celeration of the bolt relative to (a) the train car and
(b) the Earth?

Shelby Mohamed

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 ball is thrown with an initial speed $v_{i}$ at an angle $\theta_{i}$ with the horizontal. The horizontal range of the ball is $R$, and the ball reaches a maximum height $R / 6 .$ In terms of $R$ and $g$, find $(a)$ the time the ball is in motion, (b) the ball's speed at the peak of its path, (c) the initial vertical component of its velocity, (d) its initial speed, and (e) the angle $\theta_{i} .$ (f) Suppose the ball is thrown at the same initial speed found in part (d) but at the angle appropriate for reaching the maximum height. Find this height. (g) Suppose the ball is thrown at the same initial speed but at the angle necessary for maximum range. Find this range.

Shelby Mohamed

Numerade Educator

Problem 45

As some molten metal splashes, one droplet flies off to the east with initial speed $v_{i}$ 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 in Figure $\mathrm{P} 4.45 .$ In terms of $v_{i}$ and $\theta_{i}$, find the distance between the droplets as a function of time.

Shelby Mohamed

Numerade Educator

Problem 46

A ball on the end of a string is whirled around in a horizontal circle of radius $0.300 \mathrm{~m}$. The plane of the circle is $1.20 \mathrm{~m}$ above the ground. The string breaks and the ball lands $2.00 \mathrm{~m}$ (horizontally) away from the point on the ground directly beneath the ball's location when the string breaks. Find the radial acceleration of the ball during its circular motion.

Vishal Gupta

Numerade Educator

Problem 47

A projectile is fired up an incline (incline angle $\phi$ ) with an initial speed $v_{i}$ at an angle $\theta_{i}$ with respect to the horizontal $\left(\theta_{i}>\phi\right)$, as shown in Figure $\mathrm{P} 4.47$. (a) Show that the projectile travels a distance $d$ up the incline, where
$$
d=\frac{2 v_{i}^{2} \cos \theta_{i} \sin \left(\theta_{i}-\phi\right)}{g \cos ^{2} \phi}
$$

Shelby Mohamed

Numerade Educator

Problem 48

A student decides to measure the muzzle velocity of the pellets from his BB gun. He points the gun horizontally. On a vertical wall a distance $x$ away from the gun, a target is placed. The shots hit the target a vertical distance $y$ below the gun. (a) Show that the vertical displacement component of the pellets when traveling through the air is given by $y=A x^{2}$, where $A$ is a constant. (b) $\mathrm{Ex}$ press the constant $A$ in terms of the initial velocity and the free-fall acceleration. (c) If $x=8.00 \mathrm{~m}$ and $\gamma=$ $0.210 \mathrm{~m}$, what is the initial speed of the pellets?

Shelby Mohamed

Numerade Educator

Problem 48

A home run is hit in such a way that the baseball just clears a wall $21.0 \mathrm{~m}$ high, located $180 \mathrm{~m}$ from home plate. The ball is hit at an angle of $35.0^{\circ}$ to the horizontal, and air resistance is negligible. Find (a) the initial speed of the ball, (b) the time it takes the ball to reach the wall, and (c) the velocity components and the speed of the ball when it reaches the wall. (Assume the ball is hit at a height of $1.00 \mathrm{~m}$ above the ground.)

Shelby Mohamed

Numerade Educator

Problem 50

An astronaut standing on the Moon fires a gun so that the bullet leaves the barrel initially moving in a horizontal direction. (a) What must be the muzzle speed of the bullet so that it travels completely around the Moon and returns to its original location? (b) How long does this trip around the Moon take? Assume that the free-fall acceleration on the Moon is one-sixth that on the Earth.

Shelby Mohamed

Numerade Educator

Problem 51

A pendulum of length $1.00 \mathrm{~m}$ swings in a vertical plane (Fig. $4.19$ ). When the pendulum is in the two horizontal positions $\theta=90^{\circ}$ and $\theta=270^{\circ}$, its speed is $5.00 \mathrm{~m} / \mathrm{s}$.
(a) Find the magnitude of the radial acceleration and tangential acceleration for these positions. (b) Drawa vector diagram to determine the direction of the total acceleration for these two positions. (c) Calculate the magnitude and direction of the total acceleration.

Shelby Mohamed

Numerade Educator

Problem 52

A basketball player who is $2.00 \mathrm{~m}$ tall is standing on the floor $10.0 \mathrm{~m}$ from the basket, as in Figure $\mathrm{P} 4.52 .$ If he shoots the ball at a $40.0^{\circ}$ angle with the horizontal. at what initial speed must he throw so that it goes through the hoop without striking the backboard? The basket height is $8.05 \mathrm{~m}$.

Shelby Mohamed

Numerade Educator

Problem 53

A particle has velocity components
$$
v_{x}=+4 \mathrm{~m} / \mathrm{s} \quad v_{y}=-\left(6 \mathrm{~m} / \mathrm{s}^{2}\right) t+4 \mathrm{~m} / \mathrm{s}
$$
Calculate the speed of the particle and the direction. $\theta=\tan ^{-1}\left(v_{y} / v_{x}\right)$ of the velocity vector at $t=2.00 \mathrm{~s}$.

Shelby Mohamed

Numerade Educator

Problem 54

When baseball players throw the ball in from the outfield, they usually allow it to take one bounce before it reaches the infielder on the theory that the ball arrives sooner that way. Suppose that the angle at which a bounced ball leaves the ground is the same as the angle at which the outfielder launched it, as in Figure $\mathrm{P} 4.54$, but that the ball's speed after the bounce is one half of what it was before the bounce. (a) Assuming the ball is always thrown with the same initial speed, at what angle $\theta$ should the ball be thrown in order to 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 times for the one-bounce and no-bounce throws.

Shelby Mohamed

Numerade Educator

Problem 55

A boy can throw a ball a maximum horizontal distance of $40.0 \mathrm{~m}$ on a level field. How far can he throw the same ball vertically upward? Assume that his muscles give the ball the same speed in each case.

Shelby Mohamed

Numerade Educator

Problem 56

A boy can throw a ball a maximum horizontal distance of $R$ on a level field. How far can he throw the same ball vertically upward? Assume that his muscles give the ball the same speed in each case.

Mayukh Banik

Numerade Educator

Problem 57

A stone at the end of a sling is whirled in a vertical circle of radius $1.20 \mathrm{~m}$ at a constant speed $v_{i}=1.50 \mathrm{~m} / \mathrm{s}$ as in Figure $\mathrm{P} 4.57$. The center of the string is $1.50 \mathrm{~m}$ above the ground. What is the range of the stone if it is released when the sling is inclined at $80.0^{\circ}$ with the horizontal (a) at $A ?$ (b) at $B ?$ What is the acceleration of the stone (c) just before it is released at $A$ ? (d) just after it is released at $A ?$

Shelby Mohamed

Numerade Educator

Problem 58

A quarterback throws a football straight toward a receiver with an initial speed of $20.0 \mathrm{~m} / \mathrm{s}$, at an angle of $30.0^{\circ}$ above the horizontal. At that instant, the receiver is $20.0 \mathrm{~m}$ from the quarterback. In what direction and with what constant speed should the receiver run to catch the football at the level at which it was thrown?

Shelby Mohamed

Numerade Educator

Problem 59

A bomber is flying horizontally over level terrain, with a speed of $275 \mathrm{~m} / \mathrm{s}$ relative to the ground, at an altitude of $3000 \mathrm{~m}$. Neglect the effects of air resistance. (a) How far will a bomb travel horizontally between its release from the plane and its impact on the ground? (b) If the plane maintains its original course and speed, where will it be when the bomb hits the ground? (c) At what angle from the vertical should the telescopic bombsight be set so that the bomb will hit the target seen in the sight at the time of release?

Shelby Mohamed

Numerade Educator

Problem 60

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 $\mathbf{v}_{i}$ as in Figure $\mathrm{P} 4.60$.
(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?

Khoobchandra Agrawal

Khoobchandra Agrawal

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 grasp. The hawk continues on its path at the same speed for $2.00 \mathrm{~s}$ 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, find the diving speed of the hawk.
(b) What angle did the hawk make with the horizontal during its descent? (c) For how long did the mouse "enjoy" free fall?

Shelby Mohamed

Numerade Educator

Problem 62

A truck loaded with cannonball watermelons stops suddenly to avoid running over the edge of a washed-out bridge (Fig. P4.62). The quick stop causes a number of melons to fly off the truck. One melon rolls over the edge 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 edge of the road, 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?

Mayukh Banik

Numerade Educator

Problem 63

A catapult launches a rocket at an angle of $53.0^{\circ}$ above the horizontal with an initial speed of $100 \mathrm{~m} / \mathrm{s}$. The rocket engine immediately starts a burn, and for $3.00 \mathrm{~s}$ the rocket moves along its initial line of motion with an acceleration of $30.0 \mathrm{~m} / \mathrm{s}^{2} .$ Then its engine fails, and the rocket proceeds to move in free fall. Find (a) the maximum altitude reached by the rocket, (b) its total time of flight, and (c) its horizontal range.

MG

Min Gao

Numerade Educator

Problem 64

A river flows with a uniform velocity $\mathbf{v} . A$ person in a motorboat travels $1.00 \mathrm{~km}$ upstream, at which time she passes a log floating by. Always with the same throttle setting, the boater continues to travel upstream for another $60.0 \mathrm{~min}$ and then returns downstream to her starting point, which she reaches just as the same log does. Find the velocity of the river.

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 65

A car is parked on a steep incline overlooking the ocean, where the incline makes an angle of $37.0^{\circ}$ below the horizontal. The negligent driver leaves the car in neutral, and the parking brakes are defective. The car rolls from rest 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 $80.0 \mathrm{~m}$ above the ocean. Find
(a) the speed of the car when it reaches the edge of the cliff and the time it takes to get there, (b) the velocity of the car when it lands in the ocean, (c) the total time the car is in motion, and (d) the position of the car when it lands in the ocean, relative to the base of the cliff.

MG

Min Gao

Numerade Educator

Problem 66

The determined coyote is out once more to try to capture the elusive roadrunner. The coyote wears a pair of Acme jet-powered roller skates, which provide a constant horizontal acceleration of $15.0 \mathrm{~m} / \mathrm{s}^{2}$ (Fig. $\left.\mathrm{P} 4.66\right)$. The coyote starts off at rest $70.0 \mathrm{~m}$ from the edge of a cliff at the instant the roadrunner zips past him in the direction of the cliff. (a) If the roadrunner moves with constant speed, determine the minimum speed he must have to reach the cliff before the coyote. At the brink of the cliff, the roadrunner escapes by making a sudden turn, while the coyote continues straight ahead. (b) If the cliff is $100 \mathrm{~m}$ above the floor of a canyon, determine where the coyote lands in the canyon (assume his skates remain horizontal and continue to operate when he is in "flight" ). (c) Determine the components of the coyote's impact velocity.

Tanner Manwaring

Tanner Manwaring

Numerade Educator

Problem 67

A skier leaves the ramp of a skijump with a velocity of $10.0 \mathrm{~m} / \mathrm{s}, 15.0^{\circ}$ above the horizontal, as in Figure $\mathrm{P} 4.67 .$ The slope is inclined at $50.0^{\circ}$, and air resistance is negligible. Find (a) the distance from the ramp to where the jumper lands and (b) the velocity components just before the landing. (How do you think the results might be affected if air resistance were included? Note that jumpers lean forward in the shape of an airfoil, with their hands at their sides, to increase their distance. Why does this work?)

Tanner Manwaring

Tanner Manwaring

Numerade Educator

Problem 68

Two soccer players, Mary and Jane, begin running from nearly the same point at the same time. Mary runs in an easterly direction at $4.00 \mathrm{~m} / \mathrm{s}$, while Jane takes off in a direction $60.0^{\circ}$ north of east at $5.40 \mathrm{~m} / \mathrm{s}$. (a) How long is it before they are $25.0 \mathrm{~m}$ apart? (b) What is the velocity of Jane relative to Mary? (c) How far apart are they after $4.00 \mathrm{~s} ?$

Tanner Manwaring

Tanner Manwaring

Numerade Educator

Problem 69

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 70

An enemy ship is on the western side of a mountain is land, as shown in Figure P4.70. The enemy ship can maneuver 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 eastern shoreline is horizontally $300 \mathrm{~m}$ from the peak, what are the distances from the eastern shore at which a ship can be safe from the bombardment of the enemy ship?

Mayukh Banik

Numerade Educator