Essential College Physics

Andrew F. Rex, Richard Wolfson

Chapter 3

Motion in Two Dimensions - all with Video Answers

Educators

Chapter Questions

Problem 1

Identify each of the following as a vector or scalar: (a) The surface area of a floor, (b) the position of a point on Earth's surface; (c) centripetal acceleration; (d) the number of pages in this book.

Prabhu Ramji

Prabhu Ramji

Numerade Educator

Problem 2

A complete topological/weather map of a state contains the following quantities for various locations. Identify each as vector or scalar: (a) position (latitude and longitude); (b) altitude; (c) wind speed; (d) wind velocity; (e) temperature; (f) barometric pressure.

Prabhu Ramji

Numerade Educator

Problem 3

Is the magnitude of a vector a scalar or a vector? Is the direction of a vector (in two dimensions) a vector or a scalar?

Prabhu Ramji

Numerade Educator

Problem 4

In two-dimensional motion, are average speed and average velocity ever the same? Explain.

Prabhu Ramji

Numerade Educator

Problem 5

If both components of a vector are doubled, does the vector's magnitude double? Explain.

Prabhu Ramji

Numerade Educator

Problem 6

Suppose you multiply a vector by $-1 .$ How does this affect its (a) components; (b) magnitude; (c) direction?

Prabhu Ramji

Numerade Educator

Problem 7

Can you add a position vector to a velocity vector? Why or why not?

Prabhu Ramji

Numerade Educator

Problem 8

Can an object have constant speed yet changing velocity? Can it have changing speed yet constant velocity?

Prabhu Ramji

Numerade Educator

Problem 9

Does an object's average velocity during a time interval have to be equal to its instantaneous velocity at some time during the interval? Explain why or give a counter example.

Prabhu Ramji

Numerade Educator

Problem 10

You want to throw a ball over a wall using the smallest possible launch speed. Should the launch angle be less than, equal to, or greater than $45^{\circ} ?$

Prabhu Ramji

Numerade Educator

Problem 11

You have a rubber-band slingshot that you want to fire from the top of a building and reach the greatest possible horizontal distance. Should the launch angle be less than, equal to, or greater than $45^{\circ} ?$

Prabhu Ramji

Numerade Educator

Problem 12

A car rounds a circular track at constant speed. While continuing on the same track, its speed then begins to increase. What can you say about the magnitude and direction of the car's acceleration, compared with when it had a constant speed?

Prabhu Ramji

Numerade Educator

Problem 13

A propeller blade turns at a constant rate. Compare the centripetal acceleration at the blade tip with a point halfway between the rotation axis and the tip.

Prabhu Ramji

Numerade Educator

Problem 14

A $1.60-\mathrm{m}$ -tall person is standing on level ground, with the Sun $30^{\circ}$ above the horizon. The length of the person's shadow is
(a) $0.92 \mathrm{~m}$
(b) $1.85 \mathrm{~m}$
(c) $2.77 \mathrm{~m}$
(d) $3.20 \mathrm{~m}$.

Prabhu Ramji

Numerade Educator

Problem 15

If you drive $30 \mathrm{~km}$ east, then $25 \mathrm{~km}$ north, how far are you from your start?
(a) $34 \mathrm{~km}$
(b) $39 \mathrm{~km}$
(c) $46 \mathrm{~km}$;
(d) $55 \mathrm{~km}$.

Prabhu Ramji

Numerade Educator

Problem 16

The magnitude of the position vector $\vec{r}=1.7 \mathrm{~m} \hat{\imath}-4.9 \mathrm{~m} \hat{\jmath}$ is
(a) $-3.2 \mathrm{~m}$
(b) $6.6 \mathrm{~m}$
(c) $26.9 \mathrm{~m}$
(d) $5.2 \mathrm{~m}$.

Prabhu Ramji

Numerade Educator

Problem 17

The position vector $\vec{r}=-2.3 \mathrm{~m} \hat{\imath}-4.0 \mathrm{~m} \hat{\jmath}$ makes a direction angle of
(a) $60^{\circ}$
(b) $210^{\circ}$
(c) $240^{\circ} ;$
(d) $330^{\circ}$.

Prabhu Ramji

Numerade Educator

Problem 18

If you bicycle $240 \mathrm{~m}$ in the $+x$ -direction, then $360 \mathrm{~m}$ in the $-y$ -direction in a total time of 1 min, your average velocity is
(a) $10 \mathrm{~m} / \mathrm{s}$ (b) $7.2 \mathrm{~m} / \mathrm{s} ;$ (c) $4.0 \mathrm{~m} / \mathrm{s} \hat{\imath}-6.0 \mathrm{~m} / \mathrm{s} \hat{\jmath},$ (d) $8.0 \mathrm{~m} / \mathrm{s} \hat{\imath}+$ $12.0 \mathrm{~m} / \mathrm{s} \hat{\jmath}$

Prabhu Ramji

Numerade Educator

Problem 19

Your car's instantaneous velocity is $11.9 \mathrm{~m} / \mathrm{s} \hat{\imath}+19.5 \mathrm{~m} / \mathrm{s} \hat{\jmath}$. Your speed is
(a) $22.8 \mathrm{~m} / \mathrm{s}$
(b) $25.6 \mathrm{~m} / \mathrm{s}$
(c) $31.4 \mathrm{~m} / \mathrm{s}$
(d) $33.6 \mathrm{~m} / \mathrm{s}$

Prabhu Ramji

Numerade Educator

Problem 20

A running dog's velocity changes from $2.13 \mathrm{~m} / \mathrm{s} \hat{\imath}-1.91 \mathrm{~m} / \mathrm{s} \hat{\jmath}$ to
$1.25 \mathrm{~m} / \mathrm{s} \hat{\imath}+2.03 \mathrm{~m} / \mathrm{s} \hat{\jmath}$ in a time of $5.0 \mathrm{~s}$. The dog's average accel-
eration during that time interval is (a) $-0.88 \mathrm{~m} / \mathrm{s}^{2} \hat{\imath}+3.94 \mathrm{~m} / \mathrm{s}^{2} \hat{\jmath}$,
(b) $-0.18 \mathrm{~m} / \mathrm{s}^{2} \hat{\imath}+0.79 \mathrm{~m} / \mathrm{s}^{2} \hat{\jmath},(\mathrm{c})-0.18 \mathrm{~m} / \mathrm{s}^{2} \hat{\imath}+0.02 \mathrm{~m} / \mathrm{s}^{2} \hat{\jmath}$ (d) $1.13 \mathrm{~m} / \mathrm{s}^{2} \hat{\imath}+0.79 \mathrm{~m} / \mathrm{s}^{2} \hat{\jmath}$.

Prabhu Ramji

Numerade Educator

Problem 21

The maximum range (on level ground) of a projectile launched at $25.5 \mathrm{~m} / \mathrm{s}$ is $($ a) $2.8 \mathrm{~m} ;$ (b) $47.0 \mathrm{~m} ;$ (c) $66.4 \mathrm{~m}$ (d) $250 \mathrm{~m}$.

Prabhu Ramji

Numerade Educator

Problem 22

A projectile launched at $10.8 \mathrm{~m} / \mathrm{s}$ at $50^{\circ}$ above the horizontal reaches a maximum height of (a) $6.0 \mathrm{~m}$;
(b) $4.4 \mathrm{~m}$ (c) $3.4 \mathrm{~m}$ (d) $2.9 \mathrm{~m}$.

Prabhu Ramji

Numerade Educator

Problem 23

A projectile is launched horizontally from a $12.4-\mathrm{m}$ -tall building at $14.0 \mathrm{~m} / \mathrm{s} .$ How far (horizontally) from the base of the building does it strike the ground?
(a) $22.3 \mathrm{~m}$
(b) $17.7 \mathrm{~m}$;
(c) $12.4 \mathrm{~m}$
(d) $10.9 \mathrm{~m}$

Prabhu Ramji

Numerade Educator

Problem 24

A golfer hits the ball from a deep sand bunker onto a nearby green at a higher elevation. How does the ball's speed $(v)$ when it hits the green compare to its speed $\left(v_{0}\right)$ when it left the bunker?
(a) $v>v_{0}$
(b) $v=v_{0}$
(c) $v<v_{0}$
(d) cannot be determined from the information given.

Prabhu Ramji

Numerade Educator

Problem 25

A car rounds a circular track of radius $875 \mathrm{~m}$. What's its maximum speed if its centripetal acceleration must not exceed $3.50 \mathrm{~m} / \mathrm{s}^{2} ?$ (a) $250 \mathrm{~m} / \mathrm{s}$; (b) $55.3 \mathrm{~m} / \mathrm{s} ;$ (c) $47.5 \mathrm{~m} / \mathrm{s}$; (d) $27.6 \mathrm{~m} / \mathrm{s}$.

Prabhu Ramji

Numerade Educator

Problem 26

The Moon orbits Earth in a nearly circular orbit of radius $3.84 \times 10^{8} \mathrm{~m}$ and period 27.3 days. What is the Moon's centripetal acceleration?
(a) $2.72 \times 10^{-3} \mathrm{~m} / \mathrm{s}^{2}$
(b) $1.49 \times 10^{-2} \mathrm{~m} / \mathrm{s}^{2}$
(c) $0.108 \mathrm{~m} / \mathrm{s}^{2}$
(d) $9.80 \mathrm{~m} / \mathrm{s}^{2}$

Prabhu Ramji

Numerade Educator

Problem 27

What are all the angles in a right triangle with sides $3 \mathrm{~m}, 4 \mathrm{~m}$ and $5 \mathrm{~m} ?$

Prabhu Ramji

Numerade Educator

Problem 28

The hypotenuse of a right triangle measures $12.5 \mathrm{~cm} .$ If the three angles are $30^{\circ}, 60^{\circ},$ and $90^{\circ},$ what are the lengths of the other two sides?

Prabhu Ramji

Numerade Educator

Problem 29

Standing on level ground, a person casts a shadow $1.12 \mathrm{~m}$ long when the Sun is $55^{\circ}$ above the horizon. How tall is the person?

Prabhu Ramji

Numerade Educator

Problem 30

In On level ground, you stand $25.0 \mathrm{~m}$ from the base of a tree and determine that the treetop is at $42^{\circ}$ above the horizontal, as measured from ground level. How tall is the tree?

Prabhu Ramji

Numerade Educator

Problem 31

A right triangle has hypotenuse $25.0 \mathrm{~cm}$ and one side $19.8 \mathrm{~cm}$.
(a) Find the third side.
(b) Find the three angles of the triangle.

Prabhu Ramji

Numerade Educator

Problem 32

When driving west toward the Rocky Mountains, you see a peak $3.1^{\circ}$ above the horizon. Your GPS gives your current elevation as $1580 \mathrm{~m},$ and according to the map, you're $25.0 \mathrm{~km}$ (horizontally) from the peak. Find the peak's elevation.

Prabhu Ramji

Numerade Educator

Problem 33

A sign on a mountain road reads "Caution: 6 degree downgrade for next $5.5 \mathrm{~km} .$ " By how much will your elevation have changed after you complete that stretch of road?

Prabhu Ramji

Numerade Educator

Problem 34

It's $35 \mathrm{~km}$ by road from town to a ski resort. The resort's elevation is $1150 \mathrm{~m}$ above town. What's the road's average inclination angle?

Prabhu Ramji

Numerade Educator

Problem 35

On the map, let the $+x$ -axis point east and the $+y$ -axis north, with direction angles measured counterclockwise from the $+x$ -axis. What direction angle should you head if your destination is (a) $4.5 \mathrm{~km}$ north and $2.3 \mathrm{~km}$ west; $(\mathrm{b}) 9.9 \mathrm{~km}$ west and $3.4 \mathrm{~km}$ south; (c) $1.2 \mathrm{~km}$ east and $4.0 \mathrm{~km}$ south?

Prabhu Ramji

Numerade Educator

Problem 36

You walk $1250 \mathrm{~m}$ east and then $900 \mathrm{~m}$ south in a total time of 20 min. Compute your (a) displacement and (b) average velocity in $\mathrm{m} / \mathrm{s}$.

Prabhu Ramji

Numerade Educator

Problem 37

Problems $37-42$ concern the position vectors $\vec{r}_{1}=2.39 \mathrm{~m} \hat{\imath}-5.07 \mathrm{~m} \hat{\jmath}$ and $\vec{r}_{2}=-3.56 \mathrm{~m} \hat{\imath}+0.98 \mathrm{~m} \hat{\jmath}$.
Find the magnitude and direction of both vectors.

Prabhu Ramji

Numerade Educator

Problem 38

Concern the position vectors $\vec{r}_{1}=2.39 \mathrm{~m} \hat{\imath}-5.07 \mathrm{~m} \hat{\jmath}$ and $\vec{r}_{2}=-3.56 \mathrm{~m} \hat{\imath}+0.98 \mathrm{~m} \hat{\jmath}$.
Find $\vec{r}_{1}+\vec{r}_{2}$

Prabhu Ramji

Numerade Educator

Problem 39

Concern the position vectors $\vec{r}_{1}=2.39 \mathrm{~m} \hat{\imath}-5.07 \mathrm{~m} \hat{\jmath}$ and $\vec{r}_{2}=-3.56 \mathrm{~m} \hat{\imath}+0.98 \mathrm{~m} \hat{\jmath}$.
Find $\vec{r}_{1}-\vec{r}_{2}$

Prabhu Ramji

Numerade Educator

Problem 40

Concern the position vectors $\vec{r}_{1}=2.39 \mathrm{~m} \hat{\imath}-5.07 \mathrm{~m} \hat{\jmath}$ and $\vec{r}_{2}=-3.56 \mathrm{~m} \hat{\imath}+0.98 \mathrm{~m} \hat{\jmath}$.
Repeat the preceding two problems using graphical methods and compare with the results you computed exactly.

Prabhu Ramji

Numerade Educator

Problem 41

Concern the position vectors $\vec{r}_{1}=2.39 \mathrm{~m} \hat{\imath}-5.07 \mathrm{~m} \hat{\jmath}$ and $\vec{r}_{2}=-3.56 \mathrm{~m} \hat{\imath}+0.98 \mathrm{~m} \hat{\jmath}$.
Find the angle between the two vectors. Hint: Find the angle each makes with the $+x$ -axis.

Prabhu Ramji

Numerade Educator

Problem 42

Concern the position vectors $\vec{r}_{1}=2.39 \mathrm{~m} \hat{\imath}-5.07 \mathrm{~m} \hat{\jmath}$ and $\vec{r}_{2}=-3.56 \mathrm{~m} \hat{\imath}+0.98 \mathrm{~m} \hat{\jmath}$.
(a) Compute the vector $-\vec{r}_{2}$.
(b) Use components to show that $\vec{r}_{1}-\vec{r}_{2}$ is equivalent to $\vec{r}_{1}+\left(-\vec{r}_{2}\right) .$ (c) Show graphically that $\vec{r}_{1}-\vec{r}_{2}$ is equivalent to $\vec{r}_{1}+\left(-\vec{r}_{2}\right)$

Averell Hause

Carnegie Mellon University

Problem 43

Find the $x$ - and $y$ -components of these vectors in the $x-y$ plane:
(a) $\vec{A}$ has magnitude $6.4 \mathrm{~m}$, direction angle $80^{\circ}$; (b) $\vec{B}$ has magnitude $13 \mathrm{~m},$ direction $30^{\circ} ;$ (c) $\vec{C}$ has magnitude $10 \mathrm{~m}$ and points in the $-y$ -direction.

Prabhu Ramji

Numerade Educator

Problem 44

Position vector $\vec{r}$ has magnitude of $13.0 \mathrm{~m}$ and direction angle $210^{\circ} .$ (a) Find its components. (b) Find the components, magnitude, and direction of the vector $-2 \vec{r}$.

Prabhu Ramji

Numerade Educator

Problem 45

Add the following vectors, first graphically, then using components: $\vec{R}$ has magnitude $6.0 \mathrm{~m}$ and points in the $+x$ -direction, and $\vec{S}$ has a magnitude $9.0 \mathrm{~m}$ and direction angle $60^{\circ}$.

Averell Hause

Carnegie Mellon University

Problem 46

you drive due east at $85 \mathrm{~km} / \mathrm{h}$ for $50 \mathrm{~min}$. Then you follow a road going $60^{\circ}$ south of east, going $90 \mathrm{~km} / \mathrm{h}$ for another $30 \mathrm{~min}$. Find your total displacement.

Prabhu Ramji

Numerade Educator

Problem 47

A roller coaster goes $26 \mathrm{~m}$ along a section of track sloping $10^{\circ}$ below the horizontal, followed by a $15-\mathrm{m}$ section sloping upward at $6^{\circ},$ followed by an $18-\mathrm{m}$ section of level track. Find the net displacement.

Prabhu Ramji

Numerade Educator

Problem 48

A circular racetrack with $250 \mathrm{~m}$ radius lies in the $x-y$ plane and is centered at the origin. A car rounds the track counterclockwise starting at the point $(250 \mathrm{~m}, 0)$. Find the total distance traveled and the displacement after (a) one-quarter lap; (b) one-half lap; (c) one complete lap.

Prabhu Ramji

Numerade Educator

Problem 49

The car in the preceding problem travels at constant speed and completes one lap in 55 s. Assume the same starting point $(250 \mathrm{~m}, 0) .$ Find (a) the car's speed, (b) its average velocity for one-half lap, and (c) its instantaneous velocity at the end of one lap.

Prabhu Ramji

Numerade Educator

Problem 50

A runner completes an 800 -m race with two laps of the 400 -m track in $1 \mathrm{~min}, 54.3 \mathrm{~s}$. Find the runner's average velocity and average speed for the race.

Prabhu Ramji

Numerade Educator

Problem 51

A snail crawling on a sheet of graph paper goes from the origin to the point $x=5.6 \mathrm{~cm}, y=4.3 \mathrm{~cm}$ in $1 \mathrm{~min}$. Find its average velocity and average speed.

Prabhu Ramji

Numerade Educator

Problem 52

On the map, let the $+x$ -axis point east and the $+y$ -axis north. (a) An airplane flies at $810 \mathrm{~km} / \mathrm{h}$ northwestward direction (i.e., midway between north and west). Find the components of its velocity. (b) Repeat for the case when the plane flies due south at the same speed.

Prabhu Ramji

Numerade Educator

Problem 53

If the plane in the preceding problem took 45 s to execute the turn from a northwest heading to a south heading, what was its average acceleration during the turn?

Prabhu Ramji

Numerade Educator

Problem 54

You drive $100 \mathrm{~km} / \mathrm{h}$ east for $30 \mathrm{~min}$, then $80 \mathrm{~km} / \mathrm{h}$ north for 40 min. Find your average speed and average velocity for the entire trip.

Prabhu Ramji

Numerade Educator

Problem 55

In this problem let the $+x$ -axis be horizontal and the $+y$ -axis straight up. A rocket is launched from rest on the ground. After $55 \mathrm{~s},$ its speed is $950 \mathrm{~m} / \mathrm{s}$ in a direction $75^{\circ}$ above the horizontal. Find (a) the components of the rocket's velocity vector at that time and (b) its average acceleration for the first $55 \mathrm{~s}$ of flight.

Prabhu Ramji

Numerade Educator

Problem 56

A dog running in the $+y$ -direction at $6.7 \mathrm{~m} / \mathrm{s}$ hears its owner call from behind, and 2.5 s later it's running at the same speed in the opposite direction. Compute (a) the dog's change in velocity and (b) its average acceleration.

Prabhu Ramji

Numerade Educator

Problem 57

A baseball traveling horizontally at $32 \mathrm{~m} / \mathrm{s}$ is struck by the bat, giving it a speed of $36 \mathrm{~m} / \mathrm{s}$ in the opposite direction.
(a) Find the change in the ball's velocity.
(b) If the ball was in contact with the bat for $0.75 \mathrm{~ms}$, what was its average acceleration? (Give both the magnitude and direction.)

Prabhu Ramji

Numerade Educator

Problem 58

A car rounds a 1.25 -km-radius circular track at $90 \mathrm{~km} / \mathrm{h}$. Find the magnitude of the car's average acceleration after it has completed one-fourth of the circle.

Prabhu Ramji

Numerade Educator

Problem 59

Take the $+x$ -axis horizontal and the $+y$ -axis upward. A skier glides down a $7.5^{\circ}$ slope with acceleration $1.15 \mathrm{~m} / \mathrm{s}^{2}$ along the slope. (a) Find the components of the acceleration. (b) If the skier started from rest, find her velocity and speed after $10.0 \mathrm{~s}$.

Prabhu Ramji

Numerade Educator

Problem 60

You drive west at $75 \mathrm{~km} / \mathrm{h}$ for $20 \mathrm{~min}$, then follow a road that goes $40^{\circ}$ south of west, driving at $90 \mathrm{~km} / \mathrm{h}$ for another $40 \mathrm{~min}$. Find (a) your total displacement and (b) your average speed and average velocity for the trip.

Prabhu Ramji

Numerade Educator

Problem 61

A billiard ball approaches a side cushion at a $45^{\circ}$ angle with speed $1.80 \mathrm{~m} / \mathrm{s} .$ It rebounds at a $45^{\circ}$ angle, as shown in Figure $\mathrm{P} 3.61,$ with the same speed. (a) Find the change in the ball's velocity. (b) More realistically, the ball might lose a little speed, rebounding at $1.60 \mathrm{~m} / \mathrm{s} .$ Now what's its velocity change?

Prabhu Ramji

Numerade Educator

Problem 62

Meteorologists track a storm using radar. The radar shows a storm centered $35 \mathrm{~km}$ west of town. Ninety minutes later, it's $25 \mathrm{~km}$ north of town. Assuming the storm moved with constant velocity, find that velocity.

Prabhu Ramji

Numerade Educator

Problem 63

A baseball hit just above the ground leaves the bat at $27 \mathrm{~m} / \mathrm{s}$ at $45^{\circ}$ above the horizontal. (a) How far away does the ball strike the ground? (b) How much time is the ball in the air? (c) What is its maximum height?

Prabhu Ramji

Numerade Educator

Problem 64

Make a scale drawing of the tra jectory of the baseball in the preceding problem. Let the motion be in the $x-y$ plane, with $x$ horizontal and $y$ vertical.

Averell Hause

Carnegie Mellon University

Problem 65

A ball rolls off a horizontal table at $0.30 \mathrm{~m} / \mathrm{s}$ and strikes the ground $0.15 \mathrm{~m}$ horizontally from the base of the table.
(a) How high is the table? (b) If another ball rolls off the same table at $0.60 \mathrm{~m} / \mathrm{s},$ how far from the base of the table does it hit?

Prabhu Ramji

Numerade Educator

Problem 66

You hit a tennis ball from just above the ground. It leaves the racket at $20 \mathrm{~m} / \mathrm{s}$ going $15^{\circ}$ upward. (a) Show that the ball lands before reaching the baseline at the other end of the court, $25 \mathrm{~m}$ away. (b) If you struck the ball harder but at the same angle, what's the maximum initial speed it could have and still land in the court?

Prabhu Ramji

Numerade Educator

Problem 67

A projectile is fired horizontally at $13.4 \mathrm{~m} / \mathrm{s}$ from the edge of a 9.50-m-high cliff and strikes the ground. Find (a) the horizontal distance it traveled; (b) the elapsed time; and (c) its final velocity.

Prabhu Ramji

Numerade Educator

Problem 67

A patient with a broken leg stands using a pair of crutches. The crutches support $75 \%$ of the $78-\mathrm{kg}$ patient's weight.
(a) Find the force each crutch applies to the patient, assuming they're held vertically.
(b) Repeat with the crutches pointed slightly outward from the person's sides, each making a $15^{\circ}$ angle with the vertical.

Prabhu Ramji

Numerade Educator

Problem 68

In a cathode-ray tube, electrons are projected horizontally at $1.2 \times 10^{6} \mathrm{~m} / \mathrm{s}$ and travel a horizontal distance $8.5 \mathrm{~cm}$ across the tube. (a) How long are the electrons in flight? (b) Find the vertical distance the electrons fall (under the influence of gravity) during their flight.

Prabhu Ramji

Numerade Educator

Problem 69

A firefighter aims a hose upward at a $75^{\circ}$ angle above the horizontal. If the water emerges from the hose $1.5 \mathrm{~m}$ above the ground at a speed of $22 \mathrm{~m} / \mathrm{s},$ what height does it reach? Hint: Think of the water stream as composed of individual droplets subject to gravity in flight.

Prabhu Ramji

Numerade Educator

Problem 70

A long jumper maintains a constant horizontal velocity component of $7.50 \mathrm{~m} / \mathrm{s}$ while jumping upward with vertical component $3.85 \mathrm{~m} / \mathrm{s}$. Determine her horizontal range.

Prabhu Ramji

Numerade Educator

Problem 71

Guns on World War II battleships could shell targets $15 \mathrm{~km}$ away. Find (a) the minimum launch speed needed to achieve that range and (b) the shell's flight time under these conditions.

Prabhu Ramji

Numerade Educator

Problem 72

The fastest major league pitcher throws a ball at $45.0 \mathrm{~m} / \mathrm{s}$. If he throws the ball horizontally, how far does it drop vertically on the $18.4-\mathrm{m}$ trip to home plate?

Prabhu Ramji

Numerade Educator

Problem 73

In 1971 astronaut Alan Shepard took a golf club to the Moon, where the acceleration due to gravity is about $g / 6 .$ If he struck a ball with a speed and launch angle that would result in a $120-\mathrm{m}$ horizontal range on Earth, how far would it go?

Prabhu Ramji

Numerade Educator

Problem 74

A rifle has muzzle velocity $425 \mathrm{~m} / \mathrm{s}$. If you're $50.0 \mathrm{~m}$ from a target the same height as the rifle, at what angle above the horizontal should you aim in order to hit the target?

Prabhu Ramji

Numerade Educator

Problem 75

A gazelle attempts to leap a $2.1-\mathrm{m}$ fence. Assuming a $45^{\circ}$ takeoff angle, what's the minimum speed for the jump?

Prabhu Ramji

Numerade Educator

Problem 76

A football kicker is trying to make a field goal from 45.0 $\mathrm{m}$. If he kicks at $40^{\circ}$ above the horizontal, what minimum speed is needed to clear the 3.05 -m-high crossbar?

Prabhu Ramji

Numerade Educator

Problem 77

A soccer player takes a penalty kick $11.0 \mathrm{~m}$ from the goal. To score, she has to shoot the ball under the 2.44 -m-high crossbar. If she kicks at $19.8 \mathrm{~m} / \mathrm{s}$, what range of launch angles will result in a goal? Note: The ball is permitted to strike the ground before reaching the goal.

Prabhu Ramji

Numerade Educator

Problem 78

A soccer player $20.0 \mathrm{~m}$ from the goal stands ready to score. In the way stands a goalkeeper, $1.70 \mathrm{~m}$ tall and $5.00 \mathrm{~m}$ out from the goal, whose crossbar is at $2.44 \mathrm{~m}$ high. The striker kicks the ball toward the goal at $18 \mathrm{~m} / \mathrm{s}$. Determine whether the ball makes it over the goalkeeper and/or over the goal for each of the following launch angles (above the horizontal):
(a) $20^{\circ}$;
(b) $25^{\circ}$;
(c) $30^{\circ}$.

Prabhu Ramji

Numerade Educator

Problem 79

In the preceding problem, for what range of angles does the player score a goal - meaning the ball passes above the goalkeeper but below the crossbar?

Averell Hause

Carnegie Mellon University

Problem 80

Compute the centripetal acceleration of a point on Earth's equator, given Earth's 24 -h rotation period. (See Appendix $\mathrm{E}$ for needed data.) Compare your answer with $g=9.80 \mathrm{~m} / \mathrm{s}^{2}$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 81

Compute the centripetal acceleration of a point on Earth's surface at $38^{\circ}$ north latitude. Compare with the answer to the preceding problem.

Prabhu Ramji

Numerade Educator

Problem 82

You're designing a highway so that the maximum centripetal acceleration on a curve is no more than $1.0 \mathrm{~m} / \mathrm{s}^{2}$. What is the minimum curvature radius to accommodate a $100-\mathrm{km} / \mathrm{h}$ maximum speed?

Prabhu Ramji

Numerade Educator

Problem 83

In a loop-the-loop roller coaster, the minimum centripetal acceleration at the top of the loop is $9.8 \mathrm{~m} / \mathrm{s}^{2}$ if the car is not to fall off the track. Why? For a $7.3-\mathrm{m}$ -radius loop, what minimum speed must the car have at the top?

Prabhu Ramji

Numerade Educator

Problem 84

In 1974 , physicist Gerard K. O'Neill proposed a cylindrical space station that would rotate about its central axis to simulate gravity along the outer surfaces. For a $1.1-\mathrm{km}$ -radius cylinder, find the time for one revolution if "gravity" at the surface is to be $9.8 \mathrm{~m} / \mathrm{s}^{2}$

Prabhu Ramji

Numerade Educator

Problem 85

Charged particles round a $1.2-\mathrm{km}$ -radius circular particle accelerator at nearly the speed of light. Find (a) the period and (b) the centripetal acceleration of the charged particles.

Prabhu Ramji

Numerade Educator

Problem 86

Using astronomical data in Appendix E, find the centripetal acceleration of (a) Venus and (b) Mars in their orbits (assumed circular) around the Sun. Compare your results with Earth's acceleration found in Section $3.5 .$

Prabhu Ramji

Numerade Educator

Problem 87

A washing machine drum has a diameter of $46 \mathrm{~cm}$ and spins at 500 revolutions per minute. Find the centripetal acceleration at the drum's surface.

Prabhu Ramji

Numerade Educator

Problem 88

To simulate the extreme accelerations during launch, astronauts train in a large centrifuge. If the centrifuge diameter is $10.5 \mathrm{~m},$ what should be its rotation period to produce a centripetal acceleration of (a) $4 g$; (b) $6 g$ ?

Prabhu Ramji

Numerade Educator

Problem 89

A neutron star has $12-\mathrm{km}$ radius and rotation period $1.0 \mathrm{~s}$ What's the centripetal acceleration on its surface at the equator?

Prabhu Ramji

Numerade Educator

Problem 90

A satellite is in a 91.5-min-period circular orbit $350 \mathrm{~km}$ above Earth's surface. Find (a) the satellite's speed; (b) its centripetal acceleration.

Prabhu Ramji

Numerade Educator

Problem 91

A cyclist goes counterclockwise around a circular track with a constant speed. Which of the diagrams in Figure $\mathrm{P} 3.91$ correctly shows the cyclist's velocity and acceleration vectors when the bicycle is on the right edge of the track?

Prabhu Ramji

Numerade Educator

Problem 92

You're sitting in your yard, $10.5 \mathrm{~m}$ from your 7.20 -m-high house. The Sun is on the opposite side of the house. What angle must it be above the horizon for you to be in sunshine?

Prabhu Ramji

Numerade Educator

Problem 93

In A running track consists of two straight 80.0 -m segments and two semicircles of radius $38.2 \mathrm{~m}$ as shown in Figure GP3.93.
(a) Find the distance a runner travels in one complete lap.
(b) Find the displacement of a runner who covers half a lap, from $A$ to $B$.

Prabhu Ramji

Numerade Educator

Problem 94

A golf ball is hit on level ground at $25 \mathrm{~m} / \mathrm{s}$ and $32^{\circ}$ above the horizontal. What is its velocity (a) at the peak of its flight and
(b) when it lands? (c) How far does it travel horizontally?

Prabhu Ramji

Numerade Educator

Problem 95

In Repeat the preceding problem for a ball launched from a 7. 2 -m-tall hill.

Prabhu Ramji

Numerade Educator

Problem 96

Figure GP3.96 outlines the human circulatory system. Given the scale shown, estimate (a) the distance traveled by and (b) the net displacement of a red blood cell as it moves from the right hand to the left foot.

Averell Hause

Carnegie Mellon University

Problem 97

A radar signal detects a cruise boat $10 \mathrm{~km}$ due east of your position, traveling northward at $25 \mathrm{~km} / \mathrm{h}$. Your speedboat can go $40 \mathrm{~km} / \mathrm{h}$. (a) In what direction should you head to intercept the cruise boat? (b) How much time will it take to reach it? (c) Where will you intercept it?

Prabhu Ramji

Numerade Educator

Problem 98

A golfer hits a ball from a $1.50-\mathrm{m}$ -deep sand bunker, with speed $13.5 \mathrm{~m} / \mathrm{s}$ at an angle of $55^{\circ} .$ (a) How far does it travel horizontally before reaching the ground? (b) What is its speed when it strikes the ground? Compare with its initial speed. (c) What would be the horizontal range of an identically launched ball on level ground? Compare with your answer to part (a).

Prabhu Ramji

Numerade Educator

Problem 99

Kangaroos have made leaps with ranges of up to $12.8 \mathrm{~m}$ on level ground. (a) What is the minimum takeoff speed required for such a jump? (b) For the minimum takeoff speed you found in part (a), how much time is the kangaroo in the air?

Prabhu Ramji

Numerade Educator

Problem 100

A projectile is launched on level ground with speed $v_{0}$ at an angle $\theta$ above the horizontal. Show that its horizontal range is
$$
R=\frac{v_{0}^{2} \sin (2 \theta)}{g}
$$
Hint: You may need the trig identity $2 \sin \theta \cos \theta=\sin (2 \theta)$.

Prabhu Ramji

Numerade Educator

Problem 101

Use the equation derived in the preceding problem to answer the following: (a) What launch angle gives maximum range, for a given initial speed? (b) For a given initial speed, which other launch angle gives the same range as a $22^{\circ}$ launch angle? (c) What happens to the range for a $90^{\circ}$ launch? Why does this make sense?

Prabhu Ramji

Numerade Educator

Problem 102

A projectile is launched from the origin with speed $v_{0}$ at an angle $\theta$ above the horizontal. Show that its altitude $y$ as a function of horizontal position $x$ is
$$
y=(\tan \theta) x-\left(\frac{g}{2 v_{0}^{2} \cos ^{2}(\theta)}\right) x^{2}
$$

Prabhu Ramji

Numerade Educator

Problem 103

Use the results of the preceding problem to explain why the projectile's tra jectory is a parabola.

Prabhu Ramji

Numerade Educator

Problem 104

A basketball player stands in the corner of the court at the three-point line, $6.33 \mathrm{~m}$ from the basket, with the hoop $3.05 \mathrm{~m}$ above the floor. (a) If the player shoots the ball from a height of $2.00 \mathrm{~m}$ at $30^{\circ}$ above the horizontal, what should the launch speed be to make the basket? (b) How much would the launch speed have to increase to make the ball travel $40 \mathrm{~cm}$ farther and miss the hoop entirely?

Averell Hause

Carnegie Mellon University

Problem 105

Moon orbits Earth in 27.3 days, with orbital radius 384,000 $\mathrm{km}$. Find the centripetal acceleration of the Moon in its orbit, and compare the result with $g$. (In Chapter 9 , we'll show how Isaac Newton used this result to help understand how gravity works.)

Prabhu Ramji

Numerade Educator

Problem 106

An archer standing a horizontal distance $d$ from a tree fires an arrow toward an apple that hangs at height $h .$ At the moment the arrow is fired, the apple falls. Show that as long as the arrow's initial speed is large enough to travel a horizontal distance $d$, it will hit the apple.

Anand Jangid

Numerade Educator

Problem 107

Medical technicians use small centrifuges to isolate blood cells. A typical unit holds six test tubes, rotates at 3380 revolutions per minute, and produces a centripetal acceleration of $1600 g .$ How far are the test tubes from the rotation axis?

Prabhu Ramji

Numerade Educator