Essential University Physics Global Edition

Richard Wolfson

Chapter 3

Motion in Two and Three Dimensions - all with Video Answers

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Chapter Questions

Problem 1

When can you replace the vector sum of two velocities or accelerations by simple addition of their magnitudes? Can you do this for more than two velocities or accelerations?

Stephen Zaffke

Stephen Zaffke

Numerade Educator

Problem 2

Can two vectors of equal magnitude sum to zero? How about two vectors of unequal magnitude? Repeat for three vectors.

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 3

Can an object have a southward acceleration while moving northward? A westward acceleration while moving northward?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 4

You drop a ball from the roof while your friend gives another ball a horizontal velocity from the same height. Which of the two balls would reach the ground first?

Stephen Zaffke

Numerade Educator

Problem 5

In what sense is Equation $3.8$ really two (or three) equations?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 6

Is a projectile's speed constant throughout its parabolic trajectory?

Stephen Zaffke

Numerade Educator

Problem 7

Is there any point on a projectile's trajectory where velocity and acceleration are perpendicular?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 8

How is it possible for an object to be moving in one direction but accelerating in another?

Ahmed Shalaby

Numerade Educator

Problem 9

You're in a bus moving with constant velocity on a level road when you throw a ball straight up. When the ball returns, does it land ahead of you, behind you, or back at your hand? Explain.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 10

Which of the following are legitimate mathematical equations? Explain. (a) $v=5 \hat{i} \mathrm{~m} / \mathrm{s}$; (b) $\vec{v}=5 \mathrm{~m} / \mathrm{s}$; (c) $\vec{a}=d v / d t$; (d) $\vec{a}=d \vec{v} / d t ;$ (e) $\vec{v}=5 \hat{i} \mathrm{~m} / \mathrm{s}$

Ahmed Shalaby

Numerade Educator

Problem 11

You walk $1.57 \mathrm{~km}$ north, then $0.846 \mathrm{~km}$ east. Find (a) the magnitude of your displacement vector and (b) its direction, expressed as an angle relative to the northward direction.

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 12

An ion in a mass spectrometer follows a semicircular path of radius $15.8 \mathrm{~cm}$. What are (a) the distance it travels and (b) the magnitude of its displacement?

Stephen Zaffke

Numerade Educator

Problem 13

A migrating whale follows the coast of Mexico and California. It first travels $360 \mathrm{~km}$ northwest, then turns due north and travels another $410 \mathrm{~km}$. Determine graphically the magnitude and direction of its displacement.

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 14

Vector $\vec{A}$ has magnitude $4.0 \mathrm{~m}$ and points to the right; vector $\vec{B}$ has magnitude $3.0 \mathrm{~m}$ and points vertically upward. Find the magnitude and direction of vector $\vec{C}$ such that $\vec{A}+\vec{B}+\vec{C}=\overrightarrow{0}$.

Stephen Zaffke

Numerade Educator

Problem 15

Use unit vectors to express a displacement of $150 \mathrm{~km}$ at $35^{\circ}$ counterclockwise from the $x$-axis.

Stephen Zaffke

Numerade Educator

Problem 16

Find the magnitude of the vector $31 \hat{\imath}+56 \hat{\jmath} \mathrm{m}$, and determine its angle to the $x$-axis.

Stephen Zaffke

Numerade Educator

Problem 17

(a) What's the magnitude of $\hat{\imath}+\hat{\jmath} ?$ (b) What angle does it make with the $x$-axis?

Ahmed Shalaby

Numerade Educator

Problem 18

You're leading an international effort to save Earth from an asteroid heading toward us at $11 \mathrm{~km} / \mathrm{s}$. Your team mounts a rocket on the asteroid and fires it for $10 \mathrm{~min}$, after which the asteroid is moving at $21 \mathrm{~km} / \mathrm{s}$ at $24^{\circ}$ to its original path. In a news conference, what do you report for the magnitude of the acceleration imparted to the asteroid?

Stephen Zaffke

Numerade Educator

Problem 19

An object's velocity vector $\vec{v}$ has components related by $v_{y}=-v_{x}$ What are the possible values for the angle that $\vec{v}$ makes with the $x$-axis?

Linda Winkler

Numerade Educator

Problem 20

A car drives north at $45 \mathrm{~km} / \mathrm{h}$ for $12 \mathrm{~min}$ and then turns east and goes $6.0 \mathrm{~km}$ at $90 \mathrm{~km} / \mathrm{h}$. Finally, it goes southwest at $60 \mathrm{~km} / \mathrm{h}$ for $7.0 \mathrm{~min}$. Determine the car's (a) displacement and (b) average velocity for this trip.

Ivan Kochetkov

Numerade Educator

Problem 21

An object's velocity is $\vec{v}=c t^{3} \hat{\imath}+d \hat{\jmath}$, where $t$ is time and $c$ and $d$ are positive constants with appropriate units. What's the direction of the object's acceleration?

Prabhat Tyagi

Numerade Educator

Problem 22

A car, initially going eastward, rounds a $90^{\circ}$ curve and ends up heading southward. If the speedometer reading remains constant, what's the direction of the car's average acceleration vector?

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 23

What are (a) the average velocity and (b) the average acceleration of the tip of the 2.4-cm-long hour hand of a clock in the interval from noon to 6 PM? Use unit vector notation, with the $x$-axis pointing toward 3 and the $y$-axis toward noon.

Ahmed Shalaby

Numerade Educator

Problem 24

An object is moving with speed $v$ when it's subject to an acceleration that leaves it moving at an angle $\theta$ to its original direction of motion, with twice its original speed. Find an expression for the angle between the acceleration vector and the original direction of the object's motion.

Linda Winkler

Numerade Educator

Problem 25

An object is moving in the $x$-direction at $1.1 \mathrm{~m} / \mathrm{s}$ when it undergoes an acceleration $\vec{a}=0.52 \hat{\jmath} \mathrm{m} / \mathrm{s}^{2}$. Find its velocity vector after $5.2 \mathrm{~s}$.

Stephen Zaffke

Numerade Educator

Problem 26

You're piloting a small plane on a route directly north, but there's a wind blowing from the west at $65 \mathrm{~km} / \mathrm{h}$. If your plane's airspeed (i.e., its speed relative to the air) is $470 \mathrm{~km} / \mathrm{h}$, in what direction should you head?

Stephen Zaffke

Numerade Educator

Problem 27

You wish to row straight across a 63-m-wide river. You can row at a steady $1.3 \mathrm{~m} / \mathrm{s}$ relative to the water, and the river flows at $0.57 \mathrm{~m} / \mathrm{s}$. (a) What direction should you head? (b) How long will it take you to cross the river?

Ahmed Shalaby

Numerade Educator

Problem 28

A plane with airspeed $320 \mathrm{~km} / \mathrm{h}$ flies perpendicularly across the jet stream, its nose pointed into the jet stream at $38^{\circ}$ from the perpendicular direction of its flight. Find the speed of the jet stream.

Stephen Zaffke

Numerade Educator

Problem 29

A flock of geese is attempting to migrate due south, but the wind is blowing from the west at $4.9 \mathrm{~m} / \mathrm{s}$. If the birds can fly at $7.6 \mathrm{~m} / \mathrm{s}$ relative to the air, in which direction should they be headed?

Stephen Zaffke

Numerade Educator

Problem 30

The position of an object as a function of time is given by $\vec{r}=\left(3.2 t+1.8 t^{2}\right) \hat{\imath}+\left(1.7 t-2.4 t^{2}\right) \hat{\jmath} \mathrm{m}$, with $t$ in seconds. Find the object's acceleration vector.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 31

The position of an object as a function of time is given by $\vec{r}=\left(3.2 t+1.8 t^{2}\right) \hat{\imath}+\left(1.7 t-2.4 t^{2}\right) \hat{\jmath} \mathrm{m}$, with $t$ in seconds.

Stephen Zaffke

Numerade Educator

Problem 32

You toss an apple horizontally at $8.1 \mathrm{~m} / \mathrm{s}$ from a height of $2.8 \mathrm{~m}$. Simultaneously, you drop a peach from the same height. How long does each take to reach the ground?

Stephen Zaffke

Numerade Educator

Problem 33

A carpenter tosses a shingle horizontally off an $8.2-\mathrm{m}$-high roof at $14 \mathrm{~m} / \mathrm{s}$. (a) How long does it take the shingle to reach the ground? (b) How far does it move horizontally?

Stephen Zaffke

Numerade Educator

Problem 34

An arrow fired horizontally at $35 \mathrm{~m} / \mathrm{s}$ travels $23 \mathrm{~m}$ horizontally. From what height was it fired?

Stephen Zaffke

Numerade Educator

Problem 35

Droplets in an ink-jet printer are ejected horizontally at $12 \mathrm{~m} / \mathrm{s}$ and travel a horizontal distance of $1.0 \mathrm{~mm}$ to the paper. How far do they fall in this interval?

Ankur S

Numerade Educator

Problem 36

Protons drop $1.1 \mu \mathrm{m}$ over the $1.5-\mathrm{km}$ length of a particle accelerator. What's their approximate average speed?

Stephen Zaffke

Numerade Educator

Problem 37

If you can hit a golf ball $170 \mathrm{~m}$ on Earth, how far can you hit it on the Moon? (Your answer will be an underestimate because it neglects air resistance on Earth.)

Stephen Zaffke

Numerade Educator

Problem 38

China's high-speed rail network calls for a minimum turn radius of $7.0 \mathrm{~km}$ for $350-\mathrm{km} / \mathrm{h}$ trains. What's the magnitude of a train's acceleration in this case?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 39

The minute hand of a clock is $8.50 \mathrm{~cm}$ long. Find the magnitude of the acceleration of its tip.

Stephen Zaffke

Numerade Educator

Problem 40

How fast would a car have to round a 50 -m-radius turn for its acceleration to be numerically equal to that of gravity?

Stephen Zaffke

Numerade Educator

Problem 41

Determine the acceleration of the Moon, which completes a nearly circular orbit of $384.4 \mathrm{Mm}$ radius in $27.3$ days.

Prashant Bana

Numerade Educator

Problem 42

Global Positioning System (GPS) satellites circle Earth at altitudes of approximately $20,000 \mathrm{~km}$, where the gravitational acceleration has $5.8 \%$ of its surface value. To the nearest hour, what's the orbital period of the GPS satellites?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 43

Pilots of high-performance aircraft risk loss of consciousness if they undergo accelerations exceeding about $5 \mathrm{~g}$. For a military jet flying at $2470 \mathrm{~km} / \mathrm{h}$ (about twice the speed of sound), what's the minimum radius for a turn that will keep the acceleration below $5 g$ ?

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 44

You're windsurfing at $6.28 \mathrm{~m} / \mathrm{s}$ when a gust hits, accelerating your sailboard at $0.714 \mathrm{~m} / \mathrm{s}^{2}$ at $48.8^{\circ}$ to your original direction. If the gust lasts $5.42 \mathrm{~s}$, what's the magnitude of the board's displacement during this time?

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 45

You're windsurfing at $5.68 \mathrm{~m} / \mathrm{s}$ when a gust hits, accelerating your sailboard at $62.5^{\circ}$ to your original direction. The gust lasts $5.42 \mathrm{~s}$, and the board's displacement during this time is $37.2 \mathrm{~m}$. Find the magnitude of the board's acceleration during the gust.

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 46

A hockey puck glides across the ice at $27.7 \mathrm{~m} / \mathrm{s}$, when a player whacks it with her hockey stick, giving it an acceleration of $448 \mathrm{~m} / \mathrm{s}^{2}$ at $75.0^{\circ}$ to its original direction. If the acceleration lasts $41.3 \mathrm{~ms}$, what's the magnitude of the puck's displacement during this time?

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 47

A hockey puck glides across the ice at $27.7 \mathrm{~m} / \mathrm{s}$, when a player whacks it with her hockey stick, giving it an acceleration at $64.3^{\circ}$ to its original direction. The acceleration lasts $50.3 \mathrm{~ms}$, and the puck's displacement during this time is $1.76 \mathrm{~m}$. Find the magnitude of the puck's acceleration.

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 48

An engineer is designing a flat, horizontal road for a $90-\mathrm{km} / \mathrm{h}$ speed limit. If the maximum acceleration of a vehicle on this road is $4.36 \mathrm{~m} / \mathrm{s}^{2}$, what's the minimum safe radius for curves in the road?

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 49

An engineer is designing a flat, horizontal road with a curve whose radius is $125 \mathrm{~m}$. Under dry conditions, the engineer can count on an acceleration of at least $5.0 \mathrm{~m} / \mathrm{s}^{2}$, provided by the tires of vehicles rounding the curve. What should be the posted speed limit, given to the nearest $10 \mathrm{~km} / \mathrm{h}$ ?

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 50

A jet plane is capable of an acceleration of magnitude $0.564 \mathrm{~g}$ when it turns. If the plane is flying at $988 \mathrm{~km} / \mathrm{h}$, what's the minimum turning radius for the plane?

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 51

A jet plane is capable of an acceleration of magnitude $0.612 g$ when it turns. If the plane is to make a turn of radius $8.77 \mathrm{~km}$, what's its maximum possible speed?

Stephen Zaffke

Numerade Educator

Problem 52

Vector $\vec{A}$ has magnitude $1.3 \mathrm{~m}$ and points $30^{\circ}$ clockwise from the $x$-axis. Vector $\vec{B}$ has magnitude $1.9 \mathrm{~m}$. Find the direction of $\vec{B}$ such that $\vec{A}+\vec{B}$ is in the $y$-direction.

Prabhu Ramji

Numerade Educator

Problem 53

Let $\vec{A}=17 \hat{\imath}-42 \hat{\jmath}$ and $\vec{B}=31 \hat{\jmath}+17 \hat{k}$. Find $\vec{C}$ such that $\vec{A}+\vec{B}+\vec{C}=\overrightarrow{0}$

Stephen Zaffke

Numerade Educator

Problem 54

You're a pilot beginning a $1280-\mathrm{km}$ flight to a city due south of your starting point. Your plane's airspeed (i.e., its speed relative to the air) is $846 \mathrm{~km} / \mathrm{h}$, and air traffic control says you'll have to head $11.5^{\circ}$ west of south to maintain a course due south. If the flight takes $115 \mathrm{~min}$, what are the magnitude and direction of the wind velocity?

Linda Winkler

Numerade Educator

Problem 55

A particle's position is $\vec{r}=\left(c t^{2}-2 d t^{3}\right) \hat{\imath}+\left(2 c t^{2}-d t^{3}\right) \hat{\jmath}$, where $c$ and $d$ are positive constants. Find expressions for times $t>0$ when the particle is moving in (a) the $x$-direction and (b) the $y$-direction.

Whitney Massock

Whitney Massock

Numerade Educator

Problem 56

An object moving at $50 \mathrm{~m} / \mathrm{s}$ is subject to an acceleration of magnitude $20 \mathrm{~m} / \mathrm{s}^{2}$ that lasts for $2 \mathrm{~s}$. At the end of that time interval, it's moving at $88 \mathrm{~m} / \mathrm{s}$. Estimate, to the nearest $30^{\circ}$, the angle between the initial velocity and the acceleration (i.e., is that angle closer to $0^{\circ}$, to $30^{\circ}$, to $60^{\circ}$, or to $90^{\circ}$ ?). Explain your reasoning.

Linda Winkler

Numerade Educator

Problem 57

You're designing a "cloverleaf" highway interchange. Vehicles will exit the highway and slow to a constant $72 \mathrm{~km} / \mathrm{h}$ before negotiating a circular turn. If a vehicle's acceleration is not to exceed $0.45 \mathrm{~g}$ (i.e., $45 \%$ of Earth's gravitational acceleration), then what's the minimum radius for the turn? Assume the road is flat, not banked (more on this in Chapter 5).

Justin Swantek

Numerade Educator

Problem 58

An object undergoes acceleration $2.3 \hat{\imath}+3.6 \hat{\jmath} \mathrm{m} / \mathrm{s}^{2}$ for $10 \mathrm{~s}$. At the end of this time, its velocity is $33 \hat{\imath}+15 \hat{\jmath} \mathrm{m} / \mathrm{s}$. (a) What was its velocity at the beginning of the $10-s$ interval? (b) By how much did its speed change? (c) By how much did its direction change? (d) Show that the speed change is not given by the magnitude of the acceleration multiplied by the time. Why not?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 59

The New York Wheel is the world's largest Ferris wheel. It's 183 meters in diameter and rotates once every $37.3 \mathrm{~min}$. Find the magnitudes of (a) the average velocity and (b) the average acceleration at the wheel's rim, over a $5.00-\mathrm{min}$ interval. (c) Compare

Stephen Zaffke

Numerade Educator

Problem 60

A ferryboat sails between towns directly opposite each other on a river, moving at speed $v^{\prime}$ relative to the water. (a) Find an expression for the angle it should head at if the river flows at speed $V$.
(b) What's the significance of your answer if $V>v^{\prime}$ ?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 61

The sum of two vectors, $\vec{A}+\vec{B}$, is perpendicular to their difference, $\vec{A}-\vec{B}$. How do the vectors' magnitudes compare?

Ahmed Shalaby

Numerade Educator

Problem 62

A delivery drone approaches a customer's porch, flying $8.65 \mathrm{~m}$ above the porch at $21.5 \mathrm{~km} / \mathrm{h}$. (a) At what horizontal distance from the desired landing spot should it release a package? (b) At what speed will the package hit the porch?

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 63

An object is initially moving in the $x$-direction at $5.5 \mathrm{~m} / \mathrm{s}$, when it undergoes an acceleration in the $y$-direction for a period of $22 \mathrm{~s}$. If the object moves equal distances in the $x$ - and $y$-directions during this time, what's the magnitude of its acceleration?

Stephen Zaffke

Numerade Educator

Problem 64

A particle leaves the origin with its initial velocity given by $\vec{v}_{0}=11 \hat{\imath}+18 \hat{\jmath} \mathrm{m} / \mathrm{s}$, undergoing constant acceleration $\hat{a}=-1.4 \hat{\imath}+0.27 \hat{\jmath} \mathrm{m} / \mathrm{s}^{2}$. (a) When does the particle cross the $y$-axis? (b) What's its $y$-coordinate at the time? (c) How fast is it moving and in what direction?

Stephen Zaffke

Numerade Educator

Problem 65

A kid fires a squirt gun horizontally from $1.6 \mathrm{~m}$ above the ground. It hits another kid $2.1 \mathrm{~m}$ away square in the back, $0.93 \mathrm{~m}$ above the ground. What was the water's initial speed?

Ahmed Shalaby

Numerade Educator

Problem 66

A projectile has horizontal range $R$ on level ground and reaches maximum height $h$. Find an expression for its initial speed.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 67

You throw a baseball at a $45^{\circ}$ angle to the horizontal, aiming straight at a friend who's sitting in a tree a distance $h$ above level ground. At the instant you throw your ball, your friend drops another ball. (a) Show that the two balls will collide, no matter what your ball's initial speed, provided it's greater than some minimum value. (b) Find an expression for that minimum speed.

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 68

In a chase scene, a movie stuntman runs horizontally off the flat roof of one building and lands on another roof $1.3 \mathrm{~m}$ lower. If the gap between the buildings is $3.5 \mathrm{~m}$ wide, how fast must he run to cross the gap?

Stephen Zaffke

Numerade Educator

Problem 69

The meadow jumping mouse, Zapus hudsonius, inhabits much of North America and can jump as high as $1 \mathrm{~m}$. A jumping mouse is $48.3 \mathrm{~cm}$ from a 62.0-cm-high garden fence. (a) How fast and (b) at what angle should it jump so it just clears the fence without going any higher?

Linda Winkler

Numerade Educator

Problem 70

Derive a general formula for the horizontal distance covered by a projectile launched horizontally at speed $v_{0}$ from height $h .$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 71

Consider two projectiles launched on level ground with the same speed, at angles $45^{\circ} \pm \alpha$. Show that the ratio of their flight times is $\tan \left(\alpha+45^{\circ}\right)$.

Ahmed Shalaby

Numerade Educator

Problem 72

You toss a protein bar to your hiking companion located $8.6 \mathrm{~m}$ up a $39^{\circ}$ slope, as shown in Fig. 3.24. Determine the initial velocity vector so that when the bar reaches your friend, it's moving horizontally.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 73

The table below lists position versus time for an object moving in the $x-y$ plane, which is horizontal in this case. Make a plot
mine the nature of the object's path. Then determine the magnitudes of the object's velocity and acceleration.

Stephen Zaffke

Numerade Educator

Problem 74

A projectile launched at angle $\theta$ to the horizontal reaches maximum height $h$. Show that its horizontal range is $4 h / \tan \theta$.

Ivan Kochetkov

Numerade Educator

Problem 75

As an expert witness, you're testifying in a case involving a motorcycle accident. A motorcyclist driving in a $50 \mathrm{~km} / \mathrm{h}$ zone hit a stopped car on a level road. The motorcyclist was thrown from his bike and landed $16 \mathrm{~m}$ down the road. You're asked whether he was speeding. What's your answer?

Stephen Zaffke

Numerade Educator

Problem 76

Show that, for a given initial speed, the horizontal range of a projectile is the same for launch angles $45^{\circ}+\alpha$ and $45^{\circ}-\alpha$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 77

A basketball player is horizontally $4.5 \mathrm{~m}$ from the center of the basket, which is $3.05 \mathrm{~m}$ off the ground. At what angle should the player aim the ball from a height of $2.5 \mathrm{~m}$ with a speed of $8 \mathrm{~m} / \mathrm{s}$ ?

Stephen Zaffke

Numerade Educator

Problem 78

A projectile is launched from the edge of a table, a height $h$ off the floor. It rises to a maximum height $h$ above the table and then lands on the floor a horizontal distance $2 h$ from the edge of the table. Find (a) an expression for the magnitude of the initial velocity and (b) an exact value for the launch angle.

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 79

Consider the two projectiles in GOT IT? 3.5. Suppose the $45^{\circ}$ projectile is launched with speed $v$ and that it's in the air for time $t$. Find expressions for (a) the launch speed and (b) the flight time of the $60^{\circ}$ projectile, in terms of $v$ and $t$.

Ahmed Shalaby

Numerade Educator

Problem 80

In the 2015 film The Martian, astronauts ride the Hermes spaceship between Earth and Mars. To help keep the astronauts' bodies in good shupe on the long interplanetary voyages, Hermes rotates to simulate Martian gravity. If the spacecraft's maximum diameter is $38.0 \mathrm{~m}$, what should be its rotation period (the time to complete one rotation) if the acceleration at its outer edge is to equal the gravitational acceleration at Mars (see Appendix E)?

Krystal K

Numerade Educator

Problem 81

Your car can sustain an acceleration of $0.795 \mathrm{~g}$ while turning on a dry road. You're driving at $80.0 \mathrm{~km} / \mathrm{h}$ when you spot a truck jackknifed across the road. If you swerve in a circular are, as shown in Fig. 3.25, how far from the truck do you have to start swerving to avoid the truck?

Stephen Zaffke

Numerade Educator

Problem 82

Your alpine rescue team is using a slingshot to send an emergency medical packet to climbers stranded on a ledge, as shown in Fig. 3.26; your job is to calculate the launch speed. What do you report?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 83

If you can throw a stone straight up to height $h$, what's the maximum horizontal distance you could throw it over level ground?

Ahmed Shalaby

Numerade Educator

Problem 84

In a conversion from military to peacetime use, a missile with maximum horizontal range $220 \mathrm{~km}$ is being adapted for studying Earth's upper atmosphere. What is the maximum altitude it can achieve if launched vertically?

Stephen Zaffke

Numerade Educator

Problem 85

A soccer player can kick a ball $33 \mathrm{~m}$ on level ground, with its initial velocity at $37^{\circ}$ to the horizontal. At the same initial speed and angle to the horizontal, what horizontal distance can the player kick the ball on a $17^{\circ}$ upward slope?

Stephen Zaffke

Numerade Educator

Problem 86

A diver leaves a 3-m board on a trajectory that takes her $2.5 \mathrm{~m}$ above the board and then into the water $2.8 \mathrm{~m}$ horizontally from the end of the board. At what speed and angle did she leave the board?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 87

Using calculus, you can find a function's maximum or minimum by differentiating and setting the result to zero. Do this for Equation 3.15, differentiating with respect to $\theta$, and thus verify that the maximum range occurs for $\theta=45^{\circ}$.

Ahmed Shalaby

Numerade Educator

Problem 88

You're a consulting engineer specializing in athletic facilities, and you've been asked to help design the Olympic ski jump pictured in Fig. 3.27. Skiers will leave the jump at $28 \mathrm{~m} / \mathrm{s}$ and $9.5^{\circ}$
below the horizontal, and land $55 \mathrm{~m}$ horizontally from the end of the jump. Your job is to specify the slope of the ground so skiers" trajectories make an angle of only $3.0^{\circ}$ with the ground on landing, ensuring their safety. What slope do you specify?
below the horizontal, and land $55 \mathrm{~m}$ horizontally from the end of the jump. Your job is to specify the slope of the ground so skiers" trajectories make an angle of only $3.0^{\circ}$ with the ground on landing, ensuring their safety. What slope do you specify?

Stephen Zaffke

Numerade Educator

Problem 89

An object moves with constant speed in the $x$-direction, but in the $y$-direction it's subject to an acceleration that increases linearly with time: $a(t)=b t$, where $b$ is a constant. Derive an equation analogous to Equation $3.14$, giving the object's trajectory in this situation. (Assume there's no gravity.)
Your medieval history class is constructing a trebuchet, a catapult-like weapon for hurling stones at enemy castles. The plan is to Launch stones off a 75-m-high cliff, with initial speed $36 \mathrm{~m} / \mathrm{s}$. Some members of the class think a $45^{\circ}$ Launch angle will give the maximum range, but others claim the cliff height makes a difference. What do you give for the angle that will maximize the range?

Stephen Zaffke

Numerade Educator

Problem 90

Your medieval history class is constructing a trebuchet, a catapult-like weapon for hurling stones at enemy castles. The plan is to launch stones off a 75-m-high cliff, with initial speed $36 \mathrm{~m} / \mathrm{s}$. Some members of the class think a $45^{\circ}$ launch angle will give the maximum range, but others claim the cliff height makes a difference. What do you give for the angle that will maximize the range?

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 91

Generalize Problem 90 to find an expression for the angle that will maximize the range of a projectile launched with speed $v_{0}$ from height $h$ above level ground.

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 92

(a) Show that the position of a particle on a circle of radius $R$ with its center at the origin is $\vec{r}=R(\cos \theta \hat{\imath}+\sin \theta \hat{\jmath})$, where $\theta$ is the angle the position vector makes with the $x$-axis. (b) If the particle moves with constant speed $v$ starting on the $x$-axis at $t=0$, find an expression for $\theta$ in terms of time $t$ and the period $T$ to complete a full circle. (c) Differentiate the position vector twice with respect to time to find the acceleration, and show that its magnitude is given by Equation $3.16$ and its direction is toward the center of the circle.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 93

An object moves in a circular path of radius $R$ in the $x-y$ plane, where the origin is at the center of the circle. It starts from rest at $x=R$ and goes counterclockwise, undergoing constant tangential acceleration $a_{1}$. Find expressions for $(a)$ the magnitude and (b) the direction (relative to the positive $x$-axis) of its acceleration vector when it's traversed a quarter of the circle and thus crosses the positive y-axis.

Stephen Zaffke

Numerade Educator

Problem 94

After launch, a projectile lands a horizontal distance $2 R$ from its launch point and a vertical distance $R$ below its launch point. Here $R$ is the horizontal range the projectile would have had if launched over level ground at the same launch angle. Find the Luunch angle.

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 95

Which statement characterizes the distances the students travel?
a. They're equal.
b. $C>A>B$
c. $C>B>A$
d. $B>C>A$

Stephen Zaffke

Numerade Educator

Problem 96

Which statement characterizes the students ${ }^{4}$ displacements?
a. They"re equal.
b. $C>A>B$
c. $C>B>A$
d. $B>C>A$

Stephen Zaffke

Numerade Educator

Problem 97

Which statement characterizes their average speeds?
a. They"re equal.
b. $C>A>B$
c. $C>B>A$
d. $B>C>A$

Stephen Zaffke

Numerade Educator

Problem 98

Which statement characterizes their accelerations while walking (not starting and stopping)?
a. They're equal.
b. None accelerates.
c. $A>B>C$
d. $C>B>A$
e. $\mathrm{B}>\mathrm{C}>\mathrm{A}$
f. There's not enough information to decide.

Stephen Zaffke

Numerade Educator