Essential University Physics

Richard Wolfson

Chapter 3

Motion in Two and Three Dimensions - all with Video Answers

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Chapter Questions

Problem 1

Under what conditions is the magnitude of the vector sum $\vec{A}+\vec{B}$ equal to the sum of the magnitudes of the two vectors?

Ahmed Shalaby

Ahmed Shalaby

Numerade Educator

Problem 2

Can two vectors of equal magnitude sum to zero? How about two vectors of unequal magnitude?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 3

Repeat Question 2 for three vectors.

Ahmed Shalaby

Numerade Educator

Problem 4

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

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 5

You're a passenger in a car rounding a curve. The driver claims the car isn't accelerating because the speedometer reading is unchanging. Explain why the driver is wrong.

Ahmed Shalaby

Numerade Educator

Problem 6

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

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 7

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

Ahmed Shalaby

Numerade Educator

Problem 8

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

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 9

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

Ahmed Shalaby

Numerade Educator

Problem 10

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 11

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 12

You would probably reject as unscientific any claim that Earth is flat. Yet the assumption of Section 3.5 that leads to parabolic projectile trajectories is tantamount to assuming a flat Earth. Explain.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 13

You walk west $220 \mathrm{m},$ then north $150 \mathrm{m} .$ What are the magnitude and direction of your displacement vector?

Whitney Massock

Whitney Massock

Numerade Educator

Problem 14

An ion in a mass spectrometer follows a semicircular path of radius $15.2 \mathrm{cm} .$ What are (a) the distance it travels and (b) the magnitude of its displacement?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 15

A migrating whale follows the west coast of Mexico and North America toward its summer home in Alaska. It first travels 360 km northwest to just off the coast of northern California, and then turns due north and travels $400 \mathrm{km}$ toward its destination. Determine graphically the magnitude and direction of its displacement.

Vishal Gupta

Numerade Educator

Problem 16

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

Vishal Gupta

Numerade Educator

Problem 17

Use unit vectors to express a displacement of $120 \mathrm{km}$ at $29^{\circ}$ counterclockwise from the $x$ -axis.

Ahmed Shalaby

Numerade Educator

Problem 18

Find the magnitude of the vector $34 \hat{\imath}+13 \hat{\jmath} \mathrm{m}$ and determine its angle to the $x$ -axis.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 19

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

Ankur S

Numerade Educator

Problem 20

You're leading an international effort to save Earth from an asteroid heading toward us at $15 \mathrm{km} / \mathrm{s}$. Your team mounts a rocket on the asteroid and fires it for 10 min, after which the asteroid is moving at $19 \mathrm{km} / \mathrm{s}$ at $28^{\circ}$ to its original path. In a news conference, what do you report for the magnitude of the acceleration imparted to the asteroid?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 21

An object is moving at $18 \mathrm{m} / \mathrm{s}$ at $220^{\circ}$ counterclockwise from the $x$ -axis. Find the $x$ - and $y$ -components of its velocity.

Whitney Massock

Whitney Massock

Numerade Educator

Problem 22

A car drives north at $40 \mathrm{mi} / \mathrm{h}$ for 10 min, then turns east and goes $5.0 \mathrm{mi}$ at $60 \mathrm{mi} / \mathrm{h}$. Finally, it goes southwest at $30 \mathrm{mi} / \mathrm{h}$ for 6.0 min. Determine the car's (a) displacement and (b) average velocity for this trip.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 23

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?

Ahmed Shalaby

Numerade Educator

Problem 24

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?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 25

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 26

An ice skater is gliding along at $2.4 \mathrm{m} / \mathrm{s},$ when she undergoes an acceleration of magnitude $1.1 \mathrm{m} / \mathrm{s}^{2}$ for $3.0 \mathrm{s}$. After that she's moving at $5.7 \mathrm{m} / \mathrm{s}$. Find the angle between her acceleration vector and her initial velocity. Hint: You don't need to do a complicated calculation.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 27

An object is moving in the $x$ -direction at $1.3 \mathrm{m} / \mathrm{s}$ when it undergoes an acceleration $\vec{a}=0.52 \hat{\jmath} \mathrm{m} / \mathrm{s}^{2} .$ Find its velocity vector after $4.4 \mathrm{s}$

Ahmed Shalaby

Numerade Educator

Problem 28

You're a pilot beginning a 1500 -km flight. Your plane's speed is $1000 \mathrm{km} / \mathrm{h},$ and air traffic control says you'll have to head $15^{\circ}$ west of south to maintain a southward course. If the flight takes 100 min, what's the wind velocity?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 29

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 30

A plane with airspeed $370 \mathrm{km} / \mathrm{h}$ flies perpendicularly across the jet stream, its nose pointed into the jet stream at $32^{\circ}$ from the perpendicular direction of its flight. Find the speed of the jet stream.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 31

A flock of geese is attempting to migrate due south, but the wind is blowing from the west at $5.1 \mathrm{m} / \mathrm{s}$. If the birds can fly at $7.5 \mathrm{m} / \mathrm{s}$ relative to the air, what direction should they head?

Ahmed Shalaby

Numerade Educator

Problem 32

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 33

You're sailboarding at $6.5 \mathrm{m} / \mathrm{s}$ when a wind gust hits, lasting $6.3 \mathrm{s}$ accelerating your board at $0.48 \mathrm{m} / \mathrm{s}^{2}$ at $35^{\circ}$ to your original direction. Find the magnitude and direction of your displacement during the gust.

Ahmed Shalaby

Numerade Educator

Problem 34

You toss an apple horizontally at $8.7 \mathrm{m} / \mathrm{s}$ from a height of $2.6 \mathrm{m}$ Simultaneously, you drop a peach from the same height. How long does each take to reach the ground?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 35

A carpenter tosses a shingle horizontally off an 8.8 -m-high roof at $11 \mathrm{m} / \mathrm{s} .$ (a) How long does it take the shingle to reach the ground? (b) How far does it move horizontally?

Ahmed Shalaby

Numerade Educator

Problem 36

An arrow fired horizontally at $41 \mathrm{m} / \mathrm{s}$ travels $23 \mathrm{m}$ horizontally. From what height was it fired?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 37

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?

Ahmed Shalaby

Numerade Educator

Problem 38

Protons drop $1.2 \mu \mathrm{m}$ over the $1.7-\mathrm{km}$ length of a particle accelerator. What's their approximate average speed?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 39

If you can hit a golf ball 180 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.)

Ahmed Shalaby

Numerade Educator

Problem 40

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 41

The minute hand of a clock is $7.50 \mathrm{cm}$ long. Find the magnitude of the acceleration of its tip.

Ahmed Shalaby

Numerade Educator

Problem 42

How fast would a car have to round a 75 -m-radius turn for its acceleration to be numerically equal to that of gravity?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 43

Estimate the acceleration of the Moon, which completes a nearly circular orbit of 384.4 Mm radius in 27 days.

Ahmed Shalaby

Numerade Educator

Problem 44

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 45

Two vectors $\vec{A}$ and $\vec{B}$ have the same magnitude $A$ and are at right angles. Find the magnitudes of (a) $\vec{A}+2 \vec{B}$ and (b) $3 \vec{A}-\vec{B}$.

Ahmed Shalaby

Numerade Educator

Problem 46

Vector $\vec{A}$ has magnitude $1.0 \mathrm{m}$ and points $35^{\circ}$ clockwise from the $x$ -axis. Vector $B$ has magnitude $1.8 \mathrm{m}$. Find the direction of $\vec{B}$ such that $\vec{A}+\vec{B}$ is in the $y$ -direction.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 47

Let $\vec{A}=15 \hat{\imath}-40 \hat{\jmath}$ and $\vec{B}=31 \hat{\jmath}+18 \hat{k} .$ Find $\vec{C}$ such that
$\vec{A}+\vec{B}+\vec{C}=\overrightarrow{0}$

Whitney Massock

Whitney Massock

Numerade Educator

Problem 48

A biologist looking through a microscope sees a bacterium at $\vec{r}_{1}=2.2 \hat{\imath}+3.7 \hat{\jmath}-1.2 k \mu \mathrm{m} .$ After $6.2 \mathrm{s},$ it's located at
$\vec{r}_{2}=4.6 \hat{\imath}+1.9 \hat{k} \mu \mathrm{m} .$ Find (a) its average velocity, expressed in unit vectors, and (b) its average speed.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 49

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 50

For the particle in Problem $49,$ is there any time $t>0$ when the particle is (a) at rest and (b) accelerating in the $x$ -direction? If either answer is "yes," find the time(s).

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 51

You're designing a "cloverleaf" highway interchange. Vehicles will exit the highway and slow to a constant $70 \mathrm{km} / \mathrm{h}$ before negotiating a circular turn. If a vehicle's acceleration is not to exceed $0.40 g$ (i.e., $40 \%$ 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 ).

Ahmed Shalaby

Numerade Educator

Problem 52

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 53

The New York Wheel is the world's largest Ferris wheel. It's 183 meters in diameter and rotates once every 37.3 min. Find the magnitudes of (a) the average velocity and (b) the average acceleration at the wheel's rim, over a 5.00 -min interval. (c) Compare your answer to (b) with the wheel's instantaneous accelerations.

Ahmed Shalaby

Numerade Educator

Problem 54

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 $V>v^{\prime} ?$

Vishal Gupta

Numerade Educator

Problem 55

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 56

Write an expression for a unit vector at $45^{\circ}$ clockwise from the $x$ -axis.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 57

An object is initially moving in the $x$ -direction at $4.5 \mathrm{m} / \mathrm{s}$, when it undergoes an acceleration in the $y$ -direction for a period of $18 \mathrm{s}$ If the object moves equal distances in the $x$ and $y$ -directions during this time, what's the magnitude of its acceleration?

Whitney Massock

Whitney Massock

Numerade Educator

Problem 58

A particle leaves the origin with its initial velocity given by $\vec{v}_{0}=11 \hat{\imath}+14 \hat{\jmath} \mathrm{m} / \mathrm{s},$ undergoing constant acceleration $\vec{a}=-1.2 \hat{\imath}+0.26 \hat{\jmath} \mathrm{m} / \mathrm{s}^{2} .$ (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?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 59

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 60

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 61

You throw a baseball at a $45^{\circ}$ angle to the horizontal, aiming 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.

Ahmed Shalaby

Numerade Educator

Problem 62

In a chase scene, a movie stuntman runs horizontally off the flat roof of one building and lands on another roof $1.9 \mathrm{m}$ lower. If the gap between the buildings is 4.5 m wide, how fast must he run to cross the gap?

Chasen Shaw

Numerade Educator

Problem 63

Standing on the ground $3.0 \mathrm{m}$ from a building, you want to throw a package from your $1.5-\mathrm{m}$ shoulder level to someone in a window $4.2 \mathrm{m}$ above the ground. At what speed and angle should you throw the package so it just barely clears the windowsill?

Ahmed Shalaby

Numerade Educator

Problem 64

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

Vishal Gupta

Numerade Educator

Problem 65

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 66

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.
(FIGURE CAN'T COPY)

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 67

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 of position $y$ versus $x$ to determine the nature of the object's path. Then determine the magnitudes of the object's velocity and acceleration.
$$\begin{array}{ccc}
\text { Time, } t(\mathrm{s}) & x(\mathrm{m}) & y(\mathrm{m}) \\
0 & 0 & 0 \\
0.10 & 0.65 & 0.09 \\
0.20 & 1.25 & 0.33 \\
0.30 & 1.77 & 0.73 \\
0.40 & 2.17 & 1.25 \\
0.50 & 2.41 & 1.85 \\
0.60 & 2.50 & 2.50
\end{array}$$
$$\begin{array}{ccc}
\text { Time, } t(\mathrm{s}) & x(\mathrm{m}) & y(\mathrm{m}) \\
0.70 & 2.41 & 3.15 \\
0.80 & 2.17 & 3.75 \\
0.90 & 1.77 & 4.27 \\
1.00 & 1.25 & 4.67 \\
1.10 & 0.65 & 4.91 \\
1.20 & 0.00 & 5.00
\end{array}$$

Ahmed Shalaby

Numerade Educator

Problem 68

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

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 69

As an expert witness, you're testifying in a case involving a motorcycle accident. A motorcyclist driving in a $60-\mathrm{km} / \mathrm{h}$ zone hit a stopped car on a level road. The motorcyclist was thrown from his bike and landed 39 m down the road. You're asked whether he was speeding. What's your answer?

Ahmed Shalaby

Numerade Educator

Problem 70

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 71

A basketball player is 15 ft horizontally from the center of the basket, which is $10 \mathrm{ft}$ off the ground. At what angle should the player aim the ball from a height of $8.2 \mathrm{ft}$ with a speed of $26 \mathrm{ft} / \mathrm{s} ?$.

Supratim Pal

Numerade Educator

Problem 72

Two projectiles are launched simultaneously from the same point, with different launch speeds and angles. Show that no combination of speeds and angles will permit them to land simultaneously and at the same point.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 73

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 74

The portion of a projectile's parabolic trajectory in the vicinity of the peak can be approximated as a circle. If the projectile's speed at the peak of the trajectory is $v$, formulate an argument to show that the curvature radius of the circle that approximates the parabola is $r=v^{2} / g$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 75

A jet is diving vertically downward at $1200 \mathrm{km} / \mathrm{h}$. If the pilot can withstand a maximum acceleration of $5 g$ (i.e., 5 times Earth's gravitational acceleration) before losing consciousness, at what height must the plane start a $90^{\circ}$ circular turn, from vertical to horizontal, in order to pull out of the dive? See Fig. $3.25,$ assume the speed remains constant, and neglect gravity.
(FIGURE CAN'T COPY)

Ahmed Shalaby

Numerade Educator

Problem 76

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?
(FIGURE CAN'T COPY)

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 77

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 78

In a conversion from military to peacetime use, a missile with maximum horizontal range $180 \mathrm{km}$ is being adapted for studying Earth's upper atmosphere. What is the maximum altitude it can achieve if launched vertically?

Ankur S

Numerade Educator

Problem 79

A soccer player can kick the ball $28 \mathrm{m}$ on level ground, with its initial velocity at $40^{\circ}$ to the horizontal. At the same initial speed and angle to the horizontal, what horizontal distance can the player kick the ball on a $15^{\circ}$ upward slope?

Ahmed Shalaby

Numerade Educator

Problem 80

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?

Anand Jangid

Numerade Educator

Problem 81

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 82

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 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?
(FIGURE CAN'T COPY)

Vishal Gupta

Numerade Educator

Problem 83

Differentiate the trajectory Equation 3.14 to find its slope, $\tan \theta=d y / d x,$ and show that the slope is in the direction of the projectile's velocity, as given by Equations 3.10 and 3.11

Ahmed Shalaby

Numerade Educator

Problem 84

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 85

Generalize Problem 84 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.

Ahmed Shalaby

Numerade Educator

Problem 86

(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 87

In dealing with nonuniform circular motion, as shown in Fig. 3.23, we should write Equation 3.16 as $a_{r}=v^{2} / r,$ to show that this is only the radial component of the acceleration. Recognizing that $v$ is the object's speed, which changes only in the presence of tangential acceleration, differentiate this equation with respect to time to find a relation between the magnitude of the tangential acceleration and the rate of change of the magnitude of the radial acceleration. Assume the radius stays constant.

Ahmed Shalaby

Numerade Educator

Problem 88

Repeat Problem $87,$ now generalizing to the case where not only the speed but also the radius may be changing.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 89

Alice (A), Bob (B), and Carrie (C) all start from their dorm and head for the library for an evening study session. Alice takes a straight path, while the paths Bob and Carrie follow are portions of circular arcs, as shown in Fig. $3.28 .$ Each student walks at a constant speed. All three leave the dorm at the same time, and they arrive simultaneously at the library.
(FIGURE CAN'T COPY)
Which statement characterizes the distances the students travel?
a. They're equal.
b. $C > A > B$
c. $C > B > A$
d. $B > C > A$

Vipender Yadav

Numerade Educator

Problem 90

Alice (A), Bob (B), and Carrie (C) all start from their dorm and head for the library for an evening study session. Alice takes a straight path, while the paths Bob and Carrie follow are portions of circular arcs, as shown in Fig. $3.28 .$ Each student walks at a constant speed. All three leave the dorm at the same time, and they arrive simultaneously at the library.
(FIGURE CAN'T COPY)
Which statement characterizes the students' displacements?
a. They're equal.
b. $C > A > B$
c. $C > B > A$
d. $B > C > A$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 91

Alice (A), Bob (B), and Carrie (C) all start from their dorm and head for the library for an evening study session. Alice takes a straight path, while the paths Bob and Carrie follow are portions of circular arcs, as shown in Fig. $3.28 .$ Each student walks at a constant speed. All three leave the dorm at the same time, and they arrive simultaneously at the library.
(FIGURE CAN'T COPY)
Which statement characterizes their average speeds?
a. They're equal.
b. $C > A > B$
c. $C > B > A$
d. $B > C > A$

Ahmed Shalaby

Numerade Educator

Problem 92

Alice (A), Bob (B), and Carrie (C) all start from their dorm and head for the library for an evening study session. Alice takes a straight path, while the paths Bob and Carrie follow are portions of circular arcs, as shown in Fig. $3.28 .$ Each student walks at a constant speed. All three leave the dorm at the same time, and they arrive simultaneously at the library.
(FIGURE CAN'T COPY)
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. $\quad \mathbf{B}>\mathbf{C}>\mathbf{A}$
f. There's not enough information to decide.

Guilherme Barros

Guilherme Barros

Numerade Educator