Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 5

Net Force and Two-Dimensional Motion - all with Video Answers

Educators

Chapter Questions

Problem 1

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

Averell Hause

Averell Hause

Carnegie Mellon University

Problem 2

A Small Ball A small ball rolls horizontally off the edge of a tabletop that is $1.20 \mathrm{~m}$ high. It strikes the floor at a point $1.52 \mathrm{~m}$ horizontally away from the edge of the table. (a) How long is the ball in the air? (b) What is its speed at the instant it leaves the table?

Donald Albin

Numerade Educator

Problem 3

A baseball leaves a pitcher's hand horizontally at a speed of $161 \mathrm{~km} / \mathrm{h}$. The distance to the batter is $18.3 \mathrm{~m}$. (Ignore the effect of air resistance.) (a) How long does the ball take to travel the first half of that distance? (b) The second half? (c) How far does the ball fall freely during the first half?
(d) During the second half?
(e) Why aren't the quantities in (c) and
(d) equal?

Ivan Kochetkov

Numerade Educator

Problem 4

Dart A dart is thrown horizontally with an initial speed of $10 \mathrm{~m} / \mathrm{s}$ toward point $P$, the bull's-eye on a dart board. It hits at point $Q$ on the rim, vertically below $P 0.19$ s later. (a) What is the distance $P Q ?$ (b) How far away from the dart board is the dart released?

Ajay Singhal

Numerade Educator

Problem 5

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

Donald Albin

Numerade Educator

Problem 6

Mike Powell In the 1991 World Track and Field Championships in Tokyo, Mike Powell (Fig. 5-30) jumped $8.95 \mathrm{~m}$, breaking the 23 -year long-jump record set by Bob Beamon by a full $5 \mathrm{~cm}$. Assume that Powell's speed on takeoff was $9.5 \mathrm{~m} / \mathrm{s}$ (about equal to that of a sprinter) and that $g=9.80 \mathrm{~m} / \mathrm{s}^{2}$ in Tokyo. How much less was Powell's horizontal range than the maximum possible horizontal range (neglecting the effects of air) for a particle launched at the same speed of $9.5 \mathrm{~m} / \mathrm{s}$ ?

Donald Albin

Numerade Educator

Problem 7

Catapulted A stone is catapulted at Problem 6. time $t_{1}=0$, with an initial velocity of magnitude $20.0 \mathrm{~m} / \mathrm{s}$ and at an angle of $40.0^{\circ}$ above the horizontal. What are the magnitudes of the (a) horizontal and (b) vertical components of its displacement from the catapult site at $t_{2}=1.10$ s? Repeat for the (c) horizontal and (d) vertical components at $t_{3}=1.80 \mathrm{~s}$, and for the (e) horizontal and (f) vertical components at $t_{4}=5.00 \mathrm{~s}$.

Donald Albin

Numerade Educator

Problem 8

Golf Ball A golf ball is struck at ground level. The speed of the golf ball as a function of the time is shown in Fig. $5-31$, where $t=0$ at the instant the ball is struck. (a) How far does the golf ball travel horizontally before returning to ground level? (b) What is the maximum height above ground level attained by the ball?

Donald Albin

Numerade Educator

Problem 9

. Fast Bullets A rifle that shoots bullets at $460 \mathrm{~m} / \mathrm{s}$ is to be aimed at a target $45.7 \mathrm{~m}$ away and level with the rifle. How high above the target must the rifle barrel be pointed so that the bullet hits the target?

Donald Albin

Numerade Educator

Problem 10

Slow-Pitch The pitcher in a slow-pitch softball game releases the ball at a point $3.0 \mathrm{ft}$ above ground level. A stroboscopic plot of the position of the ball is shown in Fig. $5-32$, where the readings are $0.25 \mathrm{~s}$ apart and the ball is released at $t=0 .$ (a) What is the initial speed of the ball? (b) What is the speed of the ball at the instant it reaches its maximum height above ground level? (c) What is that maximum height?

Averell Hause

Carnegie Mellon University

Problem 11

Maximum Height Show that the maximum height reached by a projectile is $y^{\max }=\left(v_{1} \sin \theta_{1}\right)^{2} / 2 g$.

Donald Albin

Numerade Educator

Problem 12

You Throw a Ball You throw a ball toward a wall with a speed of $25.0 \mathrm{~m} / \mathrm{s}$ and at an angle of $40.0^{\circ}$
above the horizontal (Fig. $5-33$ ). The wall is $22.0 \mathrm{~m}$ from the release point of the ball. (a) How far above the release point does the ball hit the wall? (b) What are the horizontal and vertical components of its velocity as it hits the wall? (c) When it hits, has it passed the highest point on its trajectory?

Donald Albin

Numerade Educator

Problem 13

Shot into the Air A ball is shot from the ground into the air. At a height of $9.1 \mathrm{~m}$. Its velocity is observed to be $\vec{v}=(7.6 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+$ $(6.1 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$ (i horizontal, $\hat{\mathrm{j}}$ upward). (a) To what maximum height does the ball rise? (b) What total horizontal distance does the ball travel? What are (c) the magnitude and
(d) the direction of the ball's velocity just before it hits the ground?

Donald Albin

Numerade Educator

Problem 14

Two Seconds Later Two seconds after being projected from ground level, a projectile is displaced $40 \mathrm{~m}$ horizontally and $53 \mathrm{~m}$ vertically above its point of projection. What are the (a) horizontal and
(b) vertical components of the initial velocity of the projectile? (c) At the instant the projectile achieves its maximum height above ground level, how far is it displaced horizontally from its point of projection?

Donald Albin

Numerade Educator

Problem 15

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

Donald Albin

Numerade Educator

Problem 16

Launching Speed The launching speed of a certain projectile is five times the speed it has at its maximum height. Calculate the elevation angle $\theta_{1}$ at launching.

Donald Albin

Numerade Educator

Problem 17

Airplane and Decoy A certain airplane has a speed of $290.0 \mathrm{~km} / \mathrm{h}$ and is diving at an angle of $30.0^{\circ}$ below the horizontal when the pilot releases a radar decoy (Fig. $5-34$ ). The horizontal distance between the release point and the point where the decoy strikes the ground is $700 \mathrm{~m}$.
(a) How long is the decoy in the air?
(b) How high was the released point?

Donald Albin

Numerade Educator

Problem 18

Soccer Ball A soccer ball is Problem 17 . kicked from the ground with an initial speed of $19.5 \mathrm{~m} / \mathrm{s}$ at an upward angle of $45^{\circ} .$ A player $55 \mathrm{~m}$ away in the direction of the kick starts running to meet the ball at that instant. What must be his average speed if he is to meet the ball just before it hits the ground? Neglect air resistance.
19. Stairway A ball rolls horizontally off the top of a stairway with a speed of $1.52 \mathrm{~m} / \mathrm{s}$. The steps are $20.3 \mathrm{~cm}$ high and $20.3 \mathrm{~cm}$ wide. Which step does the ball hit first?

Donald Albin

Numerade Educator

Problem 19

. Stairway A ball rolls horizontally off the top of a stairway with a speed of $1.52 \mathrm{~m} / \mathrm{s}$. The steps are $20.3 \mathrm{~cm}$ high and $20.3 \mathrm{~cm}$ wide. Which step does the ball hit first?

Averell Hause

Carnegie Mellon University

Problem 20

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

Averell Hause

Carnegie Mellon University

Problem 21

Airplane An airplane, diving at an angle of $53.0^{\circ}$ with the vertical, releases a projectile at an altitude of $730 \mathrm{~m}$. The projectile hits the ground $5.00$ s after being released. (a) What is the speed of the aircraft? (b) How far did the projectile travel horizontally during its flight? What were the (c) horizontal and (d) vertical components of its velocity just before striking the ground?

Donald Albin

Numerade Educator

Problem 22

Tennis Match During a tennis match, a player serves the ball at $23.6 \mathrm{~m} / \mathrm{s}$, with the center of the ball leaving the racquet horizontally $2.37 \mathrm{~m}$ above the court surface. The net is $12 \mathrm{~m}$ away and $0.90 \mathrm{~m}$ high. When the ball reaches the net, (a) does the ball clear it and (b) what is the distance between the center of the ball and the top of the net? Suppose that, instead, the ball is served as before but now it leaves the racquet at $5.00^{\circ}$ below the horizontal. When the ball reaches the net, (c) does the ball clear it and (d) what now is the distance between the center of the ball and the top of the net?

Ceren Uzun

Texas Tech University

Problem 23

The Batter A batter hits a pitched ball when the center of the ball is $1.22 \mathrm{~m}$ above the ground. The ball leaves the bat at an angle of $45^{\circ}$ with the ground. With that launch, the ball should have a horizontal range (returning to the launch level) of $107 \mathrm{~m} .$ (a) Does the ball clear a 7.32-m-high fence that is $97.5 \mathrm{~m}$ horizontally from the launch point? (b) Either way, find the distance between the top of the fence and the center of the ball when the ball reaches the fence.

Donald Albin

Numerade Educator

Problem 24

Detective Story In a detective story, a body is found $4.6 \mathrm{~m}$ from the base of a building and $24 \mathrm{~m}$ below an open window. (a) Assuming the victim left that window horizontally, what was the victim's speed just then? (b) Would you guess the death to be accidental? Explain your answer.

Donald Albin

Numerade Educator

Problem 25

Football Kicker A football kicker can give the ball an initial speed of $25 \mathrm{~m} / \mathrm{s}$. Within what two elevation angles must he kick the ball to score a field goal from a point $50 \mathrm{~m}$ in front of goalposts whose horizontal bar is $3.44 \mathrm{~m}$ above the ground? (If you want to work this out algebraically, use $\sin ^{2} \theta+\cos ^{2} \theta=1$ to get a relation between $\tan ^{2} \theta$ and $1 / \cos ^{2} \theta$, substitute, and then solve the resulting quadratic equation.)

Donald Albin

Numerade Educator

Problem 26

Position Vector for an Electron The position vector for an electron is $\vec{r}=(5.0 \mathrm{~m}) \hat{\mathrm{i}}-(3.0 \mathrm{~m}) \hat{\mathrm{j}}$. (a) Find the magnitude of $\vec{r}$. (b) Sketch the vector on a coordinate system.

Donald Albin

Numerade Educator

Problem 27

Watermelon Seed A watermelon seed has the following coordinates: $x=-5.0 \mathrm{~m}$ and $y=8.0 \mathrm{~m} .$ Find its position vector (a) in unit-vector notation and as (b) a magnitude and (c) an angle relative to the positive direction of the $x$ axis. (d) Sketch the vector on a coordinate system. If the seed is moved to the coordinates $(3.00 \mathrm{~m}$, $0 \mathrm{~m}$ ), what is its displacement (e) in unit-vector notation and as (f) a magnitude and $(\mathrm{g})$ an angle relative to the positive direction of the $x$ axis?

Donald Albin

Numerade Educator

Problem 28

Radar Station A radar station detects an airplane approaching directly from the east. At first observation, the range to the plane is $360 \mathrm{~m}$ at $40^{\circ}$ above the horizon. The airplane is tracked for another $123^{\circ}$ in the vertical east-west plane, the range at final contact being $790 \mathrm{~m}$. See Fig. 5-35. Find the displacement of the airplane during the period of observation.

Anand Jangid

Numerade Educator

Problem 29

Position Vector for a Proton The position vector for a proton is initially $\vec{r}_{1}=(5.0 \mathrm{~m}) \hat{\mathrm{i}}+(-6.0 \mathrm{~m}) \hat{\mathrm{j}}$ and then later is $\vec{r}_{2}=$
$(-2.0 \mathrm{~m}) \hat{\mathrm{i}}+(6.0 \mathrm{~m}) \hat{\mathrm{j}}$. (a) What is the proton's displacement vector, and (b) to what axis (if any) is that vector parallel?

Donald Albin

Numerade Educator

Problem 30

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

Donald Albin

Numerade Educator

Problem 31

Figure $\quad 5-36$ shows the path taken by my drunk skunk over level ground, from initial point $i$ to final point $f$. The angles are $\theta_{1}=30.0^{\circ}, \quad \theta_{2}=50.0^{\circ}$, and $\theta_{3}=$
$80.0^{\circ}$, and the distances are $d_{1}=$ $5.00 \mathrm{~m}, d_{2}=8.00 \mathrm{~m}$, and $d_{3}=12.0$
m. In magnitude-angle notation, what is the skunk's displacement from $i$ to $f ?$

Donald Albin

Numerade Educator

Problem 32

Figure 5-37 gives the path of a squirrel moving about on level ground, from point $A$ (at time $t_{1}=0$ ), to points $B$ (at $t_{2}=5.00$
$\min ), \quad C \quad$ (at $\quad t_{3}=10.0$
min), and finally $D$ (at $t_{4}=15.0 \mathrm{~min}$ ). Consider
the average velocities of the squirrel from point $A$ to each of the other three points. (a) Of those three average velocities, FIGURE 5-37 = Problem 32 . which has the least magnitude, and what is the average velocity in magnitude-angle notation? (b) Which has the greatest magnitude, and what is the average velocity in magnitudeangle notation?

Donald Albin

Numerade Educator

Problem 33

A train moving at a constant speed of $60.0 \mathrm{~km} / \mathrm{h}$ moves east for $40.0$ min. then in a direction $50.0^{\circ}$ east of north for $20.0 \mathrm{~min}$, and finally west for $50.0 \mathrm{~min}$. What is the average velocity of the train during this trip?

Donald Albin

Numerade Educator

Problem 34

An ion's position vector is initially $\vec{r}_{1}=$ $(5.0 \mathrm{~m}) \hat{\mathrm{i}}+(-6.0 \mathrm{~m}) \hat{\mathrm{j}}$, and $10 \mathrm{~s}$ later it is $\vec{r}_{2}=(-2.0 \mathrm{~m}) \hat{\mathrm{i}}+(8.0 \mathrm{~m}) \hat{\mathrm{j}}$
What is its average velocity during the $10 \mathrm{~s}$ ?

Donald Albin

Numerade Educator

Problem 35

The position of an electron is given by $\vec{r}(t)=[(3.00 \mathrm{~m} / \mathrm{s}) t] \hat{\mathrm{i}}+\left[\left(-4.00 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}\right] \hat{\mathrm{j}} .$ (a) What is the elec-
tron's velocity $\vec{v}(t) ?$ At $t=2.00 \mathrm{~s}$, what is $\vec{v}(\mathrm{~b})$ in unit-vector notation and as (c) a magnitude and (d) an angle relative to the positive direction of the $x$ axis?

Donald Albin

Numerade Educator

Problem 36

$A$ is $90 \mathrm{~km}$ west of oasis $B$. A camel leaves oasis $A$ and during a $50 \mathrm{~h}$ period walks $75 \mathrm{~km}$ in a direction $37^{\circ}$ north of east. The camel then walks toward the south a distance of $65 \mathrm{~km}$ in a $35 \mathrm{~h}$ period after which it rests for $5.0 \mathrm{~h}$. (a) What is the camel's displacement with respect to oasis $A$ after resting? (b) What is the camel's average velocity from the time it leaves oasis $A$ until it finishes resting? (c) What is the camel's average speed from the time it leaves oasis $A$ until it finishes resting? (d) If the camel is able to go without water for five days $(120 \mathrm{~h})$, what must its average velocity be after resting if it is to reach oasis $B$ just in time?

Donald Albin

Numerade Educator

Problem 37

You are to ride a jet-cycle over a lake, starting from rest at point $1:$ First, moving at $30^{\circ}$ north of due east:
1. Increase your speed at $0.400 \mathrm{~m} / \mathrm{s}^{2}$ for $6.00 \mathrm{~s}$. 2. With whatever speed you then have, move for $8.00 \mathrm{~s}$.
3. Then slow at $0.400 \mathrm{~m} / \mathrm{s}^{2}$ for $6.00 \mathrm{~s}$.
Immediately next, moving due west:
4. Increase your speed at $0.400 \mathrm{~m} / \mathrm{s}^{2}$ for $5.00 \mathrm{~s}$.
5. With whatever speed you then have, move for $10.0 \mathrm{~s}$.
6. Then slow at $0.400 \mathrm{~m} / \mathrm{s}^{2}$ until you stop.
In magnitude-angle notation, what then is your average velocity for the trip from point 1 ?

Donald Albin

Numerade Educator

Problem 38

A Proton A proton initially has $\vec{v}_{1}=(4.0 \mathrm{~m} / \mathrm{s}) \mathrm{i}+(-2.0 \mathrm{~m} / \mathrm{s}) \mathrm{j}$
and then $4.0 \mathrm{~s}$ later has $\vec{v}_{2}=(-2.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(-2.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$. For that
$4.0 \mathrm{~s}$. what is the proton's average acceleration $\langle\vec{a}\rangle$ (a) in unit-vector notation and (b) as a magnitude and a direction?

Donald Albin

Numerade Educator

Problem 39

In $x y$ Plane The position $\vec{r}$ of a particle moving in an $x y$ plane is given by $\vec{r}(t)=\left[\left(2.00 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3}-(5.00 \mathrm{~m} / \mathrm{s}) t\right] \hat{\mathrm{i}}+$
$\left[(6.00 \mathrm{~m})-\left(7.00 \mathrm{~m} / \mathrm{s}^{4}\right) t^{4}\right] \hat{\mathrm{j}} .$ Calculate (a) $\vec{r},(\mathrm{~b}) \vec{v}$, and $(\mathrm{c}) \vec{a}$ for
$t=2.00 \mathrm{~s}$

Donald Albin

Numerade Educator

Problem 40

An iceboat sails across the surface of a frozen lake with constant acceleration produced by the wind. At a certain instant the boat's velocity is $\vec{v}_{1}=(6.30 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(-8.42 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$. Three seconds
later, because of a wind shift, the boat is instantaneously at rest. What is its average acceleration for this 3 s interval?

Donald Albin

Numerade Educator

Problem 41

Particle Leaves Origin A particle leaves the origin with an initial velocity $\vec{v}_{1}=(3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}$ and a constant acceleration $\vec{a}=$ $\left(-1.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(-0.500 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$. When the particle reaches its
maximum $x$ coordinate, what are (a) its velocity and (b) its position vector?

Donald Albin

Numerade Educator

Problem 42

$A$ Particle $B$ Particle $A$ moves along the line $y=30 \mathrm{~m}$ with a constant velocity $\vec{v}$ of magnitude $3.0 \mathrm{~m} / \mathrm{s}$ and directed parallel to the positive $x$ axis (Fig. $5-38$ ). Particle $B$ starts at the origin with zero speed and constant acceleration $\vec{a}$ (of magnitude $0.40 \mathrm{~m} / \mathrm{s}^{2}$ ) at the same instant that particle $A$ passes the $y$ axis. What angle $\theta$ between $\vec{a}$ and the positive $y$ axis would result in a colli- $\quad$ FIGURE $5-38=$ sion between these two particles? (If $\quad$ Problem $42 .$ your computation involves an equation with a term such as $t^{4}$, substitute $u=t^{2}$ and then consider solving the resulting quadratic equation to get $u$.)

Donald Albin

Numerade Educator

Problem 43

A particle starts from the origin at $t=0$ with a velocity of $\vec{v}_{1}=(8.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$ and moves in the $x y$ plane with a constant acceleration of $\vec{a}=\left(4.0 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(2.0 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$. At the
instant the particle's $x$ coordinate is $29 \mathrm{~m}$, what are (a) its $y$ coordinate and (b) its speed?

Donald Albin

Numerade Educator

Problem 44

A moderate wind accelerates a smooth pebble over a horizontal $x y$ plane with a constant acceleration
$$
\vec{a}=\left(5.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(7.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}
$$
At time $t=0$, its velocity is $(4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}$. In magnitude-angle notation, what is its velocity when it has been displaced by $12.0 \mathrm{~m}$ parallel to the $x$ axis?

Donald Albin

Numerade Educator

Problem 45

A particle moves so that its position as a function of time is $\vec{r}(t)=(1 \mathrm{~m}) \hat{\mathrm{i}}+\left[\left(4 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}\right] \hat{\mathrm{j}}$. Write expressions
for (a) its velocity and (b) its acceleration as functions of time.

Donald Albin

Numerade Educator

Problem 46

What is the magnitude of the acceleration of a sprinter running at $10 \mathrm{~m} / \mathrm{s}$ when rounding a turn with a radius of $25 \mathrm{~m}$ ?

Jose Carlos

Numerade Educator

Problem 47

A sprinter runs at $9.2 \mathrm{~m} / \mathrm{s}$ around a circular track with a centripetal acceleration of magnitude $3.8 \mathrm{~m} / \mathrm{s}^{2}$.
(a) What is the track radius?
(b) What is the period of the motion?

Donald Albin

Numerade Educator

Problem 48

A rotating fan completes 1200 revolutions every minute. Consider the tip of a blade, at a radius of $0.15 \mathrm{~m}$. (a) Through what distance does the tip move in one revolution? What are (b) the tip's speed and (c) the magnitude of its acceleration? (d) What is the period of the motion?

Ceren Uzun

Texas Tech University

Problem 49

An Earth Satellite An Earth satellite moves in a circular orbit $640 \mathrm{~km}$ above Earth's surface with a period of $98.0 \mathrm{~min}$. What are
(a) the speed and (b) the magnitude of the centripetal acceleration of the satellite?

Donald Albin

Numerade Educator

Problem 50

A carnival merry-go-round rotates about a vertical axis at a constant rate. A passenger standing on the edge of the merry-go-round has a constant speed of $3.66 \mathrm{~m} / \mathrm{s}$. For each of the following instantaneous situations, state how far the passenger is from the center of the merry-go-round, and in which direction.
(a) The passenger has an acceleration of $1.83 \mathrm{~m} / \mathrm{s}^{2}$, east. (b) The passenger has an acceleration of $1.83 \mathrm{~m} / \mathrm{s}^{2}$, south.

Donald Albin

Numerade Educator

Problem 51

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

Ceren Uzun

Texas Tech University

Problem 52

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

Jose Carlos

Numerade Educator

Problem 53

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

Averell Hause

Carnegie Mellon University

Problem 54

When a large star becomes a supernova, its core may be compressed so tightly that it becomes a neutron star, with a radius of about $20 \mathrm{~km}$ (about the size of the San Francisco area). If a neutron star rotates once every second, (a) what is the speed of a

Donald Albin

Numerade Educator

Problem 55

A carnival Ferris wheel has a $15 \mathrm{~m}$ radius and completes five turns about its horizontal axis every minute. (a) What is the period of the motion? What is the centripetal acceleration of a passenger at (b) the highest point and (c) the lowest point, assuming the passenger is at a $15 \mathrm{~m}$ radius?

Donald Albin

Numerade Educator

Problem 56

A Particle at Constant Speed $\mathrm{A}$ particle $P$ travels with constant speed on a circle of radius $r=3.00$ $\mathrm{m}$ (Fig. 5-39) and completes one revolution in $20.0 \mathrm{~s}$. The particle passes through $O$ at time $t=0 .$ State the following vectors in magnitude-angle notation (angle relative to the positive direction of $x$ ). With respect to $O$, find the particle's position vector at the times $t$ of (a) $5.00 \mathrm{~s}$,
(b) $7.50 \mathrm{~s}$, and (c) $10.0 \mathrm{~s}$. (d) For the FIGURE 5 -39 = $5.00 \mathrm{~s}$ interval from the end of the $\quad$ Problem $56 .$ fifth second to the end of the tenth second, find the particle's displacement. (e) For the same interval, find its average velocity. Find its velocity at (f) the beginning and
(g) the end of that $5.00$ s interval. Next, find the acceleration at
(h) the beginning and (i) the end of that interval.

Donald Albin

Numerade Educator

Problem 57

A boy whirls a stone in a horizontal circle of radius $1.5 \mathrm{~m}$ and at height $2.0 \mathrm{~m}$ above level ground. The string breaks, and the stone flies off horizontally and strikes the ground after traveling a horizontal distance of $10 \mathrm{~m}$. What is the magnitude of the centripetal acceleration of the stone while in circular motion?

Averell Hause

Carnegie Mellon University

Problem 58

A cat rides a merry-go-round while turning with uniform circular motion. At time $t_{1}=2.00 \mathrm{~s}$, the cat's velocity is
$$
\vec{v}_{1}=(3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}
$$
measured on a horizontal $x y$ coordinate system. At time $t_{2}=5.00 \mathrm{~s}$, its velocity is
$$
\vec{v}_{2}=(-3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(-4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}
$$
What are (a) the magnitude of the cat's centripetal acceleration and
(b) the cat's average acceleration during the time interval $t_{2}-t_{1}$ ?

Donald Albin

Numerade Educator

Problem 59

A particle moves horizontally in uniform circular motion, over a horizontal $x y$ plane. At one instant, it moves through the point at coordinates $(4.00 \mathrm{~m}, 4.00 \mathrm{~m})$ with a velocity of $(-5.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}$ and an acceleration of $\left(12.5 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$. What are
the coordinates of the center of the circular path?

Donald Albin

Numerade Educator

Problem 60

in Orbit Although the planet Mars orbits the Sun in a Kepler ellipse with an eccentricity of $0.09$, we can approximate its orbit by a circle. If you have faith in Newton's laws then you must conclude that there is an invisible centripetal force holding Mars in orbit. The data on the orbit of Mars around the sun are shown in the Fig. $5-40$. (a) Calculate the magnitude of the centripetal force needed to hold Mars in its circular orbit. Please use the proper number of significant figures. (b) What is the direction of the force as Mars orbits around the Sun? (c) What is the most likely source of this force?
(d) Could this force have anything in common with the force that attracts objects to the Earth?

Donald Albin

Numerade Educator

Problem 61

A boy and a girl are tossing an apple back and forth between them. Figure $5-41$ shows the path the apple followed when watched by an observer looking on from the side. The apple is moving from left to right. Five points are marked on the path. Ignore air resistance. FIGURE $5-41$ w
(a) Make a copy of this figure. At each of the Problem 61 . marked points, draw an arrow that indicates the magnitude and direction of the force on the apple when it passes through that point. (b) Make a second copy of the figure. This time, at each marked point, place an arrow indicating the magnitude and direction of the apple's velocity at the instant it passes that point.
(c) Did you change your answer to the first question after solving the second? If so, explain what you were thinking at first and why you changed it.

Donald Albin

Numerade Educator

Problem 62

The Cut Pendulum A pendulum (i.e., a string with a ball at the end) is set swinging by holding it at the point marked $\mathrm{A}$ in Fig. $5-42 a$

Donald Albin

Numerade Educator

Problem 63

Projectile Graphs A popgun is angled so that it shoots a small dense ball through the air as shown in Fig. $5-43 a$.
(a) Sketch the path that the ball will follow on the figure. For the graphs shown in Fig. 5 $43 b$, the horizontal axis represents the time. The vertical axis is unspecified. For each of the FIGURE $5-43 a=$ Problem 63 . following quantities, select the letter of the graph that could provide a correct graph of the quantity for the ball in the situation shown (if the vertical axis were assigned the proper units). Use the $x$ and $y$ coordinates shown in the picture. The arrow heads point in the positive direction. If none of the graphs could work, write $\mathrm{N}$. The time graphs begin just after the ball leaves the gun.
(b) $y$ coordinate
(c) $x$ -component of the velocity
(d) $y$ -component of the net force
(e) $y$ -component of the velocity
(f) $x$ coordinate
(g) $y$ -component of the acceleration
(h) $x$ -component of the net force

Donald Albin

Numerade Educator

Problem 64

In the demonstration discussed in Section $5-2$, two identical objects were dropped, one straight down and the other shot off to the side by a spring. Both objects seemed to hit the ground at about the same time. Explain why this happens in terms of the physics we have learned. Does it matter how fast we
shoot the one launched sideways? How would the outcome of this experiment change if the objects had different masses? (Hint: See Fig. 5-5.)

Donald Albin

Numerade Educator

Problem 65

. Billiards over the Edge Two identical billiard balls are labeled $A$ and $B$. Maryland Fats places ball $A$ at the very edge of the table.

Donald Albin

Numerade Educator

Problem 66

A heavy projectile is thrown and follows a path something like the one shown in Fig. 5 -
46. For each of the quantities in the list
(a)-(d) below, select a direction from the list $(\mathrm{A}-\mathrm{G})$ that describes it. If you think that none of the choices apply, FIGURE 5-46 = write $\mathrm{N}$. Problem 66 .
Quantities:
(a) The projectile's velocity when it is at the highest point
(b) The force on the projectile when it is part way up
(c) The force on the projectile when it is at the highest point
(d) The projectile's acceleration when it is part way down Choices:
A. Points straight up
B. Points straight down
C. Points directly to the left
D. Points directly to the right
E. Is equal to zero
F. Points somewhat upward and to the right
G. Points somewhat upward and to the left
N. None of the above

Donald Albin

Numerade Educator

Problem 67

In C. S. Forster's novel Lieutenant Hornblower (set in the early $1800 \mathrm{~s}$ ), a British naval vessel tries to sneak by a Spanish garrison. The ship passes as far away from the Spanish guns as it can -a distance $s$. The Spanish gunner knows that his gun has a muzzle velocity whose magnitude is equal to $v_{1}$.
(a) Once the gun is fired, what controls the motion of the cannonball? Write the equations that determine the vector position of the cannonball after it leaves the cannon. You may ignore air resistance. (b) Suppose the gunner inclines his gun upward at an angle $\theta$ to the horizontal. Solve the equations you have written in part (a) to obtain expressions that can be evaluated to give the position of the cannonball at any time, $t$. (c) If the gunner wants the cannonball to hit the ship, he must choose his angle correctly. Explain how he can calculate the correct angle. (Again, you may ignore air resistance.) (d) If the muzzle velocity of the cannonball has a magnitude of $100 \mathrm{~m} / \mathrm{s}$ and the ship is a distance of half a kilometer away, find the angle the gunner should use. (Take $g$ to be $10 \mathrm{~m} / \mathrm{s}^{2}$.)

Donald Albin

Numerade Educator

Problem 68

A person shoved out of a window makes just as good a projectile as a golf ball rolling off a table.

Donald Albin

Numerade Educator

Problem 69

In this problem and the one that follows you will be asked to use VideoPoint, VideoGraph, or some other video analysis program and a spreadsheet to explore and analyze the nature of a projectile launch depicted in a digital movie. If you use VideoPoint, one appropriate movie has filename PASCO106. In this movie a small ball of mass $9.5 \mathrm{~g}$ is launched at an angle, $\theta$, with respect to the horizontal. Your instructor may suggest an alternative file for your use. Open the movie PASCO106. For simplicity you might want to set the origin in the video analysis at the location of the ball at time $t=$
0. Also, for immediate visual feedback on your results you should

Yaw Asomani

Numerade Educator

Problem 70

In this problem you will use VideoPoint, VideoGraph, or some other video analysis program and a spreadsheet to explore and analyze the nature of a projectile launch depicted in a digital movie. If you use VideoPoint, one appropriate movie has filename PASCO106. In this movie a small ball of mass $9.5 \mathrm{~g}$ is launched at an angle, $\theta$, with respect to the horizontal. Your instructor may suggest an alternative file for your use.

Open the movie PASCO106. Use the VideoPoint software and spreadsheet modeling to find the equation that describes: the horizontal motion $x$ vs. $t$ and the equations that describe the vertical motion $y$ vs. $t$.
(a) Hand in the printout of your two models. Place your name, date and section # on it, and answer questions (b) through (d) at the bottom of the page. (b) According to your horizontal model, what is the equation that describes the horizontal position of the ball, $x$, as a function of time? What is its horizontal acceleration, $a_{x} ?$ What is its initial horizontal velocity, $v_{1 x}$ ?
(c) According to your vertical model, what is the equation that describes the vertical position, $y$, of the ball as a function of time? What is the value of the ball's vertical acceleration, $a_{y} ?$ What is its initial vertical velocity, $v_{1 y} ?$
(d) Use the components $v_{1 x}$ and $v_{1 y}$ to compute the initial speed of the ball. What is the launch angle with respect to the horizontal?
(e) Compare your answer to part (d) to your approximation from part (a) of the previous problem.

Manish Jain

Numerade Educator

Problem 71

A large metallic asteroid strikes Earth and quickly digs a crater into the rocky material below ground level by launching rocks upward and outward. The following table gives five pairs of launch speeds and angles (from the horizontal) for such rocks, based on a model of crater formation. (Other rocks, with intermediate speeds and angles, are also launched.) Suppose that you are at $x=20 \mathrm{~km}$ when the asteroid strikes the ground at time $t_{1}=$ 0 and position $x=0$ (Fig. 5-48). (a) At $t_{2}=20 \mathrm{~s}$, what are the $x$ and $y$ coordinates of the rocks headed in your direction from launches $A$ through $E ?$ (b) Plot these coordinates and then sketch a curve through the points to include rocks with intermediate launch speeds and angles. The curve should give you an idea of what you would see as you look up into the approaching rocks and what dinosaurs must have seen during asteroid strikes long ago.

Donald Albin

Numerade Educator