Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 4

Motion in Two and Three Dimensions - all with Video Answers

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Chapter Questions

Problem 1

The position vector for an electron is $\vec{r}=(5.0 \mathrm{~m}) \hat{\mathrm{i}}-$ $(3.0 \mathrm{~m}) \hat{\mathrm{j}}+(2.0 \mathrm{~m}) \hat{\mathrm{k}}$. (a) Find the magnitude of $\vec{r}$. (b) Sketch the vector on a right-handed coordinate system.

Averell Hause

Averell Hause

Carnegie Mellon University

Problem 2

-1 The position vector for an electron is $\vec{r}=(5.0 \mathrm{~m}) \hat{\mathrm{i}}-$ $(3.0 \mathrm{~m}) \hat{\mathrm{j}}+(2.0 \mathrm{~m}) \hat{\mathrm{k}}$. (a) Find the magnitude of $\vec{r}$. (b) Sketch the vector on a right-handed coordinate system.
-2 A watermelon seed has the following coordinates: $x=-5.0 \mathrm{~m}$, $y=8.0 \mathrm{~m}$, and $z=0 \mathrm{~m} .$ 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 right-handed coordinate system. If the seed is moved to the $x y z$ coordinates $(3.00 \mathrm{~m},$, $0 \mathrm{~m}, 0 \mathrm{~m}$ ), what is its displacement (e) in unit-vector notation and as
(f) a magnitude and (g) an angle relative to the positive $x$ direction?

Suzanne W.

Numerade Educator

Problem 3

A positron undergoes a displacement $\Delta \vec{r}=2.0 \hat{\mathrm{i}}-3.0 \hat{\mathrm{j}}+6.0 \hat{\mathrm{k}}$,
ending with the position vector $\vec{r}=3.0 \hat{\mathrm{j}}-4.0 \hat{\mathrm{k}}$, in meters. What was the positron's initial position vector?
w.4 The minute hand of a wall clock measures $10 \mathrm{~cm}$ from its tip to the axis about which it rotates. The magnitude and angle of the displacement vector of the tip are to be determined for three time intervals. What are the (a) magnitude and (b) angle from a quarter after the hour to half past, the (c) magnitude and (d) angle for the next half hour, and the (e) magnitude and (f) angle for the hour after that?

Jose Carlos

Numerade Educator

Problem 4

The minute hand of a wall clock measures $10 \mathrm{~cm}$ from its tip to the axis about which it rotates. The magnitude and angle of the displacement vector of the tip are to be determined for three time intervals. What are the (a) magnitude and (b) angle from a quarter after the hour to half past, the (c) magnitude and (d) angle for the next half hour, and the (e) magnitude and (f) angle for the hour after that?

Cyra Jelle Calleja

Cyra Jelle Calleja

Numerade Educator

Problem 5

A train at a constant $60.0 \mathrm{~km} / \mathrm{h}$ moves east for $40.0 \mathrm{~min}$, then in a direction $50.0^{\circ}$ east of due north for $20.0 \mathrm{~min}$, and then west for $50.0 \mathrm{~min}$. What are the (a) magnitude and (b) angle of its average velocity during this trip?

Jose Carlos

Numerade Educator

Problem 6

An electron's position is given by $\vec{r}=3.00 t \hat{\mathrm{i}}-4.00 t \hat{\mathrm{j}}+2.00 \hat{\mathrm{k}}$,
with $t$ in seconds and $\vec{r}$ in meters. (a) In unit-vector notation, what js the electron's velocity $\vec{v}(t) ?$ At $t=2.00 \mathrm{~s}$, what is $\vec{v}(\mathrm{~b})$ in unitvector notation and as (c) a magnitude and (d) an angle relative to the positive direction of the $x$ axis?

Jose Carlos

Numerade Educator

Problem 7

An ion's position vector is initially $\vec{r}=5.0 \hat{\mathrm{i}}-6.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}$, and $10 \mathrm{~s}$ later it is $\vec{r}=-2.0 \hat{\mathrm{i}}+8.0 \hat{\mathrm{j}}-2.0 \hat{\mathrm{k}}$, all in meters. In unit-
vector notation, what is its $\vec{v}_{\text {avg }}$ during the $10 \mathrm{~s}$ ?

Averell Hause

Carnegie Mellon University

Problem 8

A plane flies $483 \mathrm{~km}$ east from city $A$ to city $B$ in $45.0 \mathrm{~min}$ and then $966 \mathrm{~km}$ south from city $B$ to city $C$ in $1.50 \mathrm{~h}$. For the total trip, what are the (a) magnitude and (b) direction of the plane's displacement, the (c) magnitude and (d) direction of its average velocity, and (e) its average speed?

Ceren Uzun

Texas Tech University

Problem 9

Figure 4-30 gives the path of a squirrel moving about on level ground, from point $A$ (at time $t=0$ ), to points $B$ (at $t=5.00 \mathrm{~min}), C$
(at $t=10.0 \mathrm{~min})$, and finally $D$
(at $t=15.0 \mathrm{~min}$ ). Consider the average velocities of the squirrel from point $A$ to each of the other three points. Of them, what are the (a) magnitude and (b) angle of the one with the least magnitude and the (c) magnitude and (d) angle of the one with the greatest magnitude?

Rashmi Sinha

Numerade Educator

Problem 10

The position vector $\vec{r}=5.00 t \hat{\mathrm{i}}+\left(e t+f t^{2}\right) \hat{\mathrm{j}}$ locates a
particle as a function of time $t$. Vector $\vec{r}$ is in meters, $t$ is in seconds, and factors $e$ and $f$ are constants. Figure $4-31$ gives the angle $\theta$ of the particle's direction of travel as a function of $t(A$ is measured from the positive $x$ direction). What are (a) $e$ and (b) $f$, including units?

Ceren Uzun

Texas Tech University

Problem 11

The position $\vec{r}$ of a particle moving in an $x y$ plane is given by $\vec{r}=\left(2.00 t^{3}-5.00 t\right) \hat{\mathrm{i}}+\left(6.00-7.00 t^{4}\right) \hat{\mathrm{j}}$, with $\vec{r}$ in meters and $\mathrm{t}$
in seconds. In unit-vector notation, calculate (a) $\vec{r},(\mathrm{~b}) \vec{v}$, and $(\mathrm{c}) \vec{a}$ for $t=2.00 \mathrm{~s}$ (d) What is the angle between the positive direction of the $x$ axis and a line tangent to the particle's path at $t=2.00 \mathrm{~s}$ ?

Averell Hause

Carnegie Mellon University

Problem 12

At one instant a bicyclist is $40.0 \mathrm{~m}$ due east of a park's flagpole, going due south with a speed of $10.0 \mathrm{~m} / \mathrm{s}$. Then $30.0 \mathrm{~s}$ later, the cyclist is $40.0 \mathrm{~m}$ due north of the flagpole, going due east with a speed of $10.0 \mathrm{~m} / \mathrm{s}$. For the cyclist in this $30.0$ s interval, what are the
(a) magnitude and (b) direction of the displacement, the (c) magnitude and (d) direction of the average velocity, and the (e) magnitude and (f) direction of the average acceleration?

Cyra Jelle Calleja

Cyra Jelle Calleja

Numerade Educator

Problem 13

A particle moves so that its position (in meters) as a function of time (in seconds) is $\vec{r}=\hat{\mathrm{i}}+4 t^{2 \hat{\mathrm{j}}}+t \hat{\mathrm{k}}$. Write expressions for (a) its velocity and (b) its acceleration as functions of time.

Jose Carlos

Numerade Educator

Problem 14

A proton initially has $\vec{v}=4.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}}$ and then
$4.0 \mathrm{~s}$ later has $\vec{v}=-2.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{j}}+5.0 \hat{\mathrm{k}}$ (in meters per second). For
that $4.0 \mathrm{~s}$, what are (a) the proton's average acceleration $\vec{a}_{\text {avg }}$ in unitvector notation, (b) the magnitude of $\vec{a}_{\text {avg }}$, and $($ c) the angle between $\vec{a}_{\text {avg }}$ and the positive direction of the $x$ axis?

Jose Carlos

Numerade Educator

Problem 15

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

Vishal Gupta

Numerade Educator

Problem 16

The velocity $\vec{v}$ of a particle moving in the $x y$ plane is given by $\vec{v}=\left(6.0 t-4.0 t^{2}\right) \hat{\mathrm{i}}+8.0 \hat{\mathrm{j}}$, with $\vec{v}$ in meters per second and $t(>0)$ in seconds. (a) What is the acceleration when $t=3.0 \mathrm{~s}$ ?
(b) When (if ever) is the acceleration zero? (c) When (if ever) is the velocity zero?
(d) When (if ever) does the speed equal $10 \mathrm{~m} / \mathrm{s} ?$

Jose Carlos

Numerade Educator

Problem 17

A cart is propelled over an $x y$ plane with acceleration components $a_{x}=4.0 \mathrm{~m} / \mathrm{s}^{2}$ and $a_{y}=-2.0 \mathrm{~m} / \mathrm{s}^{2} .$ Its initial velocity has components $v_{0 x}=8.0 \mathrm{~m} / \mathrm{s}$ and $v_{0 y}=12 \mathrm{~m} / \mathrm{s}$. In unit-vector notation, what is the velocity of the cart when it reaches its greatest $y$ coordinate? ..18 A moderate wind accelerates a pebble over a horizontal $x y$ plane with a constant acceleration $\vec{a}=\left(5.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(7.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$.

Jose Carlos

Numerade Educator

Problem 18

A moderate wind accelerates a 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 $t=0$, the velocity is $(4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{i}}$. What are the (a) magnitude and (b) angle of its velocity when it has been displaced by $12.0 \mathrm{~m}$ parallel to the $x$ axis?

Jose Carlos

Numerade Educator

Problem 19

The acceleration of a particle moving only on a horizontal $x y$ plane is given by $\vec{a}=3 t \hat{i}+4 t \hat{j}$, where $\vec{a}$ is in meters per secondsquared and $t$ is in seconds. At $t=0$, the position vector $\vec{r}=(20.0 \mathrm{~m}) \hat{\mathrm{i}}+(40.0 \mathrm{~m}) \hat{\mathrm{j}}$ locates the particle, which then has the
velocity vector $\vec{v}=(5.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(2.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$. At $t=4.00 \mathrm{~s}$, what
are (a) its position vector in unit-vector notation and (b) the angle between its direction of travel and the positive direction of the $x$ axis?

Jose Carlos

Numerade Educator

Problem 20

In Fig. 4-32, 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 parallel to the $x$ axis. At the instant particle $\bar{A}$ passes the $y$ axis, particle $B$ leaves the origin with a zero initial speed and a constant acceleration $\vec{a}$ of magnitude $0.40 \mathrm{~m} / \mathrm{s}^{2}$. What angle $\theta$ between $\vec{a}$ and the positive direction of the $y$ axis would result in a collision?

Averell Hause

Carnegie Mellon University

Problem 21

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 $P, 0.19 \mathrm{~s}$ 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 22

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 from the table edge. (a) How long is the ball in the air? (b) What is its speed at the instant it leaves the table?

Jose Carlos

Numerade Educator

Problem 23

A projectile is fired horizontally from a gun that is $45.0 \mathrm{~m}$ above flat ground, emerging from the gun with a speed of $250 \mathrm{~m} / \mathrm{s}$. (a) How long does the projectile remain in the air? (b) At what horizontal distance from the firing point does it strike the ground? (c) What is the magnitude of the vertical component of its velocity as it strikes the ground?

Averell Hause

Carnegie Mellon University

Problem 24

In the 1991 World Track and Field Championships in Tokyo, Mike Powell jumped $8.95 \mathrm{~m}$, breaking by a full $5 \mathrm{~cm}$ the 23-year long-jump record set by Bob Beamon. 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 range than the maximum possible range for a particle launched at the same speed?

Ceren Uzun

Texas Tech University

Problem 25

The current world-record motorcycle jump is $77.0 \mathrm{~m}$, set by Jason Renie. Assume that he left the take-off ramp at $12.0^{\circ}$ to the horizontal and that the take-off and landing heights are the same. Neglecting air drag, determine his take-off speed.

Averell Hause

Carnegie Mellon University

Problem 26

A stone is catapulted at time $t=0$, 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 $t=1.10 \mathrm{~s} ?$ Repeat for the (c) horizontal and
(d) vertical components at $t=1.80 \mathrm{~s}$, and for the (e) horizontal and (f) vertical components at $t=5.00 \mathrm{~s}$

Ceren Uzun

Texas Tech University

Problem 27

A certain airplane has a speed of $290.0 \mathrm{~km} / \mathrm{h}$ and is diving at an angle of $\theta=30.0^{\circ}$ below the horizontal when the pilot releases a radar decoy (Fig. 4-33). The horizontal distance between the release point and the point where the decoy strikes the ground is $d=$ $700 \mathrm{~m}$. (a) How long is the decoy in the air? (b) How high was the release point?

Jose Carlos

Numerade Educator

Problem 28

In Fig. 4-34, a stone is projected at a cliff of height $h$ with an initial speed of $42.0 \mathrm{~m} / \mathrm{s}$ directed at angle $\theta_{0}=60.0^{\circ}$ above the horizontal. The stone strikes at $A$, $5.50 \mathrm{~s}$ after launching. Find (a) the height $h$ of the cliff, (b) the speed of the stone just before impact at $A$, and (c) the maximum height $H$ reached above the ground.

Jose Carlos

Numerade Educator

Problem 29

A projectile's launch speed is five times its speed at maximum height. Find launch angle $\theta_{0}$.

Averell Hause

Carnegie Mellon University

Problem 30

A soccer ball is 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?

Ceren Uzun

Texas Tech University

Problem 31

In a jump spike, a volleyball player slams the ball from overhead and toward the opposite floor. Controlling the angle of the spike is difficult. Suppose a ball is spiked from a height of $2.30$ $\mathrm{m}$ with an initial speed of $20.0 \mathrm{~m} / \mathrm{s}$ at a downward angle of $18.00^{\circ} .$ How much farther on the opposite floor would it have landed if the downward angle were, instead, $8.00^{\circ} ?$

Prabhat Tyagi

Numerade Educator

Problem 32

You throw a ball toward a wall at speed $25.0 \mathrm{~m} / \mathrm{s}$ and at angle $\theta_{0}=40.0^{\circ}$ above the horizontal (Fig. 4-35). The wall is distance $d=$ $22.0 \mathrm{~m}$ from the release point of the ball. (a) How far above the release point does the ball hit the wall? What are the (b) horizontal and (c) vertical components of its velocity as it hits the wall? (d) When it hits, has it passed the highest point on its trajectory?

Ceren Uzun

Texas Tech University

Problem 33

A plane, diving with constant speed 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 $5.00 \mathrm{~s}$ after release. (a) What is the speed of the plane? (b) How far does the projectile travel horizontally during its flight? What are the (c) horizontal and (d) vertical components of its velocity just before striking the ground?

Jose Carlos

Numerade Educator

Problem 34

A trebuchet was a hurling machine built to attack the walls of a castle under siege. A large stone could be hurled against a wall to break apart the wall. The machine was not placed near the

Donald Albin

Numerade Educator

Problem 35

A rifle that shoots bullets at $460 \mathrm{~m} / \mathrm{s}$ is to be aimed at a target $45.7 \mathrm{~m}$ away. If the center of the target is level with the rifle, how high above the target must the rifle barrel be pointed so that the bullet hits dead center?

Jose Carlos

Numerade Educator

Problem 36

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 37

A lowly high diver pushes off horizontally with a speed of $2.00 \mathrm{~m} / \mathrm{s}$ from the platform edge $10.0 \mathrm{~m}$ above the surface of the water. (a) At what horizontal distance from the edge is the diver $0.800 \mathrm{~s}$ after pushing off? (b) At what vertical distance above the surface of the water is the diver just then?
(c) At what horizontal distance from the edge does the diver strike the water?

Averell Hause

Carnegie Mellon University

Problem 38

A golf ball is struck at ground level. The speed of the golf ball as a function of the time is shown in Fig. $4-36$, where $t=0$ at the instant the ball is struck. The scaling on the vertical axis is set by $v_{a}=19 \mathrm{~m} / \mathrm{s}$ and $v_{b}=31 \mathrm{~m} / \mathrm{s}$
(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?

Ceren Uzun

Texas Tech University

Problem 39

In Fig. $4-37$, a ball is thrown leftward from the left edge of the roof, at height $h$ above the ground. The ball hits the ground $1.50 \mathrm{~s}$ later, at distance $d=25.0 \mathrm{~m}$ from the building and at angle $\theta=60.0^{\circ}$ with the horizontal. (a) Find $h$. (Hint: One way is to reverse the motion, as if on video.) What are the (b) magnitude and (c) angle relative to the horizontal of the velocity at which the ball is thrown? (d) Is the angle above or below the horizontal?

Averell Hause

Carnegie Mellon University

Problem 40

Suppose that a shot putter can put a shot at the worldclass speed $v_{0}=15.00 \mathrm{~m} / \mathrm{s}$ and at a height of $2.160 \mathrm{~m}$. What horizontal distance would the shot travel if the launch angle $\theta_{0}$ is
(a) $45.00^{\circ}$ and (b) $42.00^{\circ}$ ? The answers indicate that the angle of $45^{\circ}$, which maximizes the range of projectile motion, does not maximize the horizontal distance when the launch and landing are at different heights.

Averell Hause

Carnegie Mellon University

Problem 41

Upon spotting an insect on a twig overhanging water, an archer fish squirts water drops at the insect to knock it into the water (Fig. $4-38$ ). Although the fish sees the insect along a straight-line path at angle $\phi$ and distance $d$, a drop must be launched at a different angle $\theta_{0}$ if its parabolic path is to intersect the insect. If $\phi=36.0^{\circ}$ and $d=0.900 \mathrm{~m}$,
what launch angle $\theta_{0}$ is required for the drop to be at the top of the parabolic path when it reaches the insect?

Averell Hause

Carnegie Mellon University

Problem 42

In 1939 or 1940 , Emanuel Zacchini took his humancannonball act to an extreme: After being shot from a cannon, he soared over three Ferris wheels and into a net (Fig. 4-39). Assume that he is launched with a speed of $26.5 \mathrm{~m} / \mathrm{s}$ and at an angle of $53.0^{\circ}$.
(a) Treating him as a particle, calculate his clearance over the first wheel. (b) If he reached maximum height over the middle wheel, by how much did he clear it? (c) How far from the cannon should the net's center have been positioned (neglect air drag)?

Averell Hause

Carnegie Mellon University

Problem 43

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

Jose Carlos

Numerade Educator

Problem 44

A baseball leaves a pitcher's hand horizontally at a speed of $161 \mathrm{~km} / \mathrm{h}$. The distance to the batter is $18.3 \mathrm{~m}$. (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?

Ceren Uzun

Texas Tech University

Problem 45

In Fig. 4-40, a ball is launched with a velocity of magnitude $10.0 \mathrm{~m} / \mathrm{s}$, at an angle of $50.0^{\circ}$ to the horizontal. The launch point is at the base of a ramp of horizontal length $d_{1}=6.00 \mathrm{~m}$ and height $d_{2}=3.60 \mathrm{~m}$. A plateau is located at the top of the ramp. (a) Does the ball land on the ramp or the plateau? When it lands, what are the (b) magnitude and (c) angle of its displacement from the launch point?

Averell Hause

Carnegie Mellon University

Problem 46

In basketball, hang is an illusion in which a player seems to weaken the gravitational acceleration while in midair. The illusion depends much on a skilled player's ability to rapidly shift the ball between hands during the flight, but it might also be supported by the longer horizontal distance the player travels in the upper part of the jump than in the lower part. If a player jumps with an initial speed of $v_{0}=7.00 \mathrm{~m} / \mathrm{s}$ at an angle of $\theta_{0}=35.0^{\circ}$, what percent of the jump's range does the player spend in the upper half of the jump (between maximum height and half maximum height $) ?$

Donald Albin

Numerade Educator

Problem 47

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) At the fence, what is the distance between the fence top and the ball center?

Averell Hause

Carnegie Mellon University

Problem 48

In Fig. 4-41, a ball is thrown up onto a roof, landing $4.00 \mathrm{~s}$ later at height $h=20.0 \mathrm{~m}$ above the release level. The ball's path just before landing is angled at $\theta=60.0^{\circ}$ with the roof. (a) Find the horizontal distance $d$ it travels. (See the hint to Problem $39 .$ What are the
(b) magnitude and (c) angle (relative to the horizontal) of the ball's initial velocity?

Mukesh Devi

Numerade Educator

Problem 49

A football kicker can give the ball an initial speed of $25 \mathrm{~m} / \mathrm{s}$. What are the (a) least and (b) greatest elevation angles at which he can 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?

Averell Hause

Carnegie Mellon University

Problem 50

Two seconds after being projected from ground level, a projectile is displaced $40 \mathrm{~m}$ horizontally and $53 \mathrm{~m}$ vertically above its launch point. 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 the launch point?

Ceren Uzun

Texas Tech University

Problem 51

A skilled skier knows to jump upward before reaching a downward slope. Consider a jump in which the launch speed is $v_{0}=10 \mathrm{~m} / \mathrm{s}$, the launch angle is $\theta_{0}=11.3^{\circ}$, the initial course is approximately flat, and the steeper track has a slope of $9.0^{\circ}$. Figure $4-42 a$ shows a prejump that allows the skier to land on the top portion of the steeper track. Figure $4-42 b$ shows a jump at the edge of the steeper track. In Fig. $4-42 a$, the skier lands at approximately the launch level. (a) In the landing, what is the angle $\phi$ between the skier's path and the slope? In Fig. $4-42 b,(\mathrm{~b})$ how far below the launch level does the skier land and (c) what is $\phi ?$ (The greater fall and greater $\phi$ can result in loss of control in the landing.)

Animesh Raj

Numerade Educator

Problem 52

A ball is to be shot from level ground toward a wall at distance $ x$ (Fig. $4-43 a$ ). Figure $4-43 b$ shows the $y$ component $v_{y}$ of the ball's velocity just as it would reach the wall, as a function of that distance $x$. The scaling is set by $v_{y s}=5.0 \mathrm{~m} / \mathrm{s}$ and $x_{s}=20 \mathrm{~m} .$ What is the launch angle?

Ceren Uzun

Texas Tech University

Problem 53

In Fig. $4-44$, a baseball is hit at a height $h=1.00 \mathrm{~m}$ and then caught at the same height. It travels alongside a wall, moving up past the top of the wall $1.00 \mathrm{~s}$ after it is hit and then down past the top of the wall $4.00 \mathrm{~s}$ later, at distance $D=50.0 \mathrm{~m}$ farther along the wall. (a) What horizontal distance is traveled by the ball from hit to catch? What are the (b) magnitude and (c) angle (relative to the horizontal) of the ball's velocity just after being hit? (d) How high is the wall?

Averell Hause

Carnegie Mellon University

Problem 54

Go A ball is to be shot from level ground with a certain speed. Figure $4-45$ shows the range $R$ it will have versus the launch angle $\theta_{0}$. The value of $\theta_{0}$ determines the flight time; let $t_{\max }$ represent the maximum flight time. What is the least speed the ball will have during its flight if $\theta_{0}$ is chosen such that the flight time is $0.500 t_{\max } ?$

Averell Hause

Carnegie Mellon University

Problem 55

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 56

An Earth satellite moves in a circular orbit $640 \mathrm{~km}$ (uniform circular motion) 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?

Ceren Uzun

Texas Tech University

Problem 57

A carnival merry-go-round rotates about a vertical axis at a constant rate. A man standing on the edge has a constant speed of $3.66 \mathrm{~m} / \mathrm{s}$ and a centripetal acceleration $\vec{a}$ of magnitude $1.83 \mathrm{~m} / \mathrm{s}^{2}$ Position vector $\vec{r}$ locates him relative to the rotation axis. (a) What is the magnitude of $\vec{r}$ ? What is the direction of $\vec{r}$ when $\vec{a}$ is directed (b) due east and (c) due south?

Jose Carlos

Numerade Educator

Problem 58

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 59

A woman rides a carnival Ferris wheel at radius $15 \mathrm{~m}$, completing five turns about its horizontal axis every minute. What are (a) the period of the motion, the (b) magnitude and (c) direction of her centripetal acceleration at the highest point, and the (d) magnitude and (e) direction of her centripetal acceleration at the lowest point?

Averell Hause

Carnegie Mellon University

Problem 60

A centripetal-acceleration addict rides in uniform circular motion with radius $r=3.00 \mathrm{~m}$. At one instant his acceleration is $\vec{a}=\left(6.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(-4.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$. At that instant, what are the val-
ues of (a) $\vec{v} \cdot \vec{a}$ and (b) $\vec{r} \times \vec{a}$ ?

Ceren Uzun

Texas Tech University

Problem 61

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 particle on the star's equator and (b) what is the magnitude of the particle's centripetal acceleration? (c) If the neutron star rotates faster, do the answers to (a) and (b) increase, decrease, or remain the same?

Averell Hause

Carnegie Mellon University

Problem 62

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

Jose Carlos

Numerade Educator

Problem 63

At $t_{1}=2.00 \mathrm{~s}$, the acceleration of a particle in counterclockwise circular motion is $\left(6.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(4.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$. It moves at
constant speed. At time $t_{2}=5.00 \mathrm{~s}$, the particle's acceleration is $\left(4.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(-6.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$. What is the radius of the path taken
by the particle if $t_{2}-t_{1}$ is less than one period?

Donald Albin

Numerade Educator

Problem 64

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 $-5.00 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s}$
and an acceleration of $+12.5 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}^{2} .$ What are the (a) $x$ and (b) $y$ coordinates of the center of the circular path?

Jose Carlos

Numerade Educator

Problem 65

A purse at radius $2.00 \mathrm{~m}$ and a wallet at radius $3.00 \mathrm{~m}$ travel in uniform circular motion on the floor of a merry-go-round as the ride turns. They are on the same radial line. At one instant, the acceleration of the purse is $\left(2.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(4.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$. At that instant
and in unit-vector notation, what is the acceleration of the wallet?

Averell Hause

Carnegie Mellon University

Problem 66

A particle moves along a circular path over a horizontal $x y$ coordinate system, at constant speed. At time $t_{1}=4.00 \mathrm{~s}$, it is at point $(5.00 \mathrm{~m}, 6.00 \mathrm{~m})$ with velocity $(3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$ and acceleration in the
positive $x$ direction. At time $t_{2}=10.0 \mathrm{~s}$, it has velocity $(-3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}$ and acceleration in the positive $y$ direction. What are the (a) $x$ and
(b) $y$ coordinates of the center of the circular path if $t_{2}-t_{1}$ is less than one period?

Ceren Uzun

Texas Tech University

Problem 67

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 during the circular motion?

Averell Hause

Carnegie Mellon University

Problem 68

A cat rides a merry-go-round 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 $t_{2}=5.00 \mathrm{~s}$, the cat's 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 $t_{2}-t_{1}$, which is less than one period?

Averell Hause

Carnegie Mellon University

Problem 69

A cameraman on a pickup truck is traveling westward at $20 \mathrm{~km} / \mathrm{h}$ while he records a cheetah that is moving westward $30 \mathrm{~km} / \mathrm{h}$ faster than the truck. Suddenly, the cheetah stops, turns, and then runs at $45 \mathrm{~km} / \mathrm{h}$ eastward, as measured by a suddenly nervous crew member who stands alongside the cheetah's path. The change in the animal's velocity takes $2.0 \mathrm{~s}$. What are the (a) magnitude and (b) direction of the animal's acceleration according to the cameraman and the (c) magnitude and (d) direction according to the nervous crew member?

Averell Hause

Carnegie Mellon University

Problem 70

A boat is traveling upstream in the positive direction of an $x$ axis at $14 \mathrm{~km} / \mathrm{h}$ with respect to the water of a river. The water is flowing at $9.0 \mathrm{~km} / \mathrm{h}$ with respect to the ground. What are the (a) magnitude and (b) direction of the boat's velocity with respect to the ground? A child on the boat walks from front to rear at $6.0 \mathrm{~km} / \mathrm{h}$ with respect to the boat. What are the (c) magnitude and
(d) direction of the child's velocity with respect to the ground?

Ceren Uzun

Texas Tech University

Problem 71

A suspicious-looking man runs as fast as he can along a moving sidewalk from one end to the other, taking $2.50 \mathrm{~s}$. Then security agents appear, and the man runs as fast as he can back along the sidewalk to his starting point, taking $10.0 \mathrm{~s}$. What is the ratio of the man's running speed to the sidewalk's speed?

Jose Carlos

Numerade Educator

Problem 72

A rugby player runs with the ball directly toward his opponent's goal, along the positive direction of an $x$ axis. He can legally pass the ball to a teammate as long as the ball's velocity relative to the field does not have a positive $x$ component. Suppose the player runs at speed $4.0 \mathrm{~m} / \mathrm{s}$ relative to the field while he passes the ball with velocity $\vec{v}_{B P}$ relative to himself. If $\vec{v}_{B P}$ has magnitude $6.0 \mathrm{~m} / \mathrm{s}$, what is the smallest angle it can have for the pass to be legal?

Averell Hause

Carnegie Mellon University

Problem 73

Two highways intersect as shown in Fig. $4-46$. At the instant shown, a police car $P$ is distance $d_{P}=800 \mathrm{~m}$ from the intersection and moving at speed $v_{P}=80 \mathrm{~km} / \mathrm{h} .$ Motorist $M$ is distance $d_{M}=$ $600 \mathrm{~m}$ from the intersection and moving at speed $v_{M}=60 \mathrm{~km} / \mathrm{h}$.

Averell Hause

Carnegie Mellon University

Problem 74

After flying for 15 min in a wind blowing $42 \mathrm{~km} / \mathrm{h}$ at an angle of $20^{\circ}$ south of east, an airplane pilot is over a town that is $55 \mathrm{~km}$ due north of the starting point. What is the speed of the airplane relative to the air?

Ceren Uzun

Texas Tech University

Problem 75

A train travels due south at $30 \mathrm{~m} / \mathrm{s}$ (relative to the ground) in a rain that is blown toward the south by the wind. The path of each raindrop makes an angle of $70^{\circ}$ with the vertical, as measured by an observer stationary on the ground. An observer on the train, however, sees the drops fall perfectly vertically. Determine the speed of the raindrops relative to the ground.

Averell Hause

Carnegie Mellon University

Problem 76

A light plane attains an airspeed of $500 \mathrm{~km} / \mathrm{h}$. The pilot sets out for a destination $800 \mathrm{~km}$ due north but discovers that the plane must be headed $20.0^{\circ}$ east of due north to fly there directly. The plane arrives in $2.00 \mathrm{~h}$. What were the (a) magnitude and (b) direction of the wind velocity?

Ceren Uzun

Texas Tech University

Problem 77

Snow is falling vertically at a constant speed of $8.0 \mathrm{~m} / \mathrm{s}$. At what angle from the vertical do the snowflakes appear to be falling as viewed by the driver of a car traveling on a straight, level road with a speed of $50 \mathrm{~km} / \mathrm{h} ?$

Averell Hause

Carnegie Mellon University

Problem 78

In the overhead view of Fig. $4-47$, Jeeps $P$ and $B$ race along straight lines, across flat terrain, and past stationary border guard $A$. Relative to the guard, $B$ travels at a constant speed of $20.0 \mathrm{~m} / \mathrm{s}$, at the angle $\theta_{2}=30.0^{\circ}$. Relative to the guard, $P$ has accelerated from rest at a constant rate of $0.400 \mathrm{~m} / \mathrm{s}^{2}$ at the angle $\theta_{1}=60.0^{\circ} .$ At a certain time during the acceleration, $P$ has a speed of $40.0 \mathrm{~m} / \mathrm{s}$. At that time, what are the (a) magnitude and (b) direction of the velocity of $P$ relative to $B$ and the (c) magnitude and (d) direction of the acceleration of $P$ relative to $B^{?} ?$

Averell Hause

Carnegie Mellon University

Problem 79

Two ships, $A$ and $B$, leave port at the same time. Ship $A$ travels northwest at 24 knots, and ship $B$ travels at 28 knots in a direction $40^{\circ}$ west of south. $(1 \mathrm{knot}=1$ nautical mile per hour; see Appendix D.) What are the (a) magnitude and (b) direction of the velocity of ship $A$ relative to $B ?$ (c) After what time will the ships be 160 nautical miles apart? (d) What will be the bearing of $B$ (the direction of $B$ 's position) relative to $A$ at that time?

Averell Hause

Carnegie Mellon University

Problem 80

A $200-\mathrm{m}$ -wide river flows due east at a uniform speed of $2.0 \mathrm{~m} / \mathrm{s} .$ A boat with a speed of $8.0 \mathrm{~m} / \mathrm{s}$ relative to the water leaves the south bank pointed in a direction $30^{\circ}$ west of north. What are the (a) magnitude and (b) direction of the boat's velocity relative to the ground? (c) How long does the boat take to cross the river?

Ceren Uzun

Texas Tech University

Problem 81

$A$ is located $4.0 \mathrm{~km}$ north and $2.5 \mathrm{~km}$ east of ship $B$. Ship $A$ has a velocity of $22 \mathrm{~km} / \mathrm{h}$ toward the south, and ship $B$ has a velocity of $40 \mathrm{~km} / \mathrm{h}$ in a direction $37^{\circ}$ north of east. (a) What is the velocity of $A$ relative to $B$ in unit-vector notation with $\hat{\mathrm{i}}$ toward the east? (b) Write an expression (in terms of $\hat{\mathrm{i}}$ and $\hat{\mathrm{j}}$ ) for the position of $A$ relative to $B$ as a function of $t$, where $t=0$ when the ships are in the positions described above. (c) At what time is the separation between the ships least? (d) What is that least separation?

Averell Hause

Carnegie Mellon University

Problem 82

A 200-m-wide river has a uniform flow speed of $1.1 \mathrm{~m} / \mathrm{s}$ through a jungle and toward the east. An explorer wishes to

Averell Hause

Carnegie Mellon University

Problem 83

A woman who can row a boat at $6.4 \mathrm{~km} / \mathrm{h}$ in still water faces a long, straight river with a width of $6.4 \mathrm{~km}$ and a current of $3.2 \mathrm{~km} / \mathrm{h}$. Let $\hat{\text { i }}$ point directly across the river and $\hat{\mathrm{j}}$ point directly downstream. If she rows in a straight line to a point directly opposite her starting position, (a) at what angle to î must she point the boat and
(b) how long will she take? (c) How long will she take if, instead, she rows $3.2 \mathrm{~km}$ down the river and then back to her starting point? (d) How long if she rows $3.2 \mathrm{~km} u p$ the river and then back to her starting point? (e) At what angle to $\hat{\mathbf{i}}$ should she point the boat if she wants to cross the river in the shortest possible time? (f) How long is that shortest time?

Animesh Raj

Numerade Educator

Problem 84

In Fig. $4-48 a$, a sled moves in the negative $x$ direction at constant speed $v_{s}$ while a ball of ice is shot from the sled with a velocity $\vec{v}_{0}=v_{0 \mathrm{r}} \hat{\mathrm{i}}+v_{0 \mathrm{j}} \hat{\mathrm{j}}$ relative to the sled. When the ball lands, its horizontal displacement $\Delta x_{b g}$ relative to the ground (from its launch position to its landing position) is measured. Figure $4-48 b$ gives $\Delta x_{b g}$ as a function of $v_{s} .$ Assume the ball lands at approximately its launch height. What are the values of (a) $v_{0 x}$ and (b) $v_{0 y}$ ? The ball's displacement $\Delta x_{b s}$ relative to the sled can also be measured. Assume that the sled's velocity is not changed when the ball is shot. What is $\Delta x_{b s}$ when $v_{s}$ is (c) $5.0 \mathrm{~m} / \mathrm{s}$ and (d) $15 \mathrm{~m} / \mathrm{s}$ ?

Averell Hause

Carnegie Mellon University

Problem 85

You are kidnapped by political-science majors (who are upset because you told them 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 $2.0 \mathrm{~min}$, 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?

Averell Hause

Carnegie Mellon University

Problem 86

A radar station detects an airplane approaching directly from the east. At first observation, the airplane is at distance $d_{1}=360 \mathrm{~m}$ from the station and at angle $\theta_{1}=40^{\circ}$ above the horizon (Fig. $4-49$ ). The airplane is tracked through an angular change $\Delta \theta=123^{\circ}$ in the vertical east-west plane; its distance is then $d_{2}=790 \mathrm{~m}$. Find the
(a) magnitude and (b) direction of the airplane's displacement during this period.

Ceren Uzun

Texas Tech University

Problem 87

A baseball is hit at ground level. The ball reaches its maximum height above ground level $3.0 \mathrm{~s}$ after being hit. Then $2.5 \mathrm{~s}$ after reaching its maximum height, the ball barely clears a fence that is $97.5 \mathrm{~m}$ from where it was hit. Assume the ground is level. (a) What maximum height above ground level is reached by the ball? (b) How high is the fence? (c) How far beyond the fence does the ball strike the ground?

Averell Hause

Carnegie Mellon University

Problem 88

Long flights at midlatitudes in the Northern Hemisphere encounter the jet stream, an eastward airflow that can affect a plane's speed relative to Earth's surface. If a pilot maintains a certain speed relative to the air (the plane's airspeed), the speed relative to the surface (the plane's ground speed) is more when the flight is in the direction of the jet stream and less when the flight is opposite the jet stream. Suppose a round-trip flight is scheduled between two cities separated by $4000 \mathrm{~km}$, with the outgoing flight in the direction of the jet stream and the return flight opposite it. The airline computer advises an airspeed of $1000 \mathrm{~km} / \mathrm{h}$, for which the difference in flight times for the outgoing and return flights is $70.0 \mathrm{~min}$. What jet-stream speed is the computer using?

Averell Hause

Carnegie Mellon University

Problem 89

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

Jose Carlos

Numerade Educator

Problem 90

At what initial speed must the basketball player in Fig. 4-50 throw the ball, at angle $\theta_{0}=55^{\circ}$ above the horizontal, to make the foul shot? The horizontal distances are $d_{1}=1.0 \mathrm{ft}$ and $d_{2}=14 \mathrm{ft}$, and
the heights are $h_{1}=7.0 \mathrm{ft}$ and $h_{2}=10 \mathrm{ft}$.

Averell Hause

Carnegie Mellon University

Problem 91

During volcanic eruptions, chunks of solid rock can be blasted out of the volcano; these projectiles are called volcanic bombs. Figure $4-51$ shows a cross section of $\mathrm{Mt}$. Fuji, in Japan. (a) At what initial speed would a bomb have to be ejected, at angle $\theta_{0}=35^{\circ}$ to the horizontal, from the vent at $A$ in order to fall at the foot of the volcano at $B$, at vertical distance $h=3.30 \mathrm{~km}$ and horizontal distance $d=9.40 \mathrm{~km}$ ? Ignore, for the moment, the effects of air on the bomb's travel. (b) What would be the time of flight? (c) Would the effect of the air increase or decrease your answer in (a)?

Averell Hause

Carnegie Mellon University

Problem 92

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 93

$A$ is $90 \mathrm{~km}$ due west of oasis $B$. A desert camel leaves $A$ and takes $50 \mathrm{~h}$ to walk $75 \mathrm{~km}$ at $37^{\circ}$ north of due east. Next it takes $35 \mathrm{~h}$ to walk $65 \mathrm{~km}$ due south. Then it rests for $5.0 \mathrm{~h}$. What are the (a) magnitude and (b) direction of the camel's displacement relative to $A$ at the resting point? From the time the camel leaves $A$ until the end of the rest period, what are the (c) magnitude and (d) direction of its average velocity and (e) its average speed? The camel's last drink was at $A$; it must be at $B$ no more than $120 \mathrm{~h}$ later for its next drink. If it is to reach $B$ just in time, what must be the (f) magnitude and (g) direction of its average velocity after the rest period?

Animesh Raj

Numerade Educator

Problem 94

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 $t=0$ and position $x=0$ (Fig. 4-52). (a) At $t=20 \mathrm{~s}$, 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 indicate what you would see as you look up into the approaching rocks.

Averell Hause

Carnegie Mellon University

Problem 95

Figure 4-53 shows the straight path of a particle across an $x y$ coordinate system as the particle is accelerated from rest during time interval $\Delta t_{1} .$ The acceleration is constant. The $x y$ coordinates for point $A$ are $(4.00 \mathrm{~m}, 6.00 \mathrm{~m}) ;$ those for point $B$ are $(12.0$ $\mathrm{m}, 18.0 \mathrm{~m}) .$ (a) What is the ratio $a_{y} / a_{x}$ of the acceleration components? (b) What are the coordinates of the particle if the motion is continued for another interval equal to $\Delta t_{1} ?$

Averell Hause

Carnegie Mellon University

Problem 96

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 97

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

Carnegie Mellon University

Problem 98

A particle is in uniform circular motion about the origin of an $x y$ coordinate system, moving clockwise with a period of $7.00 \mathrm{~s}$. At one instant, its position vector (measured from the origin) is $\vec{r}=(2.00 \mathrm{~m}) \hat{\mathrm{i}}-(3.00 \mathrm{~m}) \hat{\mathrm{j}}$. At that instant, what is its velocity in
unit-vector notation?

Ceren Uzun

Texas Tech University

Problem 99

In Fig. $4-54$, a lump of wet putty moves in uniform circular motion as it rides at a radius of $20.0 \mathrm{~cm}$ on the rim of a wheel rotating counterclockwise with a period of $5.00$ $\mathrm{ms}$. The lump then happens to fly off the rim at the 5 o'clock position (as if on a clock face). It leaves the rim at a height of $h=1.20 \mathrm{~m}$ from the floor and at a distance $d=2.50$ $\mathrm{m}$ from a wall. At what height on the wall does the lump hit?

Averell Hause

Carnegie Mellon University

Problem 100

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 $(6.30 \hat{\mathrm{i}}-8.42 \mathrm{j}) \mathrm{m} / \mathrm{s}$. Three seconds later, because of a wind shift, the boat is instantaneously at rest. What is its average acceleration for this $3.00 \mathrm{~s}$ interval?

Jose Carlos

Numerade Educator

Problem 101

In Fig. $4-55$, a ball is shot directly upward from the ground with an initial speed of $v_{0}=7.00 \mathrm{~m} / \mathrm{s}$ Simultaneously, a construction elevator cab begins to move upward from the ground with a constant speed of $v_{c}=3.00 \mathrm{~m} / \mathrm{s}$. What maximum height does the ball reach relative to (a) the ground and (b) the cab floor? At what rate does the speed of the ball change relative to (c) the ground and (d) the cab floor?

Averell Hause

Carnegie Mellon University

Problem 102

A magnetic field forces an electron to move in a circle with radial acceleration $3.0 \times 10^{14} \mathrm{~m} / \mathrm{s}^{2} .$ (a) What is the speed of the electron if the radius of its circular path is $15 \mathrm{~cm} ?$ (b) What is the period of the motion?

Ceren Uzun

Texas Tech University

Problem 103

In $3.50 \mathrm{~h}$, a balloon drifts $21.5 \mathrm{~km}$ north, $9.70 \mathrm{~km}$ east, and $2.88 \mathrm{~km}$ upward from its release point on the ground. Find (a) the magnitude of its average velocity and (b) the angle its average velocity makes with the horizontal.

Averell Hause

Carnegie Mellon University

Problem 104

A ball is thrown horizontally from a height of $20 \mathrm{~m}$ and hits the ground with a speed that is three times its initial speed. What is the initial speed?

Ceren Uzun

Texas Tech University

Problem 105

A projectile is launched with an initial speed of $30 \mathrm{~m} / \mathrm{s}$ at an angle of $60^{\circ}$ above the horizontal. What are the (a) magnitude and
(b) angle of its velocity $2.0 \mathrm{~s}$ after launch, and $(\mathrm{c})$ is the angle above or below the horizontal? What are the (d) magnitude and (e) angle of its velocity $5.0 \mathrm{~s}$ after launch, and $(\mathrm{f})$ is the angle above or below the horizontal?

Averell Hause

Carnegie Mellon University

Problem 106

The position vector for a proton is initially $\vec{r}=$ $5.0 \hat{\mathrm{i}}-6.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}$ and then later is $\vec{r}=-2.0 \hat{\mathrm{i}}+6.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}$, all
in meters. (a) What is the proton's displacement vector, and (b) to what plane is that vector parallel?

Ceren Uzun

Texas Tech University

Problem 107

A particle $P$ travels with constant speed on a circle of radius $r=$ $3.00 \mathrm{~m}$ (Fig. $4-56)$ and completes one revolution in $20.0 \mathrm{~s}$. The particle passes through $O$ at time $t=0 .$ State the following vectors in magnitudeangle 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 $(\mathrm{c}) 10.0 \mathrm{~s}$. (d) For the $5.00 \mathrm{~s}$ interval from the end of the fifth second to the end of the tenth second, find the particle's displacement. For that interval, find (e) its average velocity and its velocity at the (f) beginning and
(g) end. Next, find the acceleration at the (h) beginning and (i) end of that interval.

Animesh Raj

Numerade Educator

Problem 108

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 109

(a) If an electron is projected horizontally with a speed of $3.0 \times 10^{6} \mathrm{~m} / \mathrm{s}$, how far will it fall in traversing $1.0 \mathrm{~m}$ of horizontal distance? (b) Does the answer increase or decrease if the initial speed is increased?

Averell Hause

Carnegie Mellon University

Problem 110

A person walks up a stalled 15 -m-long escalator in $90 \mathrm{~s}$. When standing on the same escalator, now moving, the person is carried up in $60 \mathrm{~s}$. How much time would it take that person to walk up the moving escalator? Does the answer depend on the length of the escalator?

Ceren Uzun

Texas Tech University

Problem 111

(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 Earth's rotation period have to be for objects on the equator to have a centripetal acceleration of magnitude $9.8 \mathrm{~m} / \mathrm{s}^{2} ?$

Averell Hause

Carnegie Mellon University

Problem 112

The range of a projectile depends not only on $v_{0}$ and $\theta_{0}$ but also on the value $g$ of the free-fall acceleration, which varies from place to place. In 1936 , Jesse Owens established a world's running broad jump record of $8.09 \mathrm{~m}$ at the Olympic Games at Berlin (where $g=9.8128 \mathrm{~m} / \mathrm{s}^{2}$ ). Assuming the same values of $v_{0}$ and $\theta_{0}$, by how much would his record have differed if he had competed instead in 1956 at Melbourne (where $\left.g=9.7999 \mathrm{~m} / \mathrm{s}^{2}\right)$ ?

Averell Hause

Carnegie Mellon University

Problem 113

Figure $4-57$ shows the path taken by a drunk skunk over level ground, from initial point $i$ to final point $f$. The angles are $\theta_{1}=30.0^{\circ}$, $\theta_{2}=50.0^{\circ}$, and $\theta_{3}=80.0^{\circ}$, and the
distances are $d_{1}=5.00 \mathrm{~m}, d_{2}=8.00$
$\mathrm{m}$, and $d_{3}=12.0 \mathrm{~m}$. What are the (a) magnitude and (b) angle of the skunk's displacement from $i$ to $f ?$

Averell Hause

Carnegie Mellon University

Problem 114

The position vector $\vec{r}$ of a particle moving in the $x y$ plane is $\vec{r}=2 \hat{i}+2 \sin [(\pi / 4 \mathrm{rad} / \mathrm{s}) t] \hat{\mathrm{j}}, \quad$ with
$\vec{r}$ in meters and $t$ in seconds. (a) Calculate the $x$ and $y$ components of the particle's position at $t=0,1.0,2.0,3.0$, and $4.0 \mathrm{~s}$ and sketch the particle's path in the $x y$ plane for the interval $0 \leq t \leq$ $4.0 \mathrm{~s}$. (b) Calculate the components of the particle's velocity at $t=1.0,2.0$, and $3.0 \mathrm{~s}$. Show that the velocity is tangent to the path of the particle and in the direction the particle is moving at each time by drawing the velocity vectors on the plot of the particle's path in part (a). (c) Calculate the components of the particle's acceleration at $t=1.0,2.0$, and $3.0 \mathrm{~s}$.

Averell Hause

Carnegie Mellon University

Problem 115

An electron having 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, the electron 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 the electron takes to travel the $2.00 \mathrm{~cm},(\mathrm{~b})$ the vertical distance it travels during that time, and the magnitudes of its (c) horizontal and (d) vertical velocity components as it emerges from the region.

Averell Hause

Carnegie Mellon University

Problem 116

An elevator without a ceiling is ascending with a constant speed of $10 \mathrm{~m} / \mathrm{s}$. A boy on the elevator shoots a ball directly upward, from a height of $2.0 \mathrm{~m}$ above the elevator floor, just as the elevator floor is $28 \mathrm{~m}$ above the ground. The initial speed of the ball with respect to the elevator is $20 \mathrm{~m} / \mathrm{s}$. (a) What maximum height above the ground does the ball reach? (b) How long does the ball take to return to the elevator floor?

Animesh Raj

Numerade Educator

Problem 117

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 the
(a) magnitude and (b) angle (relative to the horizontal) of the ball's initial velocity?

Averell Hause

Carnegie Mellon University

Problem 118

An airport terminal has a moving sidewalk to speed passengers through a long corridor. Larry does not use the moving sidewalk; he takes $150 \mathrm{~s}$ to walk through the corridor. Curly, who simply stands on the moving sidewalk, covers the same distance in $70 \mathrm{~s}$. Moe boards the sidewalk and walks along it. How long does Moe take to move through the corridor? Assume that Larry and Moe walk at the same speed.

Jose Carlos

Numerade Educator

Problem 119

A wooden boxcar is moving along a stra?ght ralroad track at speed $v_{1}$. A sniper fires a bullet (initial speed $v_{2}$ ) at it from a high-powered rifle. The bullet passes through both lengthwise walls of the car, its entrance and exit holes being exactly opposite each other as viewed from within the car. From what direction, relative to the track, is the bullet fired? Assume that the bullet is not deflected upon entering the car, but that its speed decreases by $20 \%$. Take $v_{1}=85 \mathrm{~km} / \mathrm{h}$ and $v_{2}=650 \mathrm{~m} / \mathrm{s}$. (Why don't you need to know the width of the boxcar?)

Animesh Raj

Numerade Educator

Problem 120

A sprinter running on a circular track has a velocity of constant magnitude $9.20 \mathrm{~m} / \mathrm{s}$ and a centripetal acceleration of magnitude $3.80 \mathrm{~m} / \mathrm{s}^{2}$. What are (a) the track radius and (b) the period of the circular motion?

Ceren Uzun

Texas Tech University

Problem 121

Suppose that a space probe can withstand the stresses of a $20 g$ acceleration. (a) What is the minimum turning radius of such a craft moving at a speed of one-tenth the speed of light? (b) How long would it take to complete a $90^{\circ}$ turn at this speed?

Averell Hause

Carnegie Mellon University

Problem 122

You are to throw a ball with a speed of $12.0 \mathrm{~m} / \mathrm{s}$ at a target that is height $h=5.00 \mathrm{~m}$ above the level at which you release the ball (Fig. 4-58). You want the ball's velocity to be horizontal at the instant it reaches the target. (a) At what angle $\theta$ above the horizontal must you throw the ball? (b) What is the horizontal distance from the release point to the target? (c) What is the speed of the ball just as it reaches the target?

Ceren Uzun

Texas Tech University

Problem 123

A projectile is fired with an initial speed $v_{0}=30.0 \mathrm{~m} / \mathrm{s}$ from level ground at a target that is on the ground, at distance $R=20.0 \mathrm{~m}$, as shown in Fig. $4-59 .$ What are the (a) least and (b) greatest launch angles that will allow the projectile to hit the target?

Averell Hause

Carnegie Mellon University

Problem 124

A graphing surprise. At time $t=0$, a burrito is launched from level ground, with an initial speed of $16.0 \mathrm{~m} / \mathrm{s}$ and launch angle $\theta_{0}$. Imagine a position vector $\vec{r}$ continuously directed from the launching point to the burrito during the flight. Graph the magnitude $r$ of the position vector for (a) $\theta_{0}=40.0^{\circ}$ and (b) $\theta_{0}=80.0^{\circ}$. For $\theta_{0}=40.0^{\circ}$, (c) when does $r$ reach its maximum value, (d) what is that value, and how far (e) horizontally and (f) vertically is the burrito from the launch point? For $\theta_{0}=80.0^{\circ},(\mathrm{g})$ when does $r$ reach its maximum value, (h) what is that value, and how far (i) horizontally and (j) vertically is the burrito from the launch point?

Paul A.

California State Polytechnic University, Pomona

Problem 125

A cannon located at sea level fires a ball with initial speed $82 \mathrm{~m} / \mathrm{s}$ and initial angle $45^{\circ} .$ The ball lands in the water after traveling a horizontal distance $686 \mathrm{~m}$. How much greater would the horizontal distance have been had the cannon been $30 \mathrm{~m}$ higher?

Averell Hause

Carnegie Mellon University

Problem 126

The magnitude of the velocity of a projectile when it is at its maximum height above ground level is $10.0 \mathrm{~m} / \mathrm{s}$. (a) What is the magnitude of the velocity of the projectile $1.00 \mathrm{~s}$ before it achieves its maximum height? (b) What is the magnitude of the velocity of the projectile $1.00 \mathrm{~s}$ after it achieves its maximum height? If we take $x=0$ and $y=0$ to be at the point of maximum height and positive $x$ to be in the direction of the velocity there, what are the (c) $x$ coordinate and (d) $y$ coordinate of the projectile $1.00 \mathrm{~s}$ before it reaches its maximum height and the (e) $x$ coordinate and (f) $y$ coordinate $1.0 \mathrm{~s}$ after it reaches its maximum height?

Animesh Raj

Numerade Educator

Problem 127

A frightened rabbit moving at $6.00 \mathrm{~m} / \mathrm{s}$ due east runs onto a large area of level ice of negligible friction. As the rabbit slides across the ice, the force of the wind causes it to have a constant acceleration of $1.40 \mathrm{~m} / \mathrm{s}^{2}$, due north. Choose a coordinate system with the origin at the rabbit's initial position on the ice and the positive $x$ axis directed toward the east. In unit-vector notation, what are the rabbit's (a) velocity and (b) position when it has slid for $3.00 \mathrm{~s}$ ?

Jose Carlos

Numerade Educator

Problem 128

The pilot of an aircraft flies due east relative to the ground in a wind blowing $20.0 \mathrm{~km} / \mathrm{h}$ toward the south. If the speed of the aircraft in the absence of wind is $70.0 \mathrm{~km} / \mathrm{h}$, what is the speed of the aircraft relative to the ground?

Ceren Uzun

Texas Tech University

Problem 129

The pitcher in a slow-pitch softball game releases the ball at a point $3.0$ ft above ground level. A stroboscopic plot of the position of the ball is shown in Fig. $4-60$, 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 130

Some state trooper departments use aircraft to enf highway speed limits. Suppose that one of the airplanes has a s of $135 \mathrm{mi} / \mathrm{h}$ in still air. It is flying straight north so that it is times directly above a north-south highway. A ground obse tells the pilot by radio that a $70.0 \mathrm{mi} / \mathrm{h}$ wind is blowing but neg to give the wind direction. The pilot observes that in spite o wind the plane can travel $135 \mathrm{mi}$ along the highway in $1.00$ other words, the ground speed is the same as if there were no
(a) From what direction is the wind blowing? (b) What is the $h$ ing of the plane; that is, in what direction does it point?

Averell Hause

Carnegie Mellon University

Problem 131

A golfer tees off from the top of a rise, giving the golf ba initial velocity of $43.0 \mathrm{~m} / \mathrm{s}$ at an angle of $30.0^{\circ}$ above the horizo The ball strikes the fairway a horizontal distance of $180 \mathrm{~m}$ fron tee. Assume the fairway is level. (a) How high is the rise above fairway? (b) What is the speed of the ball as it strikes the fairl

Jose Carlos

Numerade Educator

Problem 132

A track meet is held on a planet in a distant solar system. A shot-putter releases a shot at a point $2.0 \mathrm{~m}$ above ground level. A stroboscopic plot of the position of the shot is shown in Fig. $4-61$, where the readings are $0.50 \mathrm{~s}$ apart and the shot is released at time $t=0$. (a) What is the initial velocity of the shot in unit-vector notation? (b) What is the magnitude of the free-fall acceleration on the planet? (c) How long after it is released does the shot reach the ground? (d) If an identical throw of the shot is made on the surface of Earth, how long after it is released does it reach the ground?

Donald Albin

Numerade Educator

Problem 133

A helicopter is flying in a straight line over a level field at a constant speed of $6.20 \mathrm{~m} / \mathrm{s}$ and at a constant altitude of $9.50 \mathrm{~m}$. A package is ejected horizontally from the helicopter with an initial velocity of $12.0 \mathrm{~m} / \mathrm{s}$ relative to the helicopter and in a direction opposite the helicopter's motion. (a) Find the initial speed of the package relative to the ground. (b) What is the horizontal distance between the helicopter and the package at the instant the package strikes the ground? (c) What angle does the velocity vector of the package make with the ground at the instant before impact, as seen from the ground?

Averell Hause

Carnegie Mellon University

Problem 134

A car travels around a flat circle on the ground, at a constant speed of $12.0 \mathrm{~m} / \mathrm{s}$. At a certain instant the car has an acceleration of $3.00 \mathrm{~m} / \mathrm{s}^{2}$ toward the east. What are its distance and direction from the center of the circle at that instant if it is traveling (a) clockwise around the circle and (b) counterclockwise around the circle?

Supratim Pal

Numerade Educator

Problem 135

You throw a ball from a cliff with an initial velocity of $15.0 \mathrm{~m} / \mathrm{s}$ at an angle of $20.0^{\circ}$ below the horizontal. Find (a) its horizontal displacement and (b) its vertical displacement $2.30 \mathrm{~s}$ later.

Averell Hause

Carnegie Mellon University

Problem 136

A baseball is hit at Fenway Park in Boston at a point $0.762 \mathrm{~m}$ above home plate with an initial velocity of $33.53 \mathrm{~m} / \mathrm{s}$ directed $55.0^{\circ}$ above the horizontal. The ball is observed to clear the $11.28$ -m-high wall in left field (known as the "green monster") $5.00 \mathrm{~s}$ after it is hit, at a point just inside the left-field foulline pole. Find (a) the horizontal distance down the left-field foul line from home plate to the wall; (b) the vertical distance by which the ball clears the wall; (c) the horizontal and vertical displacements of the ball with respect to home plate $0.500 \mathrm{~s}$ before it clears the wall.

Averell Hause

Carnegie Mellon University

Problem 138

A woman can row a boat at $6.40 \mathrm{~km} / \mathrm{h}$ in still water. (a) If she is crossing a river where the current is $3.20 \mathrm{~km} / \mathrm{h}$, in what direction must her boat be headed if she wants to reach a point directly opposite her starting point? (b) If the river is $6.40 \mathrm{~km}$ wide, how long will she take to cross the river? (c) Suppose that instead of crossing the river she rows $3.20 \mathrm{~km}$ down the river and then back to her starting point. How long will she take? (d) How long will she take to row $3.20 \mathrm{~km} u p$ the river and then back to her starting point? (e) In what direction should she head the boat if she wants to cross in the shortest possible time, and what is that time?

Animesh Raj

Numerade Educator