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Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 4

Motion in Two and Three Dimensions - all with Video Answers

Educators

Chapter Questions

02:18

Problem 1

The position vector for an electron is $\vec{r}=(6.0 \mathrm{~m}) \hat{\mathrm{i}}-$ $(4.0 \mathrm{~m}) \hat{\mathrm{j}}+(3.0 \mathrm{~m}) \hat{\mathrm{k}}$. (a) Find the magnitude of $\vec{r}$. (b) Sketch the vector on a right-handed coordinate system.

Supratim Pal
Supratim Pal
Numerade Educator
04:30

Problem 2

A watermelon seed has the following coordinates: $x=-5.0 \mathrm{~m}$, $y=9.0 \mathrm{~m}$, 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 $(\mathrm{g})$ an angle relative to the positive $x$ direction?

Keshav Singh
Keshav Singh
Numerade Educator
02:01

Problem 3

An elementary particle is subjected to a displacement of $\Delta \vec{r}=2.0 \hat{\mathrm{i}}-4.0 \hat{\mathrm{j}}+8.0 \hat{\mathrm{k}}$, ending with the position vector $\vec{r}=4.0 \hat{\mathrm{j}}-5.0 \hat{\mathrm{k}}$, in meters. What was the particle's initial position vector?

Paul Gabriel
Paul Gabriel
Numerade Educator
07:47

Problem 4

The minute hand of a wall clock measures $12 \mathrm{~cm}$ 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
09:09

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
Jose Carlos
Numerade Educator
02:36

Problem 6

An electron's position is given by $\vec{r}=3.00 t \hat{\mathrm{i}}-4.00 t^{2} \hat{\mathrm{j}}+2.00 \hat{\mathrm{k}}$, with $t$ in seconds and $\vec{r}$ in meters. (a) In unit-vector notation, what is the electron's velocity $\vec{v}(t) ?$ At $t=3.00 \mathrm{~s}$, what is $\vec{v}$ (b) in unitvector notation and as (c) a magnitude and (d) an angle relative to the positive direction of the $x$ axis?

Keshav Singh
Keshav Singh
Numerade Educator
01:18

Problem 7

In a particle accelerator, the position vector of a particle is initially estimated as $\vec{r}=6.0 \hat{\mathrm{i}}-7.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}}$ and after $10 \mathrm{~s}$, it is estimated to be $\vec{r}=-3.0 \hat{\mathrm{i}}+9.0 \hat{\mathrm{j}}-3.0 \hat{\mathrm{k}}$, all in meters. In unit vector notation, what is the average velocity of the particle?

Suzanne W.
Suzanne W.
Numerade Educator
05:30

Problem 8

A plane flies $483 \mathrm{~km}$ east from city $A$ to city $B$ in $48.0 \mathrm{~min}$ 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
Ceren Uzun
Texas Tech University
05:12

Problem 9

Figure 4-21 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 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
Rashmi Sinha
Numerade Educator
05:32

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 4-22 gives the angle $\theta$ of the particle's direction of travel as a function of $t$ ( $\theta$ is measured from the positive $x$ direction). What are (a) $e$ and (b) $f$, including units?

Ceren Uzun
Ceren Uzun
Texas Tech University
04:28

Problem 11

A particle that is moving in an $x y$ plane has a position vector given by $\vec{r}=\left(3.00 t^{3}-6.00 t\right) \hat{i}+\left(7.00-8.00 t^{4}\right) \hat{j}$, where $\vec{r}$ is measured in meters and $t$ is measured in seconds. For $t=3.00 \mathrm{~s}$, in unitvector notation, find (a) $\vec{r}$, (b) $\vec{v}$, and (c) $\vec{a}$. (d) Find the angle between the positive direction of the $x$ axis and a line that is tangent to the path of the particle at $t=3.00 \mathrm{~s}$.

Averell Hause
Averell Hause
Carnegie Mellon University
08:28

Problem 12

At one instant a bicyclist is $30.0 \mathrm{~m}$ 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 $30.0 \mathrm{~s}$ 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
01:25

Problem 13

An object moves in such a way that its position (in meters) as a function of time (in seconds) is $\vec{r}=\hat{\mathrm{i}}+3 t^{2} \hat{\mathrm{j}}+t \hat{\mathrm{k}}$. Give expressions for (a) the velocity of the object and (b) the acceleration of the object as functions of time.

Jose Carlos
Jose Carlos
Numerade Educator
05:14

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

Jose Carlos
Jose Carlos
Numerade Educator
04:40

Problem 15

From the origin, a particle starts at $t=0 \mathrm{~s}$ with a velocity $\vec{v}=7.0 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s}$ and moves in the $x y$ plane with a constant acceleration of $\vec{a}=(-9.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}^{2} .$ At the time the particle reaches the maximum $x$ coordinate, what is its (a) velocity and (b) position vector?

Jose Carlos
Jose Carlos
Numerade Educator
05:04

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 $t=2.5 \mathrm{~s} ?$ (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
Jose Carlos
Numerade Educator
05:15

Problem 17

A motorbike starts from the origin and moves over an $x y$ plane with acceleration components $a_{x}=6.0 \mathrm{~m} / \mathrm{s}^{2}$ and $a_{y}=-3.0 \mathrm{~m} / \mathrm{s}^{2}$. The initial velocity of the motorbike has components $v_{0 x}=12.0 \mathrm{~m} / \mathrm{s}$ and $v_{0 y}=18.0 \mathrm{~m} / \mathrm{s}$. Find the velocity of the motorbike, in unit-vector notation, when it reaches its greatest $y$ coordinate.

Jacob Adamczyk
Jacob Adamczyk
Numerade Educator
04:49

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 $(4.00 \mathrm{~m} / \mathrm{s}) \hat{i}$. What are the (a) magnitude and (b) angle of its velocity when it has been displaced by $10.0$ $\mathrm{m}$ parallel to the $x$ axis?

Jose Carlos
Jose Carlos
Numerade Educator
05:13

Problem 19

The acceleration of a particle moving only on a horizontal $x y$ plane is given by $\vec{a}=3 t \hat{\mathrm{i}}+4 t \hat{\mathrm{j}}$, 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
Jose Carlos
Numerade Educator
04:33

Problem 20

In Fig. 4-23, 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 $A$ 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
Averell Hause
Carnegie Mellon University
04:07

Problem 21

A stone is thrown by aiming directly at the center $P$ of a picture hanging on a wall. The stone leaves from the starting point horizontally with a speed of $6.75 \mathrm{~m} / \mathrm{s}$ and strikes the target at point $Q$, which is $5.00 \mathrm{~cm}$ below $P$. Find the horizontal distance between the starting point of the stone and the target.

Donald Albin
Donald Albin
Numerade Educator
03:21

Problem 22

A small ball rolls horizontally off the edge of a tabletop that is $1.50 \mathrm{~m}$ 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 sneed at the instant it leaves the table?

Nishant Kumar
Nishant Kumar
Numerade Educator
01:40

Problem 23

A shell, which is initially located at a distance of $40.4 \mathrm{~m}$ above a horizontal plane, is fired horizontally with a muzzle velocity of $285 \mathrm{~m} / \mathrm{s}$ to strike a target on the horizontal plane. (a) How long does the projectile remain in the air? (b) At what horizontal distance from the firing point does the shell strike the plane? What are the magnitudes of the (c) horizontal and (d) vertical components of its velocity as it strikes the ground?

Averell Hause
Averell Hause
Carnegie Mellon University
01:55

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?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:34

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.

Nishant Kumar
Nishant Kumar
Numerade Educator
07:22

Problem 26

A stone is catapulted at time $t=0$, with an initial velocity of magnitude $18.0 \mathrm{~m} / \mathrm{s}$ 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 $t=1.10$ 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
Ceren Uzun
Texas Tech University
02:52

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. 4-24). 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
Jose Carlos
Numerade Educator
04:54

Problem 28

In Fig. 4-25, 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
Jose Carlos
Numerade Educator
01:29

Problem 29

A projectile's launch speed is $6.00$ times that of its speed at its maximum height. Find the launch angle $\theta_{0}$.

Averell Hause
Averell Hause
Carnegie Mellon University
03:14

Problem 30

A soccer ball is kicked from the ground with an initial speed of $21.3 \mathrm{~m} / \mathrm{s}$ 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?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:01

Problem 31

A soccer player claims that he can kick the ball over a wall of height $3.5 \mathrm{~m}$, which is $32 \mathrm{~m}$ away along a horizontal field. The player punts the ball from an elevation of $1.0 \mathrm{~m}$ and the ball is projected at an initial speed of $18 \mathrm{~m} / \mathrm{s}$ in the direction $40^{\circ}$ from the horizontal. Does the ball clear the wall?

Raj Bala
Raj Bala
Numerade Educator
05:11

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. 426). 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
Ceren Uzun
Texas Tech University
06:25

Problem 33

A defense air force plane, diving with constant speed at an angle of $52.0^{\circ}$ with the vertical, drops a shell at an altitude of $720 \mathrm{~m}$. The shell reaches the ground $6.00 \mathrm{~s}$ after its release. (a) What is the speed of the plane? (b) How far does the shell travel horizontally during its flight? What are the (c) horizontal and (d) vertical components of its velocity just before reaching the ground? Assume an $x$ axis in the direction of the horizontal motion and an upward $y$ axis.

Jose Carlos
Jose Carlos
Numerade Educator
03:21

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 wall because then arrows could reach it from the castle wall. Instead, it was positioned so that the stone hit the wall during the second half of its flight. Suppose a stone is launched with a speed of $v_{0}=30.0$ $\mathrm{m} / \mathrm{s}$ and at an angle of $\theta_{0}=40.0^{\circ}$. What is the speed of the stone if it hits the wall (a) just as it reaches the top of its parabolic path and (b) when it has descended to half that height? (c) As a percentage, how much faster is it moving in part (b) than in part (a)?

Averell Hause
Averell Hause
Carnegie Mellon University
02:33

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
Jose Carlos
Numerade Educator
06:05

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 $2.42 \mathrm{~m}$ 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
Ceren Uzun
Texas Tech University
02:53

Problem 37

From the platform edge located at $12.0 \mathrm{~m}$ above the surface of the water, a high dive champion pushes off horizontally with a speed of $2.50 \mathrm{~m} / \mathrm{s}$. (a) At what horizontal distance from the edge is the diver $0.900 \mathrm{~s}$ 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
Averell Hause
Carnegie Mellon University
05:23

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. 4-27, 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?

Ethan Deweese
Ethan Deweese
Numerade Educator
05:25

Problem 39

In Fig. 4-28, 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
Averell Hause
Carnegie Mellon University
06:36

Problem 40

Suppose that a shot putter can put a shot at the world-class 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.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:41

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. 4-29). Although the insect is located along a straightline 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?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
05:36

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. 4-30). 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
Averell Hause
Carnegie Mellon University
05:15

Problem 43

A golfer hits a golf ball into the air over level ground. The velocity of the ball at a height of $10.3 \mathrm{~m}$ is $\vec{v}=(8.6 \hat{i}+7.2 \hat{j}) \mathrm{m} / \mathrm{s}$, with $\hat{\mathrm{i}}$ horizontal and $\hat{\mathrm{j}}$ upward. Find (a) the maximum height of the ball and (b) the total horizontal distance traveled by the ball. What are the (c) magnitude and (d) angle (below the horizontal) of the ball's velocity just before it touches the ground?

Averell Hause
Averell Hause
Carnegie Mellon University
10:00

Problem 44

A baseball leaves a pitcher's hand horizontally at a speed of $153 \mathrm{~km} / \mathrm{h}$. 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 $(\mathrm{c})$ and $(\mathrm{d})$ equal?

Ceren Uzun
Ceren Uzun
Texas Tech University
02:51

Problem 45

In Fig. 4-31, 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,

Averell Hause
Averell Hause
Carnegie Mellon University
22:18

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 $v_{0}=6.00 \mathrm{~m} / \mathrm{s}$ at an angle of $\theta_{0}=35.0^{\circ}$, what percentage 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
Donald Albin
Numerade Educator
05:23

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 $7.32-\mathrm{m}$-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
Averell Hause
Carnegie Mellon University
08:13

Problem 48

In Fig. 4-32, a ball is thrown up onto a roof, landing $4.50 \mathrm{~s}$ 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
Mukesh Devi
Numerade Educator
05:36

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
Averell Hause
Carnegie Mellon University
03:37

Problem 50

Two seconds after being projected from ground level, a projectile is displaced $40 \mathrm{~m}$ horizontally and $58 \mathrm{~m}$ 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
Ceren Uzun
Texas Tech University
08:48

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 4-33a shows a prejump that allows the skier to land on the top portion of the steeper track. Figure $4-33 b$ shows a jump at the edge of the steeper track. In Fig. 4-33a, 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. 4-33b, (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.)

Averell Hause
Averell Hause
Carnegie Mellon University
03:11

Problem 52

A ball is to be shot from level ground toward a wall at distance $x$ (Fig. 4-34a). Figure 4-34b 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
Ceren Uzun
Texas Tech University
06:13

Problem 53

In Fig. 4-35, 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
Averell Hause
Carnegie Mellon University
06:26

Problem 54

A ball is to be shot from level ground with a certain speed. Figure 4-36 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 $0.500 t_{\max } ?$

Ceren Uzun
Ceren Uzun
Texas Tech University
05:02

Problem 55

A stairway has steps $18.3 \mathrm{~cm}$ high and $18.3 \mathrm{~cm}$ wide. A ball rolls horizontally off the top of the stairway with a speed of $1.00 \mathrm{~m} / \mathrm{s}$. Which step does the ball hit first?

Averell Hause
Averell Hause
Carnegie Mellon University
03:57

Problem 56

An Earth satellite moves in a circular orbit $750 \mathrm{~km}$ above Earth's surface with a period of $98.0 \mathrm{~min}$. What are the (a) speed and (b) magnitude of the centripetal acceleration of the satellite?

Donald Albin
Donald Albin
Numerade Educator
02:49

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
Jose Carlos
Numerade Educator
02:42

Problem 58

A rotating fan completes 1100 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
Ceren Uzun
Texas Tech University
00:55

Problem 59

Music is still available on vinyl records that are played on turntables. Such a record rotates with a period of $1.8 \mathrm{~s}$. For a record with a radius of $16 \mathrm{~cm}$, find the centripetal accceleration of a point on the edge of the record.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
04:32

Problem 60

A centripetal-acceleration addict rides in uniform circular motion with period $T=2.0 \mathrm{~s}$ and radius $r=3.50 \mathrm{~m}$. At $t_{1}$ 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 values of (a) $\vec{v} \cdot \vec{a}$ and (b) $\vec{r} \times \vec{a}$ ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:37

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
Averell Hause
Carnegie Mellon University
01:17

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 $20 \mathrm{~m}$ ?

Jose Carlos
Jose Carlos
Numerade Educator
15:58

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 $(4.00$ $\left.\mathrm{m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(-6.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$. What is the radius of the path taken by the particle if $t_{2}-t_{1}$ is less than one period?

Donald Albin
Donald Albin
Numerade Educator
02:30

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
Jose Carlos
Numerade Educator
02:07

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
Averell Hause
Carnegie Mellon University
04:44

Problem 66

A particle moves along a circular path over a horizontal $x y$ coordinate system, at constant speed. At time $t_{1}=5.00 \mathrm{~s}$, 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
Ceren Uzun
Texas Tech University
02:16

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?

Sanu Kumar
Sanu Kumar
Numerade Educator
02:13

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?

Narayan Hari
Narayan Hari
Numerade Educator
05:28

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
Averell Hause
Carnegie Mellon University
03:57

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 $8.2 \mathrm{~km} / \mathrm{h}$ 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
Ceren Uzun
Texas Tech University
02:49

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
Jose Carlos
Numerade Educator
01:46

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 $3.5 \mathrm{~m} / \mathrm{s}$ 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
Averell Hause
Carnegie Mellon University
02:42

Problem 73

Two highways intersect as shown in Fig. 4-37. 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 $v_{M}=60 \mathrm{~km} / \mathrm{h}$.
(a) In unit-vector notation, what is the velocity of the motorist with respect to the police car? (b) For the instant shown in Fig. 4-37, what is the angle between the velocity found in (a) and the line of sight between the two cars? (c) If the cars maintain their velocities, do the answers to (a) and (b) change as the cars move nearer the intersection?

Keshav Singh
Keshav Singh
Numerade Educator
04:47

Problem 74

After flying for $18 \mathrm{~min}$ 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
Ceren Uzun
Texas Tech University
01:24

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
Averell Hause
Carnegie Mellon University
06:01

Problem 76

A light plane attains an airspeed of $500 \mathrm{~km} / \mathrm{h}$. The pilot sets out for a destination $900 \mathrm{~km}$ 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
Ceren Uzun
Texas Tech University
01:15

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
Averell Hause
Carnegie Mellon University
06:06

Problem 78

In the overhead view of Fig. 4-38, 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 $25.0 \mathrm{~m} / \mathrm{s}$, 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
Averell Hause
Carnegie Mellon University
07:23

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 $(1$ 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
Averell Hause
Carnegie Mellon University
05:20

Problem 80

A $200 \mathrm{~m}$ wide river flows due east at a uniform speed of $2.5$ $\mathrm{m} / \mathrm{s}$. 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
Ceren Uzun
Texas Tech University
04:43

Problem 81

Ship $A$ 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 $\operatorname{ship} B$ 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 $\hat{i}$ 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
Averell Hause
Carnegie Mellon University
02:39

Problem 82

A $200 \mathrm{~m}$ 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 leave a small clearing on the south bank and cross the river in a powerboat that moves at a constant speed of $5.0 \mathrm{~m} / \mathrm{s}$ with respect to the water. There is a clearing on the north bank $82 \mathrm{~m}$ upstream from a point directly opposite the clearing on the south bank. (a) In what direction must the boat be pointed in order to travel in a straight line and land in the clearing on the north bank? (b) How long will the boat take to cross the river and land in the clearing?

Keshav Singh
Keshav Singh
Numerade Educator