University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 3

Motion in Two and Three Dimensions - all with Video Answers

Educators

+ 2 more educators

Chapter Questions

Problem 1

An arrow is shot horizontally with a speed of $20 \mathrm{~m} / \mathrm{s}$ from the top of a tower $60 \mathrm{~m}$ high. The time to reach the ground will be
a) $8.9 \mathrm{~s}$
c) $3.5 \mathrm{~s}$
e) $1.0 \mathrm{~s}$
b) $7.1 \mathrm{~s}$
d) $2.6 \mathrm{~s}$

Donald Albin

Donald Albin

Numerade Educator

Problem 2

A projectile is launched from the top of a building with an initial velocity of $30 \mathrm{~m} / \mathrm{s}$ at an angle of $60^{\circ}$ above the horizontal. The magnitude of its velocity at $t=5 \mathrm{~s}$ after the launch is
a) $-23.0 \mathrm{~m} / \mathrm{s}$
c) $15.0 \mathrm{~m} / \mathrm{s}$
e) $50.4 \mathrm{~m} / \mathrm{s}$
b) $7.3 \mathrm{~m} / \mathrm{s}$
d) $27.5 \mathrm{~m} / \mathrm{s}$

Donald Albin

Numerade Educator

Problem 3

A ball is thrown at an angle between $0^{\circ}$ and $90^{\circ}$ with respect to the horizontal. Its velocity and acceleration vectors are parallel to each other at
a) $0^{\circ}$
c) $60^{\circ}$
e) none of the
b) $45^{\circ}$
d) $90^{\circ}$ above

Donald Albin

Numerade Educator

Problem 4

An outfielder throws the baseball to first base, located $80 \mathrm{~m}$ away from the fielder, with a velocity of $45 \mathrm{~m} / \mathrm{s}$. At what launch angle above the horizontal should he throw the ball for the first baseman to catch the ball in $2 \mathrm{~s}$ at the same height?
a) $50.74^{\circ}$
c) $22.7^{\circ}$
e) $12.6^{\circ}$
b) $25.4^{\circ}$
d) $18.5^{\circ}$

Donald Albin

Numerade Educator

Problem 5

A $50-\mathrm{g}$ ball rolls off a countertop and lands $2 \mathrm{~m}$ from the base of the counter. A $100-\mathrm{g}$ ball rolls off the same counter top with the same speed. It lands from the base of the counter.
a) less than $1 \mathrm{~m}$
c) $2 \mathrm{~m}$
e) more than $4 \mathrm{~m}$
b) $1 \mathrm{~m}$
d) $4 \mathrm{~m}$

Donald Albin

Numerade Educator

Problem 6

For a given initial speed of an ideal projectile, there is (are) $\quad$ launch angle(s) for which the range of the projectile is the same.
a) only one
b) two different
c) more than two but a finite number of
d) only one if the angle is $45^{\circ}$ but otherwise two different
e) an infinite number of

Tyler Moulton

Numerade Educator

Problem 7

A cruise ship moves southward in still water at a speed of $20.0 \mathrm{~km} / \mathrm{h},$ while a passenger on the deck of the ship walks toward the east at a speed of $5.0 \mathrm{~km} / \mathrm{h}$. The passenger's velocity with respect to Earth is
a) $20.6 \mathrm{~km} / \mathrm{h}$, at an angle of $14.04^{\circ}$ east of south.
b) $20.6 \mathrm{~km} / \mathrm{h}$, at an angle of $14.04^{\circ}$ south of east.
c) $25.0 \mathrm{~km} / \mathrm{h}$, south.
d) $25.0 \mathrm{~km} / \mathrm{h}$, east.
e) $20.6 \mathrm{~km} / \mathrm{h}$, south.

Tyler Moulton

Numerade Educator

Problem 8

Two cannonballs are shot from different cannons at angles $\theta_{01}=20^{\circ}$ and $\theta_{02}=30^{\circ}$, respectively. Assuming ideal projectile motion, the ratio of the launching speeds, $v_{01} / v_{02},$ for which the two cannonballs achieve the same range is
a) $0.742 \mathrm{~m}$
d) $1.093 \mathrm{~m}$
b) $0.862 \mathrm{~m}$
e) $2.222 \mathrm{~m}$
c) $1.212 \mathrm{~m}$

Donald Albin

Numerade Educator

Problem 9

The acceleration due to gravity on the Moon is $1.62 \mathrm{~m} / \mathrm{s}^{2},$ approximately a sixth of the value on Earth. For a given initial velocity $v_{0}$ and a given launch angle $\theta_{0}$ the ratio of the range of an ideal projectile on the Moon to the range of the same projectile on Earth, $R_{\text {Moon }} / R_{\text {Farth }}$ will be
a) $6 \mathrm{~m}$
d) $5 \mathrm{~m}$
b) $3 \mathrm{~m}$
e) $1 \mathrm{~m}$
c) $12 \mathrm{~m}$

Donald Albin

Numerade Educator

Problem 10

A baseball is launched from the bat at an angle $\theta_{0}=30^{\circ}$ with respect to the positive $x$ -axis and with an initial speed of $40 \mathrm{~m} / \mathrm{s}$, and it is caught at the same height from which it was hit. Assuming ideal projectile motion (positive $y$ -axis upward), the velocity of the ball when it is caught is
a) $(20.00 \hat{x}+34.64 \hat{y}) \mathrm{m} / \mathrm{s}$.
b) $(-20.00 \hat{x}+34.64 \hat{y}) \mathrm{m} / \mathrm{s}$
c) $(34.64 \hat{x}-20.00 \hat{y}) \mathrm{m} / \mathrm{s}$
d) $(34.64 \hat{x}+20.00 \hat{y}) \mathrm{m} / \mathrm{s}$.

Tyler Moulton

Numerade Educator

Problem 11

In ideal projectile motion, the velocity and acceleration of the projectile at its maximum height are, respectively,
a) horizontal, vertical
c) zero, zero. downward.
d) zero, vertical downward.
b) horizontal, zero.
e) zero, horizontal.

Tyler Moulton

Numerade Educator

Problem 12

In ideal projectile motion, when the positive $y$ -axis is chosen to be vertically upward, the $y$ -component of the acceleration of the object during the ascending part of the motion and the $y$ -component of the acceleration during the descending part of the motion are, respectively,
a) positive, negative.
c) positive, positive.
b) negative, positive.
d) negative, negative.

Tyler Moulton

Numerade Educator

Problem 13

In ideal projectile motion, when the positive $y$ -axis is chosen to be vertically upward, the $y$ -component of the velocity of the object during the ascending part of the motion and the $y$ -component of the velocity during the descending part of the motion are, respectively,
a) positive, negative.
c) positive, positive.
b) negative, positive.
d) negative, negative.

Tyler Moulton

Numerade Educator

Problem 14

A ball is thrown from ground at an angle between $0^{\circ}$ and $90^{\circ} .$ Which of the following remain constant: $x, y, v_{x}, v_{p}$ $a_{x}, a_{y} ?$

Donald Albin

Numerade Educator

Problem 15

A ball is thrown straight up by a passenger in a train that is moving with a constant velocity. Where would the ball land-back in his hands, in front of him, or behind him? Does your answer change if the train is accelerating in the forward direction? If yes, how?

Tyler Moulton

Numerade Educator

Problem 16

A rock is thrown at an angle $45^{\circ}$ below the horizontal from the top of a building. Immediately after release will its acceleration be greater than, equal to, or less than the acceleration due to gravity?

Tyler Moulton

Numerade Educator

Problem 17

Three balls of different masses are thrown horizontally from the same height with different initial speeds, as shown in the figure. Rank in order, from the shortest to the longest, the times the balls take to hit the ground.

Ronald Prasad

Numerade Educator

Problem 18

To attain maximum height for the trajectory of a projectile, what angle would you choose between $0^{\circ}$ and $90^{\circ}$, assuming that you can launch the projectile with the same initial speed independent of the launch angle. Explain your reasoning.

Tyler Moulton

Numerade Educator

Problem 19

An airplane is traveling at a constant horizontal speed $v$, at an altitude $h$ above a lake when a trapdoor at the bottom of the airplane opens and a package is released (falls) from the plane. The airplane continues horizontally at the same altitude and velocity. Neglect air resistance.
a) What is the distance between the package and the plane when the package hits the surface of the lake?
b) What is the horizontal component of the velocity vector of the package when it hits the lake?
c) What is the speed of the package when it hits the lake?

Tyler Moulton

Numerade Educator

Problem 20

Two cannonballs are shot in sequence from a cannon, into the air, with the same muzzle velocity, at the same launch angle. Based on their trajectory and range, how can you tell which one is made of lead and which one is made of wood. If the same cannonballs where launched in vacuum, what would the answer be?

Tyler Moulton

Numerade Educator

Problem 21

One should never jump off a moving vehicle (train, car, bus, etc.). Assuming, however, that one does perform such a jump, from a physics standpoint, what would be the best direction to jump in order to minimize the impact of the landing? Explain.

Tyler Moulton

Numerade Educator

Problem 22

A boat travels at a speed of $v_{\mathrm{BW}}$ relative to the water in a river of width $D .$ The speed at which the water is flowing is $v_{\mathrm{W}}$
a) Prove that the time required to cross the river to a point
exactly opposite the starting point and then to return is $T_{1}=2 D / \sqrt{v_{B W}^{2}-v_{W}^{2}}$
b) Also prove that the time for the boat to travel a distance $D$ downstream and then return is $T_{1}=2 D v_{\mathrm{B}} /\left(v_{\mathrm{BW}}^{2}-v_{\mathrm{w}}^{2}\right)$

Donald Albin

Numerade Educator

Problem 23

A rocket-powered hockey puck is moving on a (frictionless) horizontal air-hockey table. The $x$ - and $y$ -components of its velocity as a function of time are presented in the graphs below. Assuming that at $t=0$ the puck is at $\left(x_{0}, y_{0}\right)=(1,2)$ draw a detailed graph of the trajectory $y(x)$.

Averell Hause

Carnegie Mellon University

Problem 24

In a three-dimensional motion, the $x-, y-$, and $z$ coordinates of the object as a function of time are given by $x(t)=\frac{\sqrt{2}}{2} t, \quad y(t)=\frac{\sqrt{2}}{2} t,$ and $z(t)=-4.9 t^{2}+\sqrt{3} t$
Describe the motion and the trajectory of the object in an $x y z$ coordinate system.

Massimo Antonelli

Massimo Antonelli

Numerade Educator

Problem 25

An object moves in the $x y$ -plane. The $x$ - and $y$ -coordinates of the object as a function of time are given by the following equations: $x(t)=4.9 t^{2}+2 t+1$ and $y(t)=3 t+2 .$ What is the velocity vector of the object as a function of time? What is its acceleration vector at a time $t=2$ s?

Tyler Moulton

Numerade Educator

Problem 26

A particle's motion is described by the following two parametric equations:
$$
\begin{array}{l}
x(t)=5 \cos (2 \pi t) \\
y(t)=5 \sin (2 \pi t)
\end{array}
$$
where the displacements are in meters and $t$ is the time, in seconds.
a) Draw a graph of the particle's trajectory (that is, a graph of $y$ versus $x$ ).
b) Determine the equations that describe the $x$ - and $y$ -components of the velocity, $v_{x}$ and $v_{y}$, as functions of time.
c) Draw a graph of the particle's speed as a function of time.

Tyler Moulton

Numerade Educator

Problem 27

In a proof-of-concept experiment for an antiballistic missile defense system, a missile is fired from the ground of a shooting range toward a stationary target on the ground. The system detects the missile by radar, analyzes in real time its parabolic motion, and determines that it was fired from a distance $x_{0}=5000 \mathrm{~m}$, with an initial speed of $600 \mathrm{~m} / \mathrm{s}$ at a launch angle $\theta_{0}=20^{\circ} .$ The defense system then calculates the required time delay measured from the launch of the missile and fires a small rocket situated at $y_{0}=500 \mathrm{~m}$ with an initial velocity of $v_{0} \mathrm{~m} / \mathrm{s}$ at a launch angle $\alpha_{0}=60^{\circ}$ in the $y z$ -plane, to intercept the missile. Determine the initial speed $v_{0}$ of the intercept rocket and the required time delay.

Donald Albin

Numerade Educator

Problem 28

A projectile is launched at an angle of $45^{\circ}$ above the horizontal. What is the ratio of its horizontal range to its maximum height? How does the answer change if the initial speed of the projectile is doubled?

Tyler Moulton

Numerade Educator

Problem 29

In a projectile motion, the horizontal range and the maximum height attained by the projectile are equal.
a) What is the launch angle?
b) If everything else stays the same, how should the launch angle, $\theta_{0},$ of a projectile be changed for the range of the projectile to be halved?

Tyler Moulton

Numerade Educator

Problem 30

An air-hockey puck has a model rocket rigidly attached to it. The puck is pushed from one corner along the long side of the $2.00-\mathrm{m}$ long air-hockey table, with the rocket pointing along the short side of the table, and at the same time the rocket is fired. If the rocket thrust imparts an acceleration of $2.00 \mathrm{~m} / \mathrm{s}^{2}$ to the puck, and the table is $1.00 \mathrm{~m}$ wide, with what minimum initial velocity should the puck be pushed to make it to the opposite short side of the table without bouncing off either long side of the table? Draw the trajectory of the puck for three initial velocities: $v<v_{\min }, v=v_{\min },$ and $v>v_{\min } .$ Neglect friction and air resistance.

Averell Hause

Carnegie Mellon University

Problem 31

On a battlefield, a cannon fires a cannonball up a slope, from ground level, with an initial velocity $v_{0}$ at an angle $\theta_{0}$ above the horizontal. The ground itself makes an angle $\alpha$ above the horizontal $\left(\alpha<\theta_{0}\right) .$ What is the range $R$ of the cannonball, measured along the inclined ground? Compare your result with the equation for the range on horizontal ground (equation 3.25 ).

Donald Albin

Numerade Educator

Problem 32

Two swimmers with a soft spot for physics engage in a peculiar race that models a famous optics experiment:
the Michelson-Morley experiment. The race takes place in a river $50.0 \mathrm{~m}$ wide that is flowing at a steady rate of $3.00 \mathrm{~m} / \mathrm{s} .$ Both swimmers start at the same point on one bank and swim at the same speed of $5.00 \mathrm{~m} / \mathrm{s}$ with respect to the stream. One of the swimmers swims directly across the river to the closest point on the opposite bank and then turns around and swims back to the starting point. The other swimmer swims along the river bank, first upstream a distance exactly equal to the width of the river and then downstream back to the starting point. Who gets back to the starting point first?

Donald Albin

Numerade Educator

Problem 33

What is the magnitude of an object's average velocity if an object moves from a point with coordinates $x=2.0 \mathrm{~m}$ $y=-3.0 \mathrm{~m}$ to a point with coordinates $x=5.0 \mathrm{~m}, y=-9.0 \mathrm{~m}$
in a time interval of $2.4 \mathrm{~s} ?$

Tyler Moulton

Numerade Educator

Problem 34

A man in search of his dog drives first 10 mi northeast, then $12 \mathrm{mi}$ straight south, and finally $8 \mathrm{mi}$ in a direction $30^{\circ}$ north of west. What are the magnitude and direction of his resultant displacement?

Tyler Moulton

Numerade Educator

Problem 35

During a jaunt on your sailboat, you sail $2.00 \mathrm{~km}$ east, $4.00 \mathrm{~km}$ southeast, and an additional distance in an unknown direction. Your final position is $6.00 \mathrm{~km}$ directly east of the starting point. Find the magnitude and direction of the third leg of your journey.

Donald Albin

Numerade Educator

Problem 36

A truck travels $3.02 \mathrm{~km}$ north and then makes a $90^{\circ}$ left turn and drives another $4.30 \mathrm{~km}$. The whole trip takes $5.00 \mathrm{~min}$
a) With respect to a two-dimensional coordinate system on the surface of Earth such that the $y$ -axis points north, what is the net displacement vector of the truck for this trip?
b) What is the magnitude of the average velocity for this trip?

Tyler Moulton

Numerade Educator

Problem 37

A rabbit runs in a garden such that the $x$ - and $y$ components of its displacement as function of times are given by $x(t)=-0.45 t^{2}-6.5 t+25$ and $y(t)=0.35 t^{2}+8.3 t+34 .$ (Both $x$ and $y$ are in meters and $t$ is in seconds.)
a) Calculate the rabbit's position (magnitude and direction) at $t=10 \mathrm{~s}$
b) Calculate the rabbit's velocity at $t=10 \mathrm{~s}$.
c) Determine the acceleration vector at $t=10 \mathrm{~s}$.

Donald Albin

Numerade Educator

Problem 38

Some rental cars have a GPS unit installed, which allows the rental car company to check where you are at all times and thus also know your speed at any time. One of these rental cars is driven by an employee in the company's lot and, during the time interval from 0 to $10 \mathrm{~s}$, is found to have a position vector as a function of time of
$$
\begin{aligned}
\vec{r}(t)=&\left((24.4 \mathrm{~m})-t(12.3 \mathrm{~m} / \mathrm{s})+t^{2}\left(2.43 \mathrm{~m} / \mathrm{s}^{2}\right)\right.\\
&\left.(74.4 \mathrm{~m})+t^{2}\left(1.80 \mathrm{~m} / \mathrm{s}^{2}\right)-t^{3}\left(0.130 \mathrm{~m} / \mathrm{s}^{3}\right)\right)
\end{aligned}
$$
a) What is the distance of this car from the origin of the coordinate system at $t=5.00 \mathrm{~s} ?$
b) What is the velocity vector as a function of time?
c) What is the speed at $t=5.00 \mathrm{~s} ?$
Extra credit: Can you produce a plot of the trajectory of the car in the $x y$ -plane?

Donald Albin

Numerade Educator

Problem 39

A skier launches off a ski jump with a horizontal velocity of $30.0 \mathrm{~m} / \mathrm{s}$ (and no vertical velocity component). What are the magnitudes of the horizontal and vertical components of her velocity the instant before she lands 2.00 s later?

Tyler Moulton

Numerade Educator

Problem 40

An archer shoots an arrow from a height of $1.14 \mathrm{~m}$ above ground with an initial velocity of $47.5 \mathrm{~m} / \mathrm{s}$ and an initial angle of $35.2^{\circ}$ above the horizontal. At what time after the release of the arrow from the bow will the arrow be
flying exactly horizontally?

Donald Albin

Numerade Educator

Problem 41

A football is punted with an initial velocity of $27.5 \mathrm{~m} / \mathrm{s}$ and an initial angle of $56.7^{\circ} .$ What is its hang time (the time until it hits the ground again)?

Donald Albin

Numerade Educator

Problem 42

You serve a tennis ball from a height of $1.8 \mathrm{~m}$ above the ground. The ball leaves your racket with a speed of $18.0 \mathrm{~m} / \mathrm{s}$ at an angle of $7.00^{\circ}$ above the horizontal. The horizontal distance from the court's baseline to the net is $11.83 \mathrm{~m},$ and the net is $1.07 \mathrm{~m}$ high. Neglect spin imparted on the ball as well as air resistance effects. Does the ball clear the net? If yes, by how much? If not, by how much did it miss?

Donald Albin

Numerade Educator

Problem 43

Stones are thrown horizontally with the same velocity from two buildings, One stone lands twice as far away from its building as the other stone. Determine the ratio of the heights of the two buildings.

Tyler Moulton

Numerade Educator

Problem 44

You are practicing throwing darts in your dorm. You stand $3.0 \mathrm{~m}$ from the wall on which the board hangs. The dart leaves your hand with a horizontal velocity at a point $2.0 \mathrm{~m}$ above the ground. The dart strikes the board at a point $1.65 \mathrm{~m}$ from the ground. Calculate:
a) the time of flight of the dart;
b) the initial speed of the dart;
c) the velocity of the dart when it hits the board.

Tyler Moulton

Numerade Educator

Problem 45

A football player kicks a ball with a speed of $22.4 \mathrm{~m} / \mathrm{s}$ at an angle of $49^{\circ}$ above the horizontal from a distance of $39 \mathrm{~m}$ from the goal line.
a) By how much does the ball clear or fall short of clearing the crossbar of the goalpost if that bar is $3.05 \mathrm{~m}$ high?
b) What is the vertical velocity of the ball at the time it reaches the goalpost?

Anand Jangid

Numerade Educator

Problem 46

An object fired at an angle of $35.0^{\circ}$ above the horizontal takes $1.50 \mathrm{~s}$ to travel the last $15.0 \mathrm{~m}$ of its vertical distance and the last $10.0 \mathrm{~m}$ of its horizontal distance. With what velocity was the object launched?

Donald Albin

Numerade Educator

Problem 47

A conveyor belt is used to move sand from one place to another in a factory. The conveyor is tilted at an angle of $14.0^{\circ}$ from the horizontal and the sand is moved without slipping at the rate of $7.00 \mathrm{~m} / \mathrm{s}$. The sand is collected in a big drum $3.00 \mathrm{~m}$ below the end of the conveyor belt. Determine the horizontal distance between the end of the conveyor belt and the middle of the collecting drum.

Tyler Moulton

Numerade Educator

Problem 48

Your friend's car is parked on a cliff overlooking the ocean on an incline that makes an angle of $17.0^{\circ}$ below the horizontal. The brakes fail, and the car rolls from rest down the incline for a distance of $29.0 \mathrm{~m}$ to the edge of the cliff, which is $55.0 \mathrm{~m}$ above the ocean, and, unfortunately, continues over the edge and lands in the ocean.
a) Find the car's position relative to the base of the cliff when the car lands in the ocean.
b) Find the length of time the car is in the air.

Tyler Moulton

Numerade Educator

Problem 49

An object is launched at a speed of $20.0 \mathrm{~m} / \mathrm{s}$ from the top of a tall tower. The height $y$ of the object as a function of the time $t$ elapsed from launch is $y(t)=-4.9 t^{2}+19.32 t+60$, where $h$ is in meters and $t$ is in seconds. Determine:
a) the height $H$ of the tower;
b) the launch angle;
c) the horizontal distance traveled by the object before it hits the ground.

Donald Albin

Numerade Educator

Problem 50

A projectile is launched at a $60^{\circ}$ angle above the horizontal on level ground. The change in its velocity between launch and just before landing is found to be $\Delta \vec{v}=\vec{v}_{\text {landing }}-\vec{v}_{\text {launch }}=-20 \hat{y} \mathrm{~m} / \mathrm{s}$. What is the initial velocity of the projectile? What is its final velocity just before landing?

Donald Albin

Numerade Educator

Problem 51

The figure shows the paths of a tennis ball your friend drops from the window of her apartment and of the rock you throw from the ground at the same instant. The rock and the ball collide at $x=50.0 \mathrm{~m}, y=10.0 \mathrm{~m}$ and $t=3.00 \mathrm{~s}$. If the ball was dropped from a height of $54.0 \mathrm{~m},$ determine the velocity of the rock initially and at the time of its collision with the ball.

Donald Albin

Numerade Educator

Problem 52

For a science fair competition, a group of high school students build a kicker-machine that can launch a golf ball from the origin with a velocity of $11.2 \mathrm{~m} / \mathrm{s}$ and initial angle of $31.5^{\circ}$ with respect to the horizontal.
a) Where will the golf ball fall back to the ground?
b) How high will it be at the highest point of its trajectory?
c) What is the ball's velocity vector (in Cartesian components) at the highest point of its trajectory?
d) What is the ball's acceleration vector (in Cartesian components) at the highest point of its trajectory?

Donald Albin

Numerade Educator

Problem 53

If you want to use a catapult to throw rocks and the maximum range you need these projectiles to have is $0.67 \mathrm{~km},$ what initial speed do your projectiles have to have as thev leave the catapult?

Donald Albin

Numerade Educator

Problem 54

What is the maximum height above ground a projectile of mass $0.79 \mathrm{~kg}$, launched from ground level, can achieve if you are able to give it an initial speed of $80.3 \mathrm{~m} / \mathrm{s} ?$

Donald Albin

Numerade Educator

Problem 55

During one of the games, you were asked to punt for your football team. You kicked the ball at an angle of $35.0^{\circ}$ with a velocity of $25.0 \mathrm{~m} / \mathrm{s}$. If your punt goes straight down the field, determine the average velocity at which the running back of the opposing team standing at $70.0 \mathrm{~m}$ from you must run to catch the ball at the same height as you released it. Assume that the running back starts running as the ball leaves your foot and that the air resistance is negligible.

Donald Albin

Numerade Educator

Problem 56

By trial and error, a frog learns that it can leap a maximum horizontal distance of $1.3 \mathrm{~m}$. If, in the course of an hour, the frog spends $20 \%$ of the time resting and $80 \%$ of the time performing identical jumps of that maximum length, in a straight line, what is the distance traveled by the frog?

Donald Albin

Numerade Educator

Problem 57

A circus juggler performs an act with balls that he tosses with his right hand and catches with his left hand. Each ball is launched at an angle of $75^{\circ}$ and reaches a maximum height of $90 \mathrm{~cm}$ above the launching height. If it takes the juggler $0.2 \mathrm{~s}$ to catch a ball with his left hand, pass it to his right hand and toss it back into the air, what is the maximum number of balls he can juggle?

Donald Albin

Numerade Educator

Problem 58

In an arcade game, a ball is launched from the corner of a smooth inclined plane. The inclined plane makes a $30.0^{\circ}$ angle with the horizontal and has a width of $w=50.0 \mathrm{~cm}$ The spring-loaded launcher makes an angle of $45.0^{\circ}$ with the lower edge of the inclined plane. The goal is to get the ball into a small hole at the opposite corner of the inclined plane. With what initial velocity should you launch the ball to achieve this goal?

Donald Albin

Numerade Educator

Problem 59

A copy-cat daredevil tries to reenact Evel Knievel's 1974 attempt to jump the Snake River Canyon in a rocket-powered motorcycle. The canyon is $L=400 . \mathrm{m}$ wide, with the opposite rims at the same height. The height of the launch ramp at one rim of the canyon is $h=8.00 \mathrm{~m}$ above the $\mathrm{rim},$ and the angle of the end of the ramp is $45.0^{\circ}$ with the horizontal.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 60

A golf ball is hit with an initial angle of $35.5^{\circ}$ with respect to the horizontal and an initial velocity of $83.3 \mathrm{mph}$. It lands a distance of $86.8 \mathrm{~m}$ away from where it was hit. $\mathrm{By}$ how much did the effects of wind resistance, spin, and so forth reduce the range of the golf ball from the ideal value?

Tyler Moulton

Numerade Educator

Problem 61

You are walking on a moving walkway in an airport. The length of the walkway is $59.1 \mathrm{~m}$. If your velocity relative to the walkway is $2.35 \mathrm{~m} / \mathrm{s}$ and the walkway moves with a velocity of $1.77 \mathrm{~m} / \mathrm{s}$, how long will it take you to reach the other end of the walkway?

Tyler Moulton

Numerade Educator

Problem 62

The captain of a boat wants to travel directly across a river that flows due east with a speed of $1.00 \mathrm{~m} / \mathrm{s}$. He starts from the south bank of the river and heads toward the north bank. The boat has a speed of $6.10 \mathrm{~m} / \mathrm{s}$ with respect to the water. What direction (in degrees) should the captain steer the boat? Note that $90^{\circ}$ is east, $180^{\circ}$ is south, $270^{\circ}$ is west, and $360^{\circ}$ is north.

Tyler Moulton

Numerade Educator

Problem 63

The captain of a boat wants to travel directly across a river that flows due east. He starts from the south bank of the river and heads toward the north bank. The boat has a speed of $5.57 \mathrm{~m} / \mathrm{s}$ with respect to the water. The captain steers the boat in the direction $315^{\circ} .$ How fast is the water flowing? Note that $90^{\circ}$ is east, $180^{\circ}$ is south, $270^{\circ}$ is west, and $360^{\circ}$ is north.

Tyler Moulton

Numerade Educator

Problem 64

The air speed indicator of a plane that took off from Detroit reads $350 . \mathrm{km} / \mathrm{h}$ and the compass indicates that it is heading due east to Boston. A steady wind is blowing due north at $40.0 \mathrm{~km} / \mathrm{h}$. Calculate the velocity of the plane with reference to the ground. If the pilot wishes to fly directly to Boston (due east) what must the compass read?

Donald Albin

Numerade Educator

Problem 65

You want to cross a straight section of a river that has a uniform current of $5.33 \mathrm{~m} / \mathrm{s}$ and is $127, \mathrm{~m}$ wide. Your motorboat has an engine that can generate a speed of $17.5 \mathrm{~m} / \mathrm{s}$ for your boat. Assume that you reach top speed right away (that is, neglect the time it takes to accelerate the boat to top speed).
a) If you want to go directly across the river with a $90^{\circ}$ angle relative to the riverbank, at what angle relative to the riverbank should you point your boat?
b) How long will it take to cross the river in this way?
c) In which direction should you aim your boat to achieve minimum crossing time?
d) What is the minimum time to cross the river?
e) What is the minimum speed of your boat that will still enable you to cross the river with a $90^{\circ}$ angle relative to the riverbank?

Donald Albin

Numerade Educator

Problem 66

During a long airport layover, a physicist father and his 8 -year-old daughter try a game that involves a moving walkway. They have measured the walkway to be $42.5 \mathrm{~m}$ long. The father has a stopwatch and times his daughter. First, the daughter walks with a constant speed in the same direction as the conveyor. It takes 15.2 s to reach the end of the walkway. Then, she turns around and walks with the same speed relative to the conveyor as before, just this time in the opposite direction. The return leg takes 70.8 s. What is the speed of the walkway conveyor relative to the terminal, and with what speed was the girl walking?

Donald Albin

Numerade Educator

Problem 67

An airplane has an air speed of $126.2 \mathrm{~m} / \mathrm{s}$ and is flying due north, but the wind blows from the northeast to the southwest at $55.5 \mathrm{~m} / \mathrm{s}$. What is the plane's actual ground speed?

Prabhat Tyagi

Numerade Educator

Problem 68

A cannon is fired from a hill $116.7 \mathrm{~m}$ high at an angle of $22.7^{\circ}$ with respect to the horizontal. If the muzzle velocity is $36.1 \mathrm{~m} / \mathrm{s}$, what is the speed of a 4.35 -kg cannonball when it hits the ground $116.7 \mathrm{~m}$ below?

Tyler Moulton

Numerade Educator

Problem 69

A baseball is thrown with a velocity of $31.1 \mathrm{~m} / \mathrm{s}$ at an angle of $\theta=33.4^{\circ}$ above horizontal. What is the horizontal component of the ball's velocity at the highest point of the ball's trajectory?

Tyler Moulton

Numerade Educator

Problem 70

A rock is thrown horizontally from the top of a building with an initial speed of $v=10.1 \mathrm{~m} / \mathrm{s}$. If it lands $d=57.1 \mathrm{~m}$ from the base of the building, how high is the building?

Tyler Moulton

Numerade Educator

Problem 71

A car is moving at a constant $19.3 \mathrm{~m} / \mathrm{s}$, and rain is falling at $8.9 \mathrm{~m} / \mathrm{s}$ straight down. What angle $\theta$ (in degrees) does the rain make with respect to the horizontal as observed by the driver?

Donald Albin

Numerade Educator

Problem 72

You passed the salt and pepper shakers to your friend at the other end of a table of height $0.85 \mathrm{~m}$ by sliding them across the table. They both missed your friend and slid off the table, with velocities of $5 \mathrm{~m} / \mathrm{s}$ and $2.5 \mathrm{~m} / \mathrm{s}$, respectively.
a) Compare the times it takes the shakers to hit the floor.
b) Compare the distance that each shaker travels from the edge of the table to the point it hits the floor.

Donald Albin

Numerade Educator

Problem 73

A box containing food supplies for a refugee camp was dropped from a helicopter flying horizontally at a constant elevation of $500 . \mathrm{m}$. If the box hit the ground at a distance of
150. m horizontally from the point of its release, what was the speed of the helicopter? With what speed did the box hit the ground?

Tyler Moulton

Numerade Educator

Problem 74

A car drives straight off the edge of a cliff that is $60.0 \mathrm{~m}$ high. The police at the scene of the accident note that the point of impact is $150 . \mathrm{m}$ from the base of the cliff. How fast was the car traveling when it went over the cliff?

Prashant Bana

Numerade Educator

Problem 75

At the end of the spring term, a high school physics class celebrates by shooting a bundle of exam papers into the town landfill with a homemade catapult. They aim for a point that is $30.0 \mathrm{~m}$ away and at the same height from which the catapult releases the bundle. The initial horizontal velocity component is $3.90 \mathrm{~m} / \mathrm{s}$. What is the initial velocity component in the vertical direction? What is the launch angle?

Tyler Moulton

Numerade Educator

Problem 76

Salmon often jump upstream through waterfalls to reach their breeding grounds. One salmon came across a waterfall $1.05 \mathrm{~m}$ in height, which she jumped in $2.1 \mathrm{~s}$ at an angle of $35^{\circ}$ to continue upstream. What was the initial speed of her jump?

Donald Albin

Numerade Educator

Problem 77

A firefighter, $60 \mathrm{~m}$ away from a burning building, directs a stream of water from a ground-level fire hose at an angle of $37^{\circ}$ above the horizontal. If the water leaves the hose at $40.3 \mathrm{~m} / \mathrm{s}$, which floor of the building will the stream of water strike? Each floor is $4 \mathrm{~m}$ high.

Donald Albin

Numerade Educator

Problem 78

A projectile leaves ground level at an angle of $68^{\circ}$ above the horizontal. As it reaches its maximum height, $H$, it has traveled a horizontal distance, $d$, in the same amount of time. What is the ratio $H / d ?$

Donald Albin

Numerade Educator

Problem 79

The McNamara Delta terminal at the Metro Detroit Airport has moving walkways for the convenience of the passengers. Robert walks beside one walkway and takes $30.0 \mathrm{~s}$ to cover its length. John simply stands on the walkway and covers the same distance in $13.0 \mathrm{~s}$. Kathy walks on the walkway with the same speed as Robert's. How long does Kathy take to complete her stroll?

Donald Albin

Numerade Educator

Problem 80

Rain is falling vertically at a constant speed of $7.00 \mathrm{~m} / \mathrm{s}$. At what angle from the vertical do the raindrops appear to be falling to the driver of a car traveling on a straight road with a speed of $60.0 \mathrm{~km} / \mathrm{h} ?$

Donald Albin

Numerade Educator

Problem 81

To determine the gravitational acceleration at the surface of a newly discovered planet, scientists perform a projectile motion experiment. They launch a small model rocket at an initial speed of $50.0 \mathrm{~m} / \mathrm{s}$ and an angle of $30.0^{\circ}$ above the horizontal and measure the (horizontal) range on flat ground to be $2165 \mathrm{~m}$. Determine the value of $g$ for the planet.

Tyler Moulton

Numerade Educator

Problem 82

A diver jumps from a $40.0 \mathrm{~m}$ high cliff into the sea. Rocks stick out of the water for a horizontal distance of $7.00 \mathrm{~m}$ from the foot of the cliff. With what minimum horizontal speed must the diver jump off the cliff in order to clear the rocks and land safely in the sea?

Tyler Moulton

Numerade Educator

Problem 83

An outfielder throws a baseball with an initial speed of $32 \mathrm{~m} / \mathrm{s}$ at an angle of $23^{\circ}$ to the horizontal. The ball leaves his hand from a height of $1.83 \mathrm{~m}$. How long is the ball in the air before it hits the ground?

Donald Albin

Numerade Educator

Problem 84

A rock is tossed off the top of a cliff of height $34.9 \mathrm{~m}$ Its initial speed is $29.3 \mathrm{~m} / \mathrm{s}$, and the launch angle is $29.9^{\circ}$ with respect to the horizontal. What is the speed with which the rock hits the ground at the bottom of the cliff?

Donald Albin

Numerade Educator

Problem 85

During the 2004 Olympic Games, a shot putter threw a shot put with a speed of $13.0 \mathrm{~m} / \mathrm{s}$ at an angle of $43^{\circ}$ above the horizontal. She released the shot put from a height of $2 \mathrm{~m}$ above the ground.
a) How far did the shot put travel in the horizontal direction?
b) How long was it until the shot put hit the ground?

Tyler Moulton

Numerade Educator

Problem 86

A salesman is standing on the Golden Gate Bridge in a traffic jam. He is at a height of $71.8 \mathrm{~m}$ above the water below. He receives a call on his cell phone that makes him so mad that he throws his phone horizontally off the bridge with a speed of $23.7 \mathrm{~m} / \mathrm{s}$
a) How far does the cell phone travel horizontally before hitting the water?
b) What is the speed with which the phone hits the water?

Tyler Moulton

Numerade Educator

Problem 87

A security guard is chasing a burglar across a rooftop, both running at $4.2 \mathrm{~m} / \mathrm{s}$. Before the burglar reaches the edge of the roof, he has to decide whether or not to try jumping to the roof of the next building, which is $5.5 \mathrm{~m}$ away and $4.0 \mathrm{~m}$ lower. If he decides to jump horizontally to get away from the guard, can he make it? Explain your answer.

Donald Albin

Numerade Educator

Problem 88

A blimp is ascending at the rate of $7.50 \mathrm{~m} / \mathrm{s}$ at a height of $80.0 \mathrm{~m}$ above the ground when a package is thrown from its cockpit horizontally with a speed of $4.70 \mathrm{~m} / \mathrm{s}$.
a) How long does it take for the package to reach the ground?
b) With what velocity (magnitude and direction) does it hit the ground?

Tyler Moulton

Numerade Educator

Problem 89

Wild geese are known for their lack of manners. One goose is flying northward at a level altitude of $h_{\mathrm{g}}=30.0 \mathrm{~m}$ above a north-south highway, when it sees a car ahead in the distance moving in the southbound lane and decides to deliver (drop) an "egg." The goose is flying at a speed of $v_{\mathrm{g}}=15.0 \mathrm{~m} / \mathrm{s},$ and the car is moving at a speed of $v_{c}=$ $100.0 \mathrm{~km} / \mathrm{h}$
a) Given the details in the figure, where the separation between the goose and the front bumper of the car, $d=$ $104.1 \mathrm{~m},$ is specified at the instant when the goose takes action, will the driver have to wash the windshield after this encounter? (The center of the windshield is $h_{c}=1.00 \mathrm{~m}$ off the ground.)
b) If the delivery is completed, what is the relative velocity of the "egg" with respect to the car at the moment of the impact?

Donald Albin

Numerade Educator

Problem 90

You are at the mall on the top step of a down escalator when you lean over laterally to see your $1.80 \mathrm{~m}$ tall physics professor on the bottom step of the adjacent up escalator. Unfortunately, the ice cream you hold in your hand falls out of its cone as you lean. The two escalators have identical angles of $40.0^{\circ}$ with the horizontal, a vertical height of $10.0 \mathrm{~m}$, and move at the same speed of $0.400 \mathrm{~m} / \mathrm{s}$. Will the ice cream land on your professor's head? Explain. If it does land on his head, at what time and at what vertical height does that happen? What is the relative speed of the ice cream with respect to the head at the time of impact?

Donald Albin

Numerade Educator

Problem 91

A basketball player practices shooting three-pointers from a distance of $7.50 \mathrm{~m}$ from the hoop, releasing the ball at a height of $2.00 \mathrm{~m}$ above ground. A standard basketball hoop's rim top is $3.05 \mathrm{~m}$ above the floor. The player shoots the ball at an angle of $48^{\circ}$ with the horizontal. At what initial speed must he shoot to make the basket?

Donald Albin

Numerade Educator

Problem 92

Wanting to invite Juliet to his party, Romeo is throwing pebbles at her window with a launch angle of $37^{\circ}$ from the horizontal. He is standing at the edge of the rose garden $7.0 \mathrm{~m}$ below her window and $10.0 \mathrm{~m}$ from the base of the wall. What is the initial speed of the pebbles?

Donald Albin

Numerade Educator

Problem 93

An airplane flies horizontally above the flat surface of a desert at an altitude of $5.00 \mathrm{~km}$ and a speed of $1000 . \mathrm{km} / \mathrm{h}$ If the airplane is to drop a care package that is supposed to hit a target on the ground, where should the plane be with respect to the target when the package is released? If the target covers a circular area with a diameter of $50.0 \mathrm{~m}$, what is the "window of opportunity" (or margin of error allowed) for the release time?

Donald Albin

Numerade Educator

Problem 94

A plane diving with constant speed at an angle of $49.0^{\circ}$ with the vertical, releases a package at an altitude of $600 . \mathrm{m}$. The package hits the ground $3.50 \mathrm{~s}$ after release. How far horizontally does the package travel?

Tyler Moulton

Numerade Educator

Problem 95

10.0 seconds after being fired, a cannonball strikes a point $500 . \mathrm{m}$ horizontally from and $100 . \mathrm{m}$ vertically above the point of launch.
a) With what initial velocity was the cannonball launched?
b) What maximum height was attained by the ball?
c) What is the magnitude and direction of the ball's velocity just before it strikes the given point?

Donald Albin

Numerade Educator

Problem 96

Neglect air resistance for the following. A soccer ball is kicked from the ground into the air. When the ball is at a height of $12.5 \mathrm{~m},$ its velocity is $(5.6 \hat{x}+4.1 \hat{y}) \mathrm{m} / \mathrm{s}$.
a) To what maximum height will the ball rise?
b) What horizontal distance will be traveled by the ball?
c) With what velocity (magnitude and direction) will it hit the ground?

Donald Albin

Numerade Educator