• Home
  • Textbooks
  • University Physics with Modern Physics
  • Motion in Two or Three Dimensions

University Physics with Modern Physics

Hugh D. Young, Roger A. Freedman

Chapter 3

Motion in Two or Three Dimensions - all with Video Answers

Educators

Chapter Questions

05:38

Problem 1

$A$ squinel has $x$ - and y-coordinates $(1.1 \mathrm{~m}, 3.4 \mathrm{~m})$ at time $f_{1}=0$ and coordinates $(5.3 \mathrm{~m},-0.5 \mathrm{~m})$ at time $t_{2}=3.0 \mathrm{~s}$. For this time interval, find (a) the components of the average velocity, and (b) the magnitude and direction of the average velocity.

Steven Emmel
Steven Emmel
University of California - Los Angeles
04:15

Problem 2

A rhinoceros is at the origin of coordinutes at time $t_{1}=0$. For the time interval from $t_{1}=0$ to $t_{2}=12.0 \mathrm{~s}$, the thino's avcrage velocity has $x$ -conponent $-3.8 \mathrm{~m} / \mathrm{s}$ and $y$ -component $4.9 \mathrm{~m} / \mathrm{s}$. At time $t_{2}=12.0 \mathrm{~s}_{6}$ (d) what are the $x$ - and y coordinutes of the rhino?
(b) How far is the rhino frorn the origin?

Steven Emmel
Steven Emmel
University of California - Los Angeles
01:00

Problem 3

A web page designer creates an animation in which in dot on a compuler screen has a position of $\vec{r}=[4.0 \mathrm{~cm}+$ $\left.\left(2.5 \mathrm{~cm} / \mathrm{s}^{2}\right) t^{2}\right] \mathrm{i}+(5.0 \mathrm{~cm} / \mathrm{s}) \mathrm{f}$. (a) Find the mugnitude sid
direction of the dot's nvernge velocity berween $t=0$ und $t=20 \mathrm{~s}$. (b) Find the magnitude und direction of the instantaneous velocity et $t=0, t=1.0 \mathrm{~s}$, sind $t=2.0 \mathrm{~s}$. (c) Sketch the dot's trajectery from $t=0$ to $t=20 \mathrm{~s}$, and show the velocities calculated in part (b).

Dominador Tan
Dominador Tan
Numerade Educator
01:40

Problem 4

If $\left.\vec{r}=b r^{2} \hat{\imath}+c t^{3}\right\}$, where $b$ and $c$ are positive conctants, when does the velocity vector make un ungle of $45.0^{\circ}$ with the $x$ and $y$ -axes?

Mitchel Vereen
Mitchel Vereen
Numerade Educator
05:38

Problem 5

A jet plane is fiying at a constant altitude. At time $t_{1}=0$ it has components of velocity $v_{x}=90 \mathrm{~m} / \mathrm{s}, v_{y}=110 \mathrm{~m} / \mathrm{s}$. At time $t_{2}=30.0 \mathrm{~s}$ the components are $v_{\mathrm{y}}=-170 \mathrm{~m} / \mathrm{s}, \mathrm{v}_{\mathrm{y}}=40 \mathrm{~m} / \mathrm{s}$.
(i) Sketch the velocity vectors at $t_{1}$ and $t_{2}$. How do these two vee. tors differ? For this time interval calculate (b) the components of the averige acceleration, and (c) the magnitude and direction of the average acceleration.

Supratim Pal
Supratim Pal
Numerade Educator
16:12

Problem 6

A dog running in an open ficld has components of velocity $v_{x}=2.6 \mathrm{~m} / \mathrm{s}$ and $v_{y}=-1.8 \mathrm{~m} / \mathrm{s}$ at $t_{1}=10.0 \mathrm{~s}$. For the time
interval from $t_{1}=10.0 \mathrm{~s}$ to $t_{2}=20.08$, the average acceleration of the dog has magnitude $0.45 \mathrm{~m} / \mathrm{s}^{3}$ and direction $31.0^{\circ}$ measared from the $+x$ -axis toward the $+y$ -axis. At $t_{2}=20.0 \mathrm{~s}_{1}(\mathrm{n})$ whet are the x- and y-ccmponents of the dog's velocity? (b) What are the magnitude and direction of the dog's veloeity?
(c) Sketch the velocity vectors at $I_{1}$ and $t_{2}$, How do these two vectors differ?

Chris Dimenichi
Chris Dimenichi
Lehigh University
12:23

Problem 7

The coordinstes of a bird flying in the xy-plane are given by $x(t)=a t$ and $y(t)=3.0 \mathrm{~m}-\beta t^{2}$, where $\alpha=2.4 \mathrm{~m} / \mathrm{s}$ and
$\beta=1.2 \mathrm{~m} / \mathrm{s}^{2} .$ (a) Sketch the path of the bird between $t=0$ and $t=20 \mathrm{k}$. (b) Calculate the velocity and accelemtion vectors of the bird as functions of time. (c) Calculate the magnitude and direction of the bird's velocity and acocleration at $t=2.0$ s. (d) Sketch the velocity and acceleration vectors at $t=2.0 \mathrm{~s}$. At this instant, is the bird speeding up, is it slowing down, ot is its speed instantaneously not chinging? Is the bird tuming? If so, in what ?irection?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:35

Problem 8

A particle moves along $\mathrm{n}$ path as shown in Fig. 338 . Betwcen points $B$ and $D$, the path is a straight line. Sketch the acceleration vectors at $A, C$, and $E$ in the cases in which (a) the particle moves with n constant speed; (b) the particle moves with a steadily increasing speed; (c) the particle moves with stearily decreasing speed.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:59

Problem 9

A physics boolk stides off a horizontal tabletop with a speed of $1.10 \mathrm{~m} / \mathrm{s}$. It strikes the floor in $0.350$ s. Ignore air resistance. Find
(b) the horizontal dis-
(a) the height of the tabletop above the floor; tance from the edge of the table to the point where the book strikes the foor, (c) the borizontal and vertical cotrponents of the book's velocity, and the magnitude and diroction of its velocity, just before the boolk reaches the fionr. (d) Draw $x-t, y-t, v_{x}-t_{1}$ and $v_{y}-t$ graphs for the motion.

Averell Hause
Averell Hause
Carnegie Mellon University
05:41

Problem 10

A military helicopter on a training mission is flying horizontally at a speed of $60.0 \mathrm{~m} / \mathrm{s}$ and accidentally drops a bomb (fortunately not armed) at an elevation of $300 \mathrm{~m}$. You can ignore air resistance. (a) How much time is requirod for the botib to reach the earth? (b) How far does it travel horizontally while falling? (c) Find the hotizontal and vertical components of its velocity just before it
bomb's motion. (e) If the velocity of the helicopter remains constint, where is the helioopter when the bomb hits the groand?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:22

Problem 11

Two crickets, Chirpy and Milada, jump from the top of a vertical cliff. Chirpy just drops and reaches the ground in $3.50 \mathrm{~s}$, whilc Milada jumps horizontally with an initial cpeed of $95.0 \mathrm{~cm} / \mathrm{s}$. How far from the base of the cliff will Milada hit the ground?

Yanlian Xin
Yanlian Xin
Numerade Educator
01:49

Problem 12

A daring $510-N$ swimmer dives off a cliff with a running horizontal leap, ws shown in Fig. 339. What must her minimum speed be just as she leaves the top of the cliff so that she will miss the ledge at the boutom, which is $1.75 \mathrm{~m}$ wide and $9.00 \mathrm{~m}$ below the top of the clifr?

Averell Hause
Averell Hause
Carnegie Mellon University
07:19

Problem 13

Leaping the River. I. $A$ car comes to a bridge during a storm and finds the bridge washed out. The driver must got to the other side, so he decldos to try leaping it with his car. The side of the road the car is on is $21.3 \mathrm{~m}$ above the river, while the opposite side is a mere $1.8 \mathrm{~m}$ above the river. The river itself is a raging torrent $61.0 \mathrm{~m}$ wide. (a) How fast should the car be traveling at the time it leaves the road in order just to clear the river and Iand safely on the opposite side? (b) What is the spood of the car just before it lands an the other side?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:02

Problem 14

A small marble Figure $3.40$ Exercise $3.14$ rolls horizontally $\longrightarrow v_{0}=?$ with speed $v_{0}$ off the top of slatform $2.75 \mathrm{~m}$ tall and foels no appesciable air resistance, On the level ground, $2.00 \mathrm{~m}$
from the base of the platform, there is a gaping hole in the ground (Fig. 3.40.) For what range of marble speeds $v_{0}$ will the marble land in the hole?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:13

Problem 15

Inside n starship at rest on the earth, al ball rolls off the top of a borizontul table and Iands a distance $D$ from the foot of the table. This starship now lands on the unexplorod Planct $X$. The commander, Captain Curious, rolls the same ball off the same table with the same initial speod as on earth and finds that it lands a distance $2.76 D$ from the foot of the table. What is the acceleration due to gravity on Planct $\mathrm{X}$ ?

Supratim Pal
Supratim Pal
Numerade Educator
07:27

Problem 16

A rookie quarterback throws a football with an initial upward velocity component of $16.0 \mathrm{~m} / \mathrm{s}$ and s horizontal velocity component of $20,0 \mathrm{~m} / \mathrm{s}$. Lgnore sir rusistance. (e) How much time is required for the football to reach the highest point of the trejectory?
(b) How high is this point? (c) How mpch time (after it is throwa) is required for the football to retum to its original level? How does this compare with the time calculated in part (c)? (d) How far has the football trirveled horizontally during this time? (c) Druw $x-t$, $y-t, v,-t$, and $v_{-}-t$ graphs for the motion.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:29

Problem 17

On level ground a shell is fired with an initial velocity of $80.0 \mathrm{~m} / \mathrm{s}$ at $60.0^{\circ}$ above the borizontal and fecls no eppreciable sir resistance. (a) Find the horizontal and vertical components of the shell's initial velocity. (b) How long does it take the shell to reach its highest point? (c) Find its maximum height above the ground.
(d) How far from its firing point does the shell land?
(c) At its highest point, find the borizontal and vertical cornponents of its accelemtion and velocity.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:26

Problem 18

A pistol that fires a signal flare gives it an initial velocity (muzale velocity) of $125 \mathrm{~m} / \mathrm{s}$ at an ungle of $55.0^{\circ}$ above the horizontal. You can ignore air resistance. Find the flare's matimum height and the distance from its firing poict to its landing point if it is fired (a) on the level salt flats of Utah, and (b) over the flat Sea of Tranquility on the Moon, where $g=1.67 \mathrm{~m} / \mathrm{s}^{2}$.

Averell Hause
Averell Hause
Carnegie Mellon University
11:19

Problem 19

A major leaguer hits a bascball so that it leaves the bat at a speed of $300 \mathrm{~m} / \mathrm{s}$ and at an angle of $36.9^{\circ}$ nbove the borizontal. You can ignore air resistnnce. (a) At what two times is the baseball at a beight of $10.0 \mathrm{~m}$ above the point at which it left the bat? (b) CalcuLate the horizontal and vertical components of the bascball's veloeity at each of the two times calculated in part (a). (c) What are the magnitude and direction of the baschall's velocity when it returns to the level at which it left the bet?

Jayashree Behera
Jayashree Behera
Numerade Educator
06:25

Problem 20

A shot putter roleases the shot some distance above the level ground with a velocity of $12.0 \mathrm{~m} / \mathrm{s}, 51.0^{\circ}$ sbove the horizontal. The shot hits the ground $2.08$ s later. You can ignore air resistance.
(a) What are the components of the shot's acoeleration while in fiight? (b) What are the components of the shot's velocity at the beginning and at the end of its trijectory? (c) How far did sbe throw the shot horizontally'1 (d) Why does the expression for $R$ in Example $3.8$ not give the correct answer for part (c)? (c) How high was the shot above the ground when she released ir? (f) Draw $x-t$, $y^{-L}, v_{x} t$, and $v_{-} t$ praphs for the motion.

Keshav Singh
Keshav Singh
Numerade Educator
05:26

Problem 21

Win the Prixe. In a carnival bocth, you win a stuffed giraffe if you toss a quarter into a small dish. The ?ish is on a shelf above the point where the quarter leaves your hand and is a horizontul distance of $2.1 \mathrm{~m}$ from this point (Fig. $3.41$ ). If you toss the coin with a velocity of $6.4 \mathrm{~m} / \mathrm{s}$ at an angle of $60^{\circ}$ above the horizontal, the coin Iands in the dish. You can ignore air resistance. (a) What is the height of the shelf above the point where the quarter leaves your hand? (b) What is the vertical component of the velocity of the quarter just before it lands in the dish?

Mitchel Vereen
Mitchel Vereen
Numerade Educator
03:44

Problem 22

Suppose the departare angle $\alpha_{0}$ in Fig. $3.26$ is $42.0^{\circ}$ and the distance d is $3.00 \mathrm{~m}$. Where will the dart and monkey meet if the initial speed of the dart is (a) $12.0 \mathrm{~m} / \mathrm{s}$ ? (b) $80 \mathrm{~m} / \mathrm{s}$ ? (c) What will hippen if the initial speed of the dart is $4.0 \mathrm{~m} / \mathrm{s}$ ? Sketch the trijeetory in each case.

Keshav Singh
Keshav Singh
Numerade Educator
11:28

Problem 23

A man stands on the roof of a $15.0-\mathrm{m}$ -tall building and throws a rock with a velocity of magnitode $30.0 \mathrm{~m} / \mathrm{s}$ st an angle of $33.0^{\circ}$ above the horizontnl. You can ignore alr reststance, Calculate (a) the maximum height above the roof reached by the rock; (b) the magnitude of the velocity of the rock just before it strikes the ground; and (c) the borizontal range from the base of the building to the point where the rock strikes the ground. (d) Draw $x-t, y-l$, $v_{x}-t_{0}$ and $v_{y}-t$ graphs for the motion.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:54

Problem 24

Firemen are shooting a streatn of water at a borning building using a high-pressure hose that shoots out the water with s speed of $25.0 \mathrm{~m} / \mathrm{s}$ as it leaves the cnd of the hose, Once it leaves the hosc, the water moves in projectile motion. The firemen adjust the angle of elevation $\alpha$ of the hose until the water takes $3.00 \mathrm{~s}$ to reach a building $45.0 \mathrm{~m}$ nway. You can ignore air resistance; assume that the end of the hose is at gnound level. (a) Find the angle of elevit tion $\alpha$, (b) Find the speed and accelerution of the water at the highest point in its trajectory, (c) How high sbove the ground does the water strike the boilding, and how fast is it moving just bcfore it bits the building?

Keshav Singh
Keshav Singh
Numerade Educator
16:24

Problem 25

$\mathrm{A} 124 \mathrm{~kg}$ balloon carrying a $22 \mathrm{~kg}$ basket is descending with fo constant dowmward velocity of $200 \mathrm{~m} / \mathrm{s} . \mathrm{A} 1.0-\mathrm{kg}$ stone is thrown from the basket with an initial velocity of $15.0 \mathrm{~m} / \mathrm{s}$ perpendicular to the path of the descending balloon, es measured relative to a persoa at rest in the basket. The persoa in the basket sees the stone hit the ground $6.00$ s after being thrown. Assume that the balloon continues its downwand descent with the same constant speed. of $20.0 \mathrm{~m} / \mathrm{s}$. (a) How high was the balloon when the rock was thrown out? (b) How high is the balloon when the rock hits the ground? (c) At the instant the rock hits the ground, bow far is it from the basket? (d) Just before the rock hits the ground, find its horizoatal and vertical velocity componcnts as measurcd by an observer (i) at rest in the basket and (ii) at rest on the ground.

GA
Gabriel A
Numerade Educator
04:05

Problem 26

A cannon, located $60.0 \mathrm{~m}$ from the base of a vertical $25.0-\mathrm{m}$ tall cliff, shooss u 15-kg shell at $43.0^{\circ}$ above the horizontal toward the cliff. (a) What must the minimum muzzle velocity be for the sbell to clear the top of the cliff? (b) The groand at the top of the cliff is lovel, with a constant elevation of $25.0 \mathrm{~m}$ above the cannon. Under the conditions of part (a), how far does the shell land past the edge of the clif??

Suzanne W.
Suzanne W.
Numerade Educator
02:14

Problem 27

An airplane is dying with n velocity of $90.0 \mathrm{~m} / \mathrm{s}$ at an angle of $23.0^{\circ}$ above the borizoutal. When the plane is $114 \mathrm{~m}$ directly above a dog that is standing on level ground, u suitcase drops out of the luggage cornpartment. Fow far from the dog will the sultcase land? You can fgnore air resistance.

Averell Hause
Averell Hause
Carnegie Mellon University
05:32

Problem 28

On your first day at work for an applinnce manufucturer, you are told to figure out what to do to the period of rotation ?uring of washer spin cycle to triple the centripetal nccelerntion. You inpress your boss by answering immediately. What do you tell her?

Guilherme Barros
Guilherme Barros
Numerade Educator
02:23

Problem 29

The earth has a radius of $6380 \mathrm{~km}$ and turns around once on Its axis in $24 \mathrm{~h}$. (o) What is the radial occeleration of an object at the earth's equator? Give your answer in $\mathrm{m} / \mathrm{s}^{2}$ and as a fraction of
g. (b) If $a_{\text {at }}$ at the cquator is greater than $g$, objccts would tly off the earth's surface and into space. (We will see the reason for this in Chapter 5.) What would the period of the earth's rotation have to be for this to occur?

Averell Hause
Averell Hause
Carnegie Mellon University
06:28

Problem 30

A model of a helicopter rotor has four blades, each $3.40 \mathrm{~m}$ long from the central shaft to the blade tip. The model is rolated in at wind tunnel at 550 rev/min, (a) What is the linear speed of the blade tip, in m/s? (b) What is the radial accelenation of the blade tip expressed as a multiple of the acceleration of gravity, $8 ?$

Chris Dimenichi
Chris Dimenichi
Lehigh University
02:54

Problem 31

In a test of a "g-iuit," a volunteet is rotated in a horizontal circle of radius $7.0 \mathrm{~m}$. What must the period of rotation be so that the ceatripetal acceleration has at magnitude of (a) $3.0 \mathrm{~g}$ ? (b) $10 \mathrm{~g}$ ?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:20

Problem 32

3.32. The radius of the earth"s orbit around the sun (assumed to be circular) is $1.50 \times 10^{4} \mathrm{~km}$, and the carth travels around this orbit in 365 days. (a) What is the magnitude of the orbital velocity of the earth, in $\mathrm{m} / \mathrm{s}$ ?
(b) What is the radial seceleration of the earth toward the sun, in $\mathrm{m} / \mathrm{s}^{2} ?$ (c) Repeat parts (a) and (b) for the motion of the planet Mercury (orbit radius $=5.79 \times 10^{7} \mathrm{~km}$, orbital period $=88.0$ days .

Vishal Gupta
Vishal Gupta
Numerade Educator
04:35

Problem 33

A Ferris wheel with rudius $14.0 \mathrm{~m}$ is turning about a horizontal axis through its center (Fig. 3.42). The linear speed of a pasecuger on the rim is constant and equal to $7.00 \mathrm{~m} / \mathrm{s}$. What are the magnitude and direction of the passenger's acceleration as she passes through (a) the lowest point in her circular motion?
(b) The highest point in her circular motion? (c) How much time does it trike the Ferris wheel to make one revolution?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:03

Problem 34

The Ferris wheel in Fig- $3.42$, which motates counterclockwise, is just starting up. At a given instant, a passenger on the rim of the wheel and passing through the lowest point of his circular motion is moving at $3.00 \mathrm{~m} / \mathrm{s}$ and is gaining speed at n rate of $0.500 \mathrm{~m} / \mathrm{s}^{2}$. (a) Find the magnifude und the direction of the passenger's nccelerition at this instant. (b) Sketch the Ferris wheel and the pashenger, showing his velocity and acceleration vectors.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:47

Problem 35

Hypergravity. At its Ames Rosearch Center, NASA uses its large "20-C" centrifuge to test the effects of very large accelerations ("hypergruvity") on test pilots and astronauts. In this device, an um $8.84 \mathrm{~m}$ long rotates about one end in a horizoutal plune, and the astronaut is strupped in at the other end. Suppose that be is aligned along the arm with his head at the outermost end. The maximum sustaincd accclerition to which hmmans are subjcotod in this machine is typically $12.5 g$ - (a) How fast must the astronaut's head
(b) What is be moving to experience this maximum acceleration? the diference between the soceleration of his head and feet if the astronaut is $2.00 \mathrm{~m}$ tall? (c) How fast in mm (rev/min) is the arm torning to produce the maximum sustained acceleration?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:26

Problem 36

A railroad flatear is traveling to the right at se speed of $13.0 \mathrm{~m} / \mathrm{s}$ relative to an obscrver standing on the ground. Someone is riding a motor scooter on the flatear (Fig. 3.43). What is the velocity (magnitude and direction) of the motor scooter relative to the flatear if its velocity relative to the observer on the ground is
(a) $18.0 \mathrm{~m} / \mathrm{s}$ to the right?
(b) $3.0 \mathrm{~m} / \mathrm{s}$ to the lef?
(c) zero?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:48

Problem 37

A "moving sidewalk" in an airport terminal building moves at $1.0 \mathrm{~m} / \mathrm{s}$ and is $35.0 \mathrm{~m}$ long. If a woman steps on at one end and walks at $1.5 \mathrm{~m} / \mathrm{s}$ relative to the moving sidewalk, how much time does she requine to reach the opposite end if she walks (a) in the same direction the sidewalk is movino
(b) In the opposite direction

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:25

Problem 38

Two piers, $A$ and $B$, are located en a river: $B$ is $1500 \mathrm{~m}$ downstrean from $A$ (Fig. 3.44). Two frieuds mast make round trips from pier $A$ to pier $B$ and return. One rows a boat at a constant speed of $4.00 \mathrm{~km} / \mathrm{h}$ relative to the water, the other walks on the shore at a constant speed of $4.00 \mathrm{~km} / \mathrm{h}$, The velocity of the river is $2.80 \mathrm{~km} / \mathrm{h}$ in the ?irection from $A$ to $B$. How much time does it tuke each person to make the round trip?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:39

Problem 39

A canoe has a velocity of $0.40 \mathrm{~m} / \mathrm{s}$ southeast relative to the earth. The canoe is on a river that is flowing $0.50 \mathrm{~m} / \mathrm{s}$ east rolative to the carth. Find the velocity (magritude and direction) of the canoe relative to the river.

Jayashree Behera
Jayashree Behera
Numerade Educator
14:37

Problem 40

An airplane pilot wishes to fly ?ue west. A wind of $80.0 \mathrm{~km} / \mathrm{h}$ (about $50 \mathrm{mi} / \mathrm{b})$ is blowing toward the south. (a) If the airpeed of the plane (its speed in still air) is $320.0 \mathrm{~km} / \mathrm{h}$ (about $200 \mathrm{mi} / \mathrm{b})$, in which direction sbould the pilot bead? (b) What is the spoed of the plane over the ground? IIlustrate with a vector diagram.

Chris Dimenichi
Chris Dimenichi
Lehigh University
03:56

Problem 41

Crossing the River I. A river fows due south with a speed of $20 \mathrm{~m} / \mathrm{s}$. A man steers a motorboat scross the river; his velocity relative to the water is $4.2 \mathrm{~m} / \mathrm{s}$ due east. The river is $800 \mathrm{~m}$ wide.
(a) What is his velocity (magmitude and direction) relative to the earth? (b) How much time is required to cross the river? (c) How far south of his starting point will be reach the opposite bank?

Averell Hause
Averell Hause
Carnegie Mellon University
View

Problem 42

Crossing the River II.
(a) In which dircction should the motorboat in Exercise $3.41$ head in order to reach a point on the opposite benk directly east from the starting point? (The boat's speed relative to the weter remains $4.2 \mathrm{~m} / \mathrm{s}$.) (b) What is the veloc. ity of the boat relative to the earth? (c) How moch time is requined to cross the river?

David Morris
David Morris
Numerade Educator
03:53

Problem 43

The nose of an ultralight plane is pointed south, and its airspeed indicator shows $35 \mathrm{~m} / \mathrm{s}$. The plane is in a $10-\mathrm{m} / \mathrm{s}$ wind blowing toward the southwest rolative to the earth. (a) In a vector addition diagram, show the relutiondip of $c_{\mathrm{m} \mu 1}$ (the velocity of the pline relative to the earth) to the two given vectors. (b) Letting $x$ be enst and $y$ be north, find the components of $\vec{v}_{P)}$ (c) Find the magnitude and direction of $\vec{v}_{\text {P)E }}$

Mitchel Vereen
Mitchel Vereen
Numerade Educator
06:41

Problem 44

A faculty model rocket moves in the $x y$ -plane (the positive $y$ direction is vertically upward). The rocket's acceleratice has components $a_{x}(t)=\alpha t^{2}$ and $a_{y}(t)=\beta-y t$, where $\alpha=2.50 \mathrm{~m} / \mathrm{s}^{4}$.
$\beta=9.00 \mathrm{ra} / \mathrm{s}^{2}$, and $\gamma=1.40 \mathrm{~m} / \mathrm{s}^{3} .$ At $t=0$ the rocket is at the
origin end has velocity $\vec{d}_{0}=v_{b} \hat{i}+v_{0}$, with $v_{\mathrm{cs}}=1.00 \mathrm{~m} / \mathrm{s}$ and $v_{0 y}=7.00 \mathrm{~m} / \mathrm{s}$ (a) Calculate the velocity and position vectors as functions of time. (b) What is the maximum height reached by the rocket? (c) Sketch the path of the rocket.
(d) What is the borizontal displacement of the rocket when it retums to $y=0 ?$

Keshav Singh
Keshav Singh
Numerade Educator
12:38

Problem 45

A rocket is fired at an angle from the top of a tower of height $h_{0}=50.0 \mathrm{~m}$. Because of the design of the engines, its position coontinates are of the form $x(t)=A+B t^{2}$ and $y(t)=C+D t^{3}$ where $A, B, C$, and $D$ are constants. Furthermore, the accelention of the rocket $1.00 \mathrm{~s}$ after firing is $\mathrm{d}=(4.00 \bar{t}+3.00 \mathrm{j}) \mathrm{m} / \mathrm{s}^{2}$. Take
the origin of coordinutes to be at the base of the tower. (a) Find the constants $A, B, C$, and $D$, including their SI units (b) At the instant after the rocket is fired, what are its acceleration vector and its velocity? (c) What are the $x$ - and y-compecents of the rocket's velocity $10.0 \mathrm{~s}$ after it is firod, and how fast is it moving? (d) What is the pesition vector of the rocket $100 \mathrm{~s}$ after it is fired?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:25

Problem 46

A bird flies in the $x y$ -plane with a velocity vector given by $\vec{v}=\left(\alpha-\beta t^{2}\right) \hat{t}+\gamma \mathrm{t}$, with $\boldsymbol{a}=2.4 \mathrm{~m} / \mathrm{s}, \boldsymbol{\beta}-1.6 \mathrm{~m} / \mathrm{s}^{3}$, and
$\gamma=4.0 \mathrm{~m} / \mathrm{s}^{2}$. The positive y direction is vertically upward. $\mathrm{At} t=0$ the bird is at the origin. (a) Culculate the poaition and acceleration vectors of the bird as functions of time. (b) What is the trird's altitude ( ( -coordinetc) as it flies over $x=0$ for the first time after $t=0 ?$

Zachary Warner
Zachary Warner
Numerade Educator
07:27

Problem 47

A test rocket is launched Figure 3.45 Problem $3.47$. by accelerating it along a $200.0-\mathrm{m}$ incline at $1.25 \mathrm{~m} / \mathrm{s}^{2}$
starting from rest at point $A$ (Figure 3.45.) The incline rises at $35.0^{\circ}$ above the horizontal, and at the instant the rocket leaves it, its cngincs turn off and it is subject only to gravity (air resistance can be ignored). Find
(a) the maximum height above the groand that the rocket reaches, and $(\mathrm{b})$ the greatest borizontnl runge of the rocket beyoad point $A$.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:24

Problem 48

3.40. Martian Athletics. In the long jurnp, an athlete launches herself at an angle above the ground and Lands at the same height, trying to travel the greatest borizontal distance, Suppose that on earth whe is in the alir for time $T$, reaches a muxiruum height $h_{3}$ and achieves a horizontal distance $D$. If she jumped in exactly the same way during a compctition on Mars, where guen is $0.379$ of its carth value, find her time in the air, muxirum beight, and borizontal distunce. Bxpress each of these three quantities in terms of its carth value. Air resistance can be neglected on both planets.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:19

Problem 49

Dynamitel A dernolition crew uses dynamite to blow an old beilding apart. Debris from the explosion flies off in all directions and is later found at distances as far as $50 \mathrm{~m}$ from the explosion. Estimate the maximum specd at which debris wus blown outward by the cxplosion. Describe any nssumptions that you make.

Artemisa Mazón
Artemisa Mazón
Numerade Educator
05:22

Problem 50

Spiraling Up. It is common to see birds of prey rising upward on thernals, The paths they take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assame a bird completes a circle of radius $8.00 \mathrm{~m}$ every $5.00 \mathrm{~s}$ and rises vertically at a rate of $3.00 \mathrm{~m} / \mathrm{s}$. Determine: (a) the speed of the bird relative to the ground (b) the bird's acceleration (magnitude and direction): and
(c) the angle between the hind's velocity vector and the horizontal.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:15

Problem 51

A jungle veterinarian with a blow-gun loaded with a tranquilizer dart and a sly $1.5-\mathrm{k}$ s monkey are each $25 \mathrm{~m}$ above the ground in trees $90 \mathrm{~m}$ apart. Just as the hunter shoots horizontally ut the monkey, the monkey drops from the tree in a vain attempt to escape being hit. What must the minimum muzzle velocity of the dart bave been for the hunter to hit the monkey before it reached the ground?

Prashant Bana
Prashant Bana
Numerade Educator
06:21

Problem 52

A movie stuntwoman drops from a helicopter that is $30.0 \mathrm{~m}$ above the ground and moving with a constant velocity whose components are $10.0 \mathrm{~m} / \mathrm{s}$ upward and $15.0 \mathrm{~m} / \mathrm{s}$ horizontal and toward the south. You can ignore air resistance. (a) Where on the ground (relafive to the pesition of the helicoptcr when she drops) should the stuntwoman have placed the foam mats that break her fall? (b) Draw $x-l, y-L, v_{x}-t$, and $v_{y}-t$ graphs of her motion.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:03

Problem 53

In fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. A pilot is practicing by dropping a canister of red dye, hoping to hit a target on the ground below. If the plane is flying in a horizontal path $90.0 \mathrm{~m}$ above the ground and with a speed of $64.0 \mathrm{~m} / \mathrm{s}$ ( $143 \mathrm{mi} / \mathrm{h})$, at what horizontal distance from the urget should the pilot release the canister? Ignore air resistance.

Ajay Singhal
Ajay Singhal
Numerade Educator
03:20

Problem 54

As a ship is approaching the dock at $45.0 \mathrm{~cm} / \mathrm{s}$, an importunt piece of landing equipment noeds to be thrown to it before it can dock. This equipment is thrown at $15.0 \mathrm{~m} / \mathrm{s}$ at $60.0^{\circ}$ above the horizoptal from the top of a tower at the edge of the water, $8.75 \mathrm{~m}$ above the ship's deck (Hig. 3.46.) For this equipuent to land at the froat of the ship, at what distance $D$ from the dock should the ship be when the equipment is thrown? Air resiatance can be neglected.

Tiannie Zhao
Tiannie Zhao
Numerade Educator
03:51

Problem 55

The Longest Home Run. According to the Guinness Book of World Records, the longest home run ever measured was hit by Roy "Dizzy" Carlyle in a minor league game. The ball traveled $188 \mathrm{~m}(618 \mathrm{ft})$ before landing on the ground outside the ballpark.
(a) Assuming the ball's initial velocity was $45^{\circ}$ above the horizontal and ignoring air resistance, what did the initial speed of the ball need to be to produce such a home run if the ball was hit at a point 0.9 m (3.0 ft) above ground level? Assume that the ground was perfectly fat.
(b) How far would the ball be above a fonce $3.0 \mathrm{~m}$ ( 10 ft) high if the fence was $116 \mathrm{~m}$ (380 ft) from home plate?

Averell Hause
Averell Hause
Carnegie Mellon University
09:47

Problem 56

A water hose is used to fill a large cylindrical storage tank of diameter $D$ and height $2 D$. The hose shoots the water at $45^{\circ}$ above the horizcatal from the same level as the base of the tank and is a distance $6 D$ away (Fig. $3.47$ ). For what range of launch speeds $\left(\mathrm{v}_{0}\right)$ will the water enter the tank? Ignore air resistance, and express your answer in terms of $D$ and $g$.

Guilherme Barros
Guilherme Barros
Numerade Educator
08:45

Problem 57

A projectile is being launched from ground level with no air resistance. You want to avoid having it enter a tempersture inversion layer in the atmosphere a beight $h$ ahove the ground. (a) What is the maximum launch speed yoa cogld give this projectile if you shot it straight up? Bxpress yoar answer in terms of $h$ and $R$. (b) Suppose the launcher available shoots projectiles at twice the maximum laanch speed you found in part (a). At what maximum angle above the horizontal whocld you 1aunch the projectile? (c) How fur (in terms of $h$ ) from the launcher does the projcctile in part (b) land?

Donald Albin
Donald Albin
Numerade Educator
07:38

Problem 58

Kicling a Ficld Goal, In U.S. football, after a touchdown the team has the opportunity to earn one more point by kicking the ball over the bar between the goal posts. The bar is $10.0 \mathrm{ft}$ above the ground, and the ball is koked from ground level, $36.0 \mathrm{ft}$ horizoutally from the bar (Iig. 3.48). Football regulations are stated in Einglish units, but convert to SI urits for this problem, (a) There is a minimum angle above the ground such that if the ball is launched below this angle, it can never clear the bar, no matter how fast it is Kicked. What is this angle? (b) If the ball is kicked at $45.0^{\circ}$ above the horizontal, what must its initial speed be if it to just clear the bar? Express your answer in $\mathrm{m} / \mathrm{s}$ and $\mathrm{km} / \mathrm{h}$.

Mitchel Vereen
Mitchel Vereen
Numerade Educator
10:25

Problem 59

A projectile is launched with speed $v_{0}$ ut an angle $\alpha_{0}$ above the horizontal. The Iminch point is a height $h$ uhove the ground.
(a) Show that if air resistance is ignored, the horizontal distance that the projcctile travels before striking the ground is
$$
x=\frac{v_{0} \cos \alpha_{0}}{g}\left(v_{0} \sin \alpha_{0}+\sqrt{v_{0}^{2} \sin ^{2} \alpha_{0}+2 g h}\right)
$$
Verify that if the launch point is at ground level so that $h=0$, this is equill to the borizontal range $R$ found in Exumple 3.8. (b) For the cnse where $v_{\mathrm{b}}=10 \mathrm{~m} / \mathrm{s}$ and $h=5.0 \mathrm{~m}$, graph $x$ as a function of 1mnch angle $\alpha_{0}$ for values of $\alpha_{0}$ fram $0^{\text {s }}$ to $90^{\circ}$. Your graph should show that $x$ is zero if $\alpha_{0}=90^{\circ}$, but $x$ is nonzero if $\alpha_{0}=0$ explain why this is so. (c) We saw in Example $3.8$ that for a projectile that lands at the same belght from which it is lannched, the horizontal mnge is maximum for $a_{0}=45^{\circ}$. For the case graphed in part $(b)$, is the angle for maximum horizoatal distance equal to, less than, or greater than $45^{\circ}$ ? (This is a general result for the situation whero a moicctile is launched from a point higher than where it lands.)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
09:12

Problem 60

Look Out! A snowball Figure 3.49 Problem rolls off a bam roof that slopes downward nt an angle of $40^{\circ}$ (Fig. 3.49). The edge of the roof is $14.0 \mathrm{~m}$ above the ground, and the snowball has a speed of $7.00 \mathrm{~m} / \mathrm{s}$ as it rolls off the roof. Ignore air resistance.
(a) How far from the edge of the barn does the snowball strike the ground if it doesn't strike anything else while falling? (b) Draw $x-t, y-t$, $v_{x}-t$, and $v_{y}-t$ graphs for the motion in part (a). (c) $\mathrm{A}$ man $1.9$ m tall is standing $4.0 \mathrm{~m}$ from the edge of the bum. Will he be hit by the snowball?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:09

Problem 61

(a) Prove that $=$ projectile lanched at angle $\alpha_{0}$ has the same horizontal range as cae Iaunched with the same speed at angle $\left(90^{\circ}-\alpha_{0}\right)$. (b) A frog jumps at a sipeed of $2.2 \mathrm{~m} / \mathrm{s}$ and Lands $25 \mathrm{~cm}$ fromits starting point. At which angles above the horizontal could it have jumped?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
25:37

Problem 62

On the Flying Trapeze. A Figure $3.50$ Problem $3.62$ new circus act is called the 'Texas Tumblers. Lovely Mary Belle swings from a trupeae, projects hersclf at an angle of $53^{n}$, and is supposed to be caught by Joe Bob, whose hands are $6.1 \mathrm{~m}$ above and $8.2 \mathrm{~m}$ boritontally from her lnunch point (Fig. 3.50). You can ignore air resistance. (a) What initial speed $v_{0}$ must Mary Belle havo just to reach Joe Bob? (b) For the initial speed calculuted in part (a), what are the magnitude and direotion of her velocity when Mary Belle reaches Joe Bob? (c) Assuming that Mary Belle has the initial speod calculatod in part (a), druw $x-t, y-t, v_{x}-t$, and $v_{y}-t$ graphs showing the motion of both tumblers. Your graphs should show the motion up until the point where Mary Belle reaches Joe Bob, (d) The night of their debut performance, Joe Bob misses ber completely as she ffies paut. How far horizontally does Mary Belle travel, from ber initial luunch point, before landing in the safety net $8.6 \mathrm{~m}$ below her starting point?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
05:10

Problem 63

Leaping the River II. A physics professor did daredevil stunts in his spare time. His last stunt was an atternpt to jump acnoss a river on a motorcycle (Fig. $3.51$ ). The takooft ramp was inclined at $53.0^{\circ}$, the river was $40.0 \mathrm{~m}$ wide, and the far bank was 15.0 m lower than the top of the rurup. The river itself was $100 \mathrm{~m}$ below the ramp. You can ignore air resistance. (a) What should his spoed have been at the top of the rarmp to have just made it to the cdge of the far banl?? (b) If his spced was only half the value found in (a), where ?id he land?

Averell Hause
Averell Hause
Carnegie Mellon University
05:58

Problem 64

A rock is thrown from the roof of a building with a velocity $v_{0}$ at an angle of $\alpha_{0}$ frotri the horizonsal. The building has belght $h$. You can ignore air resistance. Calculate the magnitude of the velocity of the rock just before it strikes the groand, and show that this speed is independent of $a_{0}$ -

Mitchel Vereen
Mitchel Vereen
Numerade Educator
09:20

Problem 65

A $5500-\mathrm{kg}$ cart carrying a vertical rocket launcher moves to the right at s constant speed of $30.0 \mathrm{~m} / \mathrm{s}$ along a horizontal track. It Iauncbes a $45.0$ -kg rocket vertically upward with an initial specd of $40.0 \mathrm{~m} / \mathrm{s}$ relative to the cart. (a) How high will the rocket go?
(b) Where, relative to the cart, will the rocket land? (c) How far does the cart move while tho rocket is in the air? (d) At what anglo, relative to the horizontal, is the rocket traveling jost as it leaves the cart, as measured by an observer at rest on the ground? (e) Sketch the rocket's trajectory as seen by an observer (i) stationary on the cant and (ii) stationary on the ground.

Guilherme Barros
Guilherme Barros
Numerade Educator
05:35

Problem 66

A $27 \mathrm{~kg}$ ball is thrown upward with an initial speed of $20.0 \mathrm{~m} / \mathrm{s}$ from the cdge of a $45.0$ -m-high cliff. At the instant the ball is thrown, a woman starts running away from the base of the cliff with a constant kpeed of $6.00 \mathrm{~m} / \mathrm{s}$. The woman runs in a straight line on level ground, and air resistance acting on the bell can be ignored. (a) At what angle above the horizontal should the ball be thrown so thant the runner will catch it just before it hits the ground, and how far does the woman run before she cutches the ball? (b) Carefully sketch the ball's trijectory as viewed by (i) a person at rest on the ground end (ii) the runner.

Zachary Warner
Zachary Warner
Numerade Educator
View

Problem 67

A $76.0-\mathrm{kg}$ boulder is rolling horizontally at the top of a vertical cliff that is $20 \mathrm{~m}$ above the surface of a lake, as shown in Fig, 3.52. The top of the vertical face of a dam is located $100 \mathrm{~m}$ from the foot of the cliff, with the top of the dam level with the surface of the water in the lake. A level plain is $25 \mathrm{~m}$ below the top of the dam. (a) What must be the minimura speed of the rock just as it leaves the cliff so it will trivel to the phain without striking the dam? (b) How far from the foot of the dam does the rock hit the plain?

Dan Ni
Dan Ni
Numerade Educator
08:00

Problem 68

Tossing Your Lanch. Heariettu is going off to her physics class, jogging down the sidewalk at $3.05 \mathrm{~m} / \mathrm{s}$. Her hushend Bruce suddenly realines that she left in such a hurry that she forgot her Iunch of bagels, so be runs to the window of their apartment, which is $43.9 \mathrm{~m}$ above the street level and directly above the sidewalk, to throw them to ber. Bruce throws thern horizontally $9.00 \mathrm{~s}$ after Henriette has passed below the window, and she catches them on the run. You can ignore tir reststance. (a) With what initial speed must Bruce throw the bagels so Henrietta can catch them just before they hit the ground? (b) Where is Henrietta when she catches the hagels?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:39

Problem 69

Two tanks are engaged in a training exercise on level ground. Tbe first tank fires a paint-fitled training round with a muzzle speed of $250 \mathrm{~m} / \mathrm{s}$ at $10.0^{\circ}$ ubove the horizontal while advancing toward the second tank with a speed of $15.0 \mathrm{~m} / \mathrm{s}$ rolntive to the ground. The second tank is retrcating at $35.0 \mathrm{~m} / \mathrm{s}$ relative to the ground, but is hit by the shell, Yoa can ignore air resistance and assome the shell hits at the same height above ground from which it was fired. Find the distance berween the tanks (a) when the round was fint fired and (b) at the time of impact.

Ajay Singhal
Ajay Singhal
Numerade Educator
02:09

Problem 70

Bangt A student sits atop a platform a distance $h$ above the ground. He throws a large firecracker horizontally with a speed $v$. However, a wind blowing perallel to the ground gives the firecracker a conatunt horizontal acceleration with magnitude $a$. This results in the firccracker reaching the ground directly under the student. Determine the height $h$ in terms of $v, a$, and $g$. You can ignore the effect of air resistance on the vertical motion.

Zachary Warner
Zachary Warner
Numerade Educator
15:32

Problem 71

A rocket is launched vertically from rest with a constant upward acceleration of $1.75 \mathrm{~m} / \mathrm{s}^{2}+$ Suddenly $22.0 \mathrm{~s}$ nfter launch, an unneeded fucl tank is jettisoned by shooting it away from the rocket, A crew member riding in the rocket measures that the initial specd of the tank is $25.0 \mathrm{~m} / \mathrm{s}$ and that it moves perpendicular to the rocket's path. The fuel tank feels no appreciable air resistance and feels only the force of gravity once it leaves the rocket, (a) How fast is the rocket moving at the instant the foel tank is jettisonod?
(b) What are the borizontal and vertical components of the fucl tank's vclocity just as it is jcttisoncd as measurod by (i) a crew member in the rocket and (ii) a tochnician standing on the ground?
(c) At what angle with respect to the horkoutal does the jetrisoned
(i) a crew member in the fuel tank initially move, is measured by rocket and (ii) a technician standing on the ground? (d) What maximum beight above the launch pad does the jcttisoned tank reach?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
11:16

Problem 72

When it is $145 \mathrm{~m}$ above tho ground, a rocket traveling vertically upward at a constunt $8.50 \mathrm{~m} / \mathrm{s}$ relative to the ground launches a secondary rocket at a speed of $12.0 \mathrm{~m} / \mathrm{s}$ at an angle of $53.0^{\circ}$ above the horizontal, both quantities being measured by an astronaut sitting in the rocket. Air resistance is too small fo worry about.
(a) Just as the secondary rocket is launched, what are the horizontal and vertical components of its velocity relative to (i) the astronaut sitting in the rocket and (ii) Mission Control on the ground?
(b) Find the initial speed and launch angle of the secondary rocket as mealured by Mission Control. (c) What taasimum height ubove the ground does the secondary rocket reach?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:50

Problem 73

In a Fourth of July celcbration, a firecracker is lumncbed from ground level with an initial velocity of $25.0 \mathrm{~m} / \mathrm{s}$ at $30.0^{\circ}$ from the vertical. At its maximum height it explodes in a starburst into many fragments, two of which travel forward initially at $20.0 \mathrm{~m} / \mathrm{s}$ at $\pm 53.0^{\circ}$ with respect to the horizontal, both quantities measured relative to the original firecracker jinst before it exploded. Witb what angles with respect to the borizontal do the two fragments initially movo right after the explosion, as measured by a spectator standing on the ground?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:41

Problem 74

In an actioa-adventure flim, the hero is supposed to throw a grenade from his car, which is going $90.0 \mathrm{~km} / \mathrm{b}$, to his enemy's eat, whick is going $110 \mathrm{~km} / \mathrm{h}$. The enemy"s car is $15.8 \mathrm{~m}$ in front of the hero's when he lets go of the grenade. If the hero throws the grenade so its initial velocity relative to him is at an angle of $45^{\circ}$ above the horizontal, what should the magnitude of the initial velocity be? The cars are both traveling in the same dircction on a level road. You can ignore air resistance. Find the magnitude of the velocity both relative to the hero and relative to the earth.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
09:01

Problem 75

A rock tied to a rope moves in the $x y$ plane. Its coordinates are given as functions of time by $x(t)=$ Rcosor $\quad y(t)=$ Rsint where $R$ and $\omega$ sre constants. (a) Show that the rock's ?istance from the origin is constant and equal to $R$ - that is, that its path is a circle of radius $R .$ (b) Show that at every point the rock's velocity is perpendicular to its poxition vector. (c) Show that the rock's accclerution is always opposite in direction to its position vector and has magnitnde $\omega^{2} R$. (d) Show that the magnimde of the rock's velocity is constant and cqual to $\omega R .$ (e) Combine the results of parts (c) and (d) to show thist the rock's acceleration has constant magnitude $v^{2} / R$.

Guilherme Barros
Guilherme Barros
Numerade Educator
16:14

Problem 76

A $400.0-\mathrm{m}$ -wide river flows from west to cast ut $300 \mathrm{~m} / \mathrm{min}$. Your boat moves at $100.0 \mathrm{~m} / \mathrm{min}$ relntive to the water no matter which direction you point it. To cross this river, you start from a dock at point $A$ on the south benk. There is a boet landing directly oppesite at point $B$ on the north bank, and also one at point $C, 75.0 \mathrm{~m}$ downstream from $B$ (Fig. 3.53). (a) Where on the north stare will you land if you point your boat perpendicolne to the water carreat, and what distance will you have traveled? (b) If you initially aim your boatdirectly toward point $C$ and do not change that bearing relative to the shore, where on the north shore will you land? (c) To reach point C: (i) at what bearing must you aim your boat, (ii) how long will it take to cross the river, (iii) what distance do you travel, nnd (iv) and what is the specd of your boat as measured by an observer standing on the river bank?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:34

Problem 77

Cyclaid. A particle moves in the $x y$ -plane. Its coordinates are given as functions of time by
$$
x(t)=R(\text { wot - sincot }) \quad y(t)=R(1-\cos \omega t)
$$
where $R$ and w are constants. (a) Sketch the trajectory of the particle. This is the trajectory of a point on the rim of a wheel that is rolling at a constant speed on a horizontal surface. The curve traced out by such a point as it moves through space is called a cycloid) (b) Determine the velocity componeats and the acceleraton components of the particle at any time $t$. (c) At which times is the purticle momerturily at rest? What are the coordinates of the particle at these times? What are the magnitude and direction of the acceleration at these times? (d) Does the magnitude of the acceleration depend on time? Cormpare to uniform circular motioa.

Keshav Singh
Keshav Singh
Numerade Educator
06:16

Problem 78

A projectile is fired from point $A$ at an angle above the hor'zontal. At its highest point, after having traveled a horizontal distance $D$ from its lamnch point, it saddenly explodes into two identical fragments thut truvel borizontally with equal but opposite velocities as measured relative to the projectile just before it exploded. If one fragment lands back at point $A$, how far from $A$ (in torms of $D$ ) docs the other fragment land?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
07:03

Problem 79

A laboratory centrifuge on earth makes $n$ rpm (rev/min) and produces an accelention of $5.00 \mathrm{~g}$ at its outer end. (a) Whet is the accelentioa (in $g^{\prime}$ s) at a point halfway out to the end? (b) This centrifuge is now used in a space capsule on the planet Mercury, where guenug is $0.378$ what it is on earth. How many rpm (in terms of $n$ ) should it make to produce 5 gurmany at its outer end?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:56

Problem 80

When a train's velocity is $120 \mathrm{~m} / \mathrm{s}$ eastward, raindrops that are falling vertically with respect to the earth make traces that are inclined $30.0^{\circ}$ to the vertical on the windows of the train. (a) What is the borizontnl component of a drop's velocity with respect to the earth? With respect to the train? (b) What is the magnitude of the velocity of the raindrop with respect to the carth? With respect to the train?

Vishal Gupta
Vishal Gupta
Numerade Educator
09:09

Problem 81

An airplane pilot sets a counpass counse due west and mainpains an airspeed of $220 \mathrm{~km} / \mathrm{h}$. After flying for $0.500 \mathrm{~h}$, she finds herself over u town $120 \mathrm{~km}$ west and $20 \mathrm{~km}$ south of her starting point. (a) Find the wind velocity (magnitude and direction). (b) If the wind velocity is $40 \mathrm{~km} / \mathrm{h}$ due south, in what direction should the pilot set her courne to trivel ?e west? Use the same airspeed of $220 \mathrm{~km} / \mathrm{h}$.

Averell Hause
Averell Hause
Carnegie Mellon University
05:32

Problem 82

An clevator is moving upward at a constant speed of $2.50 \mathrm{~m} / \mathrm{s}$. A bolt in the elevator ceiling $3.00 \mathrm{~m}$ nbove the elevator ftoor works loose and falls. (a) How long does it take for the bolt to fall to the elevator floor? Whit is the speed of the bolttust as it hits the elcvator floot (b) accoruling to an observer in the clevator?
(c) Acconding to an observer standing on one of the floor landings of the building? (d) According to the observer in purt (c), what distance did the bolt travel between the ceiling and the floor of the elevator?

Zachary Warner
Zachary Warner
Numerade Educator
09:08

Problem 83

Suppose the clevator in Problem $3.82$ sturts from rest and maintains a constant upward acceleration of $4,00 \mathrm{~m} / \mathrm{s}^{2}$, and the bolt falls out the instant the elevater begins to move. (a) How long does it take for the bolt to reach the floor of the elevator? (b) Just as it reaches the floor, how fast is the bolt moving acoording to at observer (i) in the clevator? (ii) Standing on the floor landings of the building? (c) According to cach observer in part (b), how far has the bolt traveled betweea the ceiting and floor of the elevator?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:12

Problem 84

City $A$ Hes directly west of city $B$. When there is no wind, an airtioermakes the 5550 -km round-trip fight between them in $6.60 \mathrm{~h}$ of flying time while traveling at the sarne speed in both directions. When a strong, steady $225-\mathrm{km} /$ h wind is blowing from west to east and the airliner has the sume airspeed as before, how long will the trip take?

Darren Wilson
Darren Wilson
Numerade Educator
03:06

Problem 85

In a Wodd Cup soccer match, Juan is running due north toward the goal with a speed of $8.00 \mathrm{~m} / \mathrm{s}$ relative to the ground, A tearmmate passes the ball to hitn. The bell has a vpeed of $12.0 \mathrm{~m} / \mathrm{s}$ and is moving in a dircction of $37.0^{\circ}$ enst of north, relative to the ground. What are the magnitode and dircction of the ball's velocity relative to Juan?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:27

Problem 86

A man is riding on a flatcar traveling at a constant speed of $9.10 \mathrm{~m} / \mathrm{s}$ (Fig. 3.54). He wishos to throw at bell through a stationary hoop $4.90 \mathrm{~m}$ above the beight of his hands in such a manner that the ball will move borizoatally as it passes through the hoop. He throws the ball with a speed of $10.8 \mathrm{~m} / \mathrm{s}$ with respect to himsclf. (a) What must the vertical component of the initlal velocity of the ball be? (b) How many seconds after he releases the ball will it pass throegh the hoop? (c) At what horizontal distance in front of the hoop must he release the bull? (d) When the ball leaves the man's hands, what is the direction of its velocity rolative to the frame of refcrence of the flatcar? Relative to the frame of rcference of an observer standing on the ground?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:49

Problem 87

A shotgun fires a large number of pellets upward, with some pellets traveling very nearly vertically and others as much as $1.0^{\circ}$ from the vertical. Assame that the initial speed of the pellets is uniformly $150 \mathrm{~m} / \mathrm{s}$, and ignore air resistance. (a) Within what radius from the point of firing will the pellets land? (b) If there are 1000 pellets, and they fall in a uniform distribution over a circle with the radius calculaled in part (a), what is the probability that at least one pellet will fall on the head of the person who fires the shotgon?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:32

Problem 88

A projectile is thrown from a point $P$. It moves in such a way That its distance from $P$ is alwuys increasing. Find the maximum angle above the horizontal witb which the projectile could have been thrown. You can ignore air resistance.

Averell Hause
Averell Hause
Carnegie Mellon University
09:54

Problem 89

Projectile Motion on un Incline I. A baseball is given an initial velocity with magnitude $v_{0}$ at an angle $\phi$ above the surface of an incline, which is in turn inclined at an angle $\theta$ abeve the horizontal (Fig. 3.55) (a) Calcalate the distance, measurod along the incline, fram the launch point to whene the baseball strikes the incline. Yoer answer will be in terms of $v_{0 \mathrm{v}} \& \theta$, and $\phi .$ (b) What angle \phi gives the maximual range, measured aloog the incline? (Note: You might be interested in the three different methods of solution presented by L R Lapidus in Amer Jour of Phys., Vol. 51
(1983), pp. 806 and 847 . Sec also H. A. Buckmaster in Amer. Jour. of Phys, Vol. 53 (1985), pp. $638-641$, for a thorough study of this and some similar problems.)

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
13:26

Problem 90

Refer to Challenge Problem 3.89. (a) An archer on ground that has a constant upward slope of $30.0^{\circ}$ tims at n target $600 \mathrm{~m}$ farther up the incline. The arrow in the bow and the bull's-eye ut the center of the target are each $1.50 \mathrm{~m}$ above the ground. The initial velocity of the amow just after it Ieaves the bow has magnitude $32.0 \mathrm{~m} / \mathrm{s}$. At what angle above the horizontal sbould the archer aim to hit the bull's-eye? If there are two such angles, calculate the smaller of the two. Yoa might have to solve the cquation for the angle by iteration- that is, by trial nod error. How does the angle compare to that reçuired when the ground is level, with 0 slope? (b) Repeat the above for ground that has a constant downward slope of $30.0^{\circ} .$

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:10

Problem 91

For no apparcat reason, a poodle is running at a constant speed of $v=5.00 \mathrm{~m} / \mathrm{s}$ in a circle with radius $R=2.50 \mathrm{~m}$. Let $\vec{v}_{1}$ be the velocity vector at time $t_{1}$, and let $\vec{v}_{2}$ be the velocity vector at time $t_{2}$. Consider $\Delta \vec{v}=\vec{v}_{2}-\vec{v}_{1}$ and $\Delta t=t_{2}-t_{1}$. Rocall that $\vec{d}_{\mathrm{av}}=\Delta \ddot{v} / \Delta t .$ For $\Delta t=0.5 \mathrm{~s}, 0.1 \mathrm{~s}$, and $0.05 \mathrm{~s}$, calculate the $\mathrm{mag}$nitude (to four significant figures) and direction (relative to $\left.\vec{\nabla}_{1}\right)$ of the average acceleration $\vec{d}_{\text {sx }}$. Compare your results to the general expresion for the instantaneous acceleration $\vec{d}$ for uniform circuLar motion that is derived in the text.

Shelby Mohamed
Shelby Mohamed
Numerade Educator
11:31

Problem 92

A rocket designed to place small payloads into orbit is carried to an altitude of $12.0 \mathrm{~km}$ above sea level by a converted airliner. When the airtiner is fying in a stralght line at a constant speed of $850 \mathrm{~km} / \mathrm{h}$, the rocket is droppod. After the drop, the airIiner maintains the same altitude and spced and continues to fiy in a straight line, The rocket falls for a brief time, after which its rocket motor urns on, Once its rocket motor is on, the combined effects of thrust and grivity give the rocket a constant scceleration of magnitude $3.00 \mathrm{~g}$ dirccted at an angle of $30.0^{\circ}$ above the horjzonta], For reasons of safety, the rocket should be at least $1.00 \mathrm{~km}$ in front of the airliner when it climbs through tho airliner's altitwde. Your job is to determine the minimnm time that the rocker must fall before its engine starts. You can ignore air resistance. Your answer should include (i) s diagram sbowing the fight path of both the rocket and the airliner, Inbeled at several points witi vectors for their velocities and accelerations; (ii) an $x-t$ grapl showing the motions of both the rocket and the airliner; and (ii) $y-t$ graph sbowing the motions of both the rocket and the airline:
In the diagram and the grupbs, indicate when the rocket is dropped when the rocket motor tarns on, and when the rocket climb. through the altitude of the airliner.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
06:06

Problem 93

Two students are canocing on a rivec. While heading upstream, they aceldentally drop an empry bottle overtoard. They

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator