College Physics: A Strategic Approach

Randall D. Knight, Brian Jones, Stuart Field

Chapter 3

Vectors and Motion in Two Dimensions - all with Video Answers

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Chapter Questions

Problem 1

Trace the vectors in Figure P3.1 onto your paper. Then use graphical methods to draw the vectors (a) $\vec{A}+\vec{B}$ and (b) $\vec{A}-\vec{B}$

Vishal Gupta

Vishal Gupta

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Problem 2

Trace the vectors in Figure $\mathrm{P} 3.2$ onto your paper. Then use graphical methods to draw the vectors
(a) $\vec{A}+\vec{B}$ and
(b) $\vec{A}-\vec{B}$

William Dunkerton

William Dunkerton

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Problem 3

Trace the vectors in Figure $\mathrm{P} 3.3$ onto your paper. Then draw the vector $\vec{C}$ such that $\vec{A}+\vec{B}+\vec{C}=0$.

William Dunkerton

William Dunkerton

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Problem 4

Two vectors $\vec{A}$ and $\vec{B}$ are at right angles to each other. The magnitude of $\vec{A}$ is $1 .$ What should be the length of $\vec{B}$ so that the magnitude of their vector sum is $2 ?$

William Dunkerton

William Dunkerton

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Problem 5

A position vector with magnitude $10 \mathrm{m}$ points to the right and up. Its $x$ -component is $6.0 \mathrm{m}$. What is the value of its $y$ -component?

William Dunkerton

William Dunkerton

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Problem 6

A velocity vector $40^{\circ}$ above the positive x-axis has a $y$ -component of $10 \mathrm{m} / \mathrm{s} .$ What is the value of its $x$ -component?

William Dunkerton

William Dunkerton

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Problem 7

A cannon tilted upward at $30^{\circ}$ fires a cannonball with a speed of $100 \mathrm{m} / \mathrm{s}$. At that instant, what is the component of the cannonball's velocity parallel to the ground?

William Dunkerton

William Dunkerton

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Problem 8

a. What are the $x$ - and $y$ -components of vector $\vec{E}$ of Figure $\mathrm{P} 3.8$ in terms of the angle $\theta$ and the magnitude $E ?$
b. For the same vector, what are the $x$ - and $y$ -components in terms of the angle $\phi$ and the magnitude $E ?$

William Dunkerton

William Dunkerton

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Problem 9

Draw each of the following vectors, then find its $x$ - and $y$ -components.
a. $\vec{d}=\left(100 \mathrm{m}, 45^{\circ}\right.$ below $+x$ -axis $)$
b. $\vec{v}=\left(300 \mathrm{m} / \mathrm{s}, 20^{\circ}\right.$ above $+x$ -axis $)$
c. $\vec{a}=\left(5.0 \mathrm{m} / \mathrm{s}^{2},-y\right.$ -direction $)$

William Dunkerton

William Dunkerton

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Problem 10

Draw each of the following vectors, then find its $x$ - and $y$ -components.
a. $\vec{d}=\left(2.0 \mathrm{km}, 30^{\circ}\right.$ left of $+y$ -axis )
b. $\vec{v}=(5.0 \mathrm{cm} / \mathrm{s},-x$ -direction $)$
c. $\vec{a}=\left(10 \mathrm{m} / \mathrm{s}^{2}, 40^{\circ}\right.$ left of $-y$ -axis $)$

William Dunkerton

William Dunkerton

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Problem 11

Each of the following vectors is given in terms of its $x$ - and $y$ -components. Draw the vector, label an angle that specifies the vector's direction, then find the vector's magnitude and direction.
a. $v_{x}=20 \mathrm{m} / \mathrm{s}, v_{y}=40 \mathrm{m} / \mathrm{s}$
b. $a_{x}=2.0 \mathrm{m} / \mathrm{s}^{2}, a_{y}=-6.0 \mathrm{m} / \mathrm{s}^{2}$

William Dunkerton

William Dunkerton

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Problem 12

Each of the following vectors is given in terms of its $x$ - and $y$ -components. Draw the vector, label an angle that specifies the vector's direction, then find the vector's magnitude and direction.
a. $v_{x}=10 \mathrm{m} / \mathrm{s}, v_{y}=30 \mathrm{m} / \mathrm{s}$
b. $a_{x}=20 \mathrm{m} / \mathrm{s}^{2}, a_{y}=10 \mathrm{m} / \mathrm{s}^{2}$

William Dunkerton

William Dunkerton

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Problem 13

A wildlife researcher is tracking a flock of geese. The geese fly $4.0 \mathrm{km}$ due west, then turn toward the north by $40^{\circ}$ and $\mathrm{fly}$. another $4.0 \mathrm{km} .$ How far west are they of their initial position? What is the magnitude of their displacement?

Anand Jangid

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Problem 14

Jack and Jill ran up the hill at $3.0 \mathrm{m} / \mathrm{s}$. The horizontal component of Jill's velocity vector was $2.5 \mathrm{m} / \mathrm{s}$
a. What was the angle of the hill?
b. What was the vertical component of Jill's velocity?

William Dunkerton

William Dunkerton

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Problem 15

Josh is climbing up a steep $34^{\circ}$ slope, moving at a steady 0.75 $\mathrm{m} / \mathrm{s}$ along the ground. How many meters of elevation does he gain in one minute of this climb?

William Dunkerton

William Dunkerton

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Problem 16

You begin sliding down a $15^{\circ}$ ski slope. Ignoring friction and air resistance, how fast will you be moving after $10 \mathrm{s} ?$

William Dunkerton

William Dunkerton

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Problem 17

$\mathrm{A}$ car traveling at $30 \mathrm{m} / \mathrm{s}$ runs out of gas while traveling up a $5.0^{\circ}$ slope. How far will it coast before starting to roll back down?

Anand Jangid

Numerade Educator

Problem 18

In the Soapbox Derby, young participants build nonmotorized cars with very lowfriction wheels. Cars race by rolling down a hill. The track at Akron's Derby Downs, where the national championship is held, begins with a $55-\mathrm{ft}-\mathrm{long}$ section tilted $13^{\circ}$ below horizontal.
a. What is the maximum possible acceleration of a car moving down this stretch of track?
b. If a car starts from rest and undergoes this acceleration for the full $55 \mathrm{ft}$, what is its final speed in $\mathrm{m} / \mathrm{s}$ ?

William Dunkerton

William Dunkerton

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Problem 19

$A$ piano has been pushed to the top of the ramp at the back of a moving van. The workers think it is safe, but as they walk away, it begins to roll down the ramp. If the back of the truck is $1.0 \mathrm{m}$ above the ground and the ramp is inclined at $20^{\circ},$ how much time do the workers have to get to the piano before it reaches the bottom of the ramp?

William Dunkerton

William Dunkerton

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Problem 20

In the winter sport of bobsledding, athletes push their sled along a horizontal ice surface and then hop on the sled as it starts to careen down the steeply sloped track. In one event, the sled reaches a top speed of $9.2 \mathrm{m} / \mathrm{s}$ before starting down the initial part of the track, which is sloped downward at an angle of $6.0^{\circ}$. What is the sled's speed after it has traveled the first $100 \mathrm{m} ?$

William Dunkerton

William Dunkerton

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Problem 21

A car goes around a corner in a circular arc at constant speed. Draw a motion diagram including positions, velocity vectors, and acceleration vectors.

Vishal Gupta

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Problem 22

Figure 3.33 showed the motion diagram for a rider on a Ferris wheel that was turning at a constant speed. The inset to the figure showed how to find the acceleration vector at the rightmost point. Use a similar analysis to find the rider's acceleration vector at the leftmost position of the motion diagram, then at one of the highest positions and at one of the lowest positions. Use a ruler so that your analysis is accurate.

William Dunkerton

William Dunkerton

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Problem 23

Show partial motion diagrams. For each:
a. Complete the motion diagram by adding acceleration vectors.
b. Write a physics problem for which this is the correct motion. Be imaginative! Don't forget to include enough information to make the problem complete and to state clearly what is to be found.

William Dunkerton

William Dunkerton

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Problem 24

Show partial motion diagrams. For each:
a. Complete the motion diagram by adding acceleration vectors.
b. Write a physics problem for which this is the correct motion. Be imaginative! Don't forget to include enough information to make the problem complete and to state clearly what is to be found.

William Dunkerton

William Dunkerton

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Problem 25

By choosing one of the eight labeled acceleration vectors or selecting option I: $\vec{a}=\overrightarrow{0}$.
At this instant, the particle has steady speed and is curving to the right. What is the direction of its acceleration?

William Dunkerton

William Dunkerton

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Problem 26

By choosing one of the eight labeled acceleration vectors or selecting option I: $\vec{a}=\overrightarrow{0}$.
At this instant, the particle is speeding up and curving upward. What is the direction of its acceleration?

William Dunkerton

William Dunkerton

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Problem 27

A ball is thrown horizontally from a 20-m-high building with a speed of $5.0 \mathrm{m} / \mathrm{s}$.
a. Make a sketch of the ball's trajectory.
b. Draw a graph of $v_{x},$ the horizontal velocity, as a function of time. Include units on both axes.
c. Draw a graph of $v_{y},$ the vertical velocity, as a function of time. Include units on both axes.
d. How far from the base of the building does the ball hit the ground?

William Dunkerton

William Dunkerton

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Problem 28

A ball with a horizontal speed of $1.25 \mathrm{m} / \mathrm{s}$ rolls off a bench $1.00 \mathrm{m}$ above the floor.
a. How long will it take the ball to hit the floor?
b. How far from a point on the floor directly below the edge of the bench will the ball land?

William Dunkerton

William Dunkerton

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Problem 29

A pipe discharges storm water into a creek. Water flows hor- izontally out of the pipe at $1.5 \mathrm{m} / \mathrm{s},$ and the end of the pipe is $2.5 \mathrm{m}$ above the creek. How far out from the end of the pipe is the point where the stream of water meets the creek?

William Dunkerton

William Dunkerton

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Problem 30

Un a day when the water is flowing relatively gently, water in the Niagara River is moving horizontally at $4.5 \mathrm{m} / \mathrm{s}$ before shooting over Niagara Falls. After moving over the edge, the water drops $53 \mathrm{m}$ to the water below. If we ignore air resistance, how much time does it take for the water to go from the top of the falls to the bottom? How far does the water move horizontally during this time?

William Dunkerton

William Dunkerton

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Problem 31

A running mountain lion can make a leap $10.0 \mathrm{m}$ long, reaching a maximum height of $3.0 \mathrm{m}$.
a. What is the speed of the mountain lion just as it leaves the ground?
b. At what angle does it leave the ground?

Nishant Kumar

Numerade Educator

Problem 32

A rifle is aimed horizontally at a target $50 \mathrm{m}$ away. The bullet hits the target $2.0 \mathrm{cm}$ below the aim point.
a. What was the bullet's flight time?
b. What was the bullet's speed as it left the barrel?

William Dunkerton

William Dunkerton

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Problem 33

A gray kangaroo can bound across a flat stretch of ground with each jump carrying it $10 \mathrm{m}$ from the takeoff point. If the kangaroo leaves the ground at a $20^{\circ}$ angle, what are its (a) takeoff speed and (b) horizontal speed?

William Dunkerton

William Dunkerton

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Problem 34

On the Apollo 14 mission to the moon, astronaut Alan Shepard hit a golf ball with a golf club improvised from a tool. The free-fall acceleration on the moon is $1 / 6$ of its value on
earth. Suppose he hit the ball with a speed of $25 \mathrm{m} / \mathrm{s}$ at an angle $30^{\circ}$ above the horizontal.
a. How long was the ball in flight?
b. How far did it travel?
c. Ignoring air resistance, how much farther would it travel on the moon than on earth?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 35

Emily throws a soccer ball out of her dorm window to Allison, who is waiting below to catch it. If Emily throws the ball at an angle of $30^{\circ}$ below horizontal with a speed of $12 \mathrm{m} / \mathrm{s}$, how far from the base of the dorm should Allison stand to catch the ball? Assume the vertical distance between where Emily releases the ball and Allison catches it is $6.0 \mathrm{m}$.

Chasen Shaw

Numerade Educator

Problem 36

A soccer player takes a free kick from a spot that is $20 \mathrm{m}$ from the goal. The ball leaves his foot at an angle of $32^{\circ},$ and it eventually hits the crossbar of the goal, which is $2.4 \mathrm{m}$ from the ground. At what speed did the ball leave his foot?

Prabhat Tyagi

Numerade Educator

Problem 37

Racing greyhounds are capable of rounding corners at very high speeds. A typical greyhound track has turns that are 45-m-diameter semicircles. A greyhound can run around these turns at a constant speed of $15 \mathrm{m} / \mathrm{s}$. What is its acceleration in $\mathrm{m} / \mathrm{s}^{2}$ and in units of $g ?$

William Dunkerton

William Dunkerton

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Problem 38

To withstand "g-forces" of up to 10 g's, caused by suddenly pulling out of a steep dive, fighter jet pilots train on a "human centrifuge." $10 \mathrm{g}$ 's is an acceleration of $98 \mathrm{m} / \mathrm{s}^{2}$. If the length of the centrifuge arm is $12 \mathrm{m},$ at what speed is the rider moving when she experiences 10 g's?

William Dunkerton

William Dunkerton

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Problem 39

The moon completes one (circular) orbit of the earth in 27.3 days. The distance from the earth to the moon is $3.84 \times 10^{8} \mathrm{m} .$ What is the moon's centripetal acceleration?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 40

In a roundabout (or traffic circle), cars go around a 25-m-diameter circle. If a car's tires will skid when the car
experiences a centripetal acceleration greater than $0.60 g,$ what is the maximum speed of the car in this roundabout?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 41

A particle rotates in a circle with centripetal acceleration $a=8.0 \mathrm{m} / \mathrm{s}^{2} .$ What is $a$ if
a. The radius is doubled without changing the particle's speed?
b. The speed is doubled without changing the circle's radius?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 42

Entrance and exit ramps for freeways are often circular stretches of road. As you go around one at a constant speed, you will experience a constant acceleration. Suppose you drive through an entrance ramp at a modest speed and your acceleration is 3.0 $\mathrm{m} / \mathrm{s}^{2} .$ What will be the acceleration if you double your speed?

William Dunkerton

William Dunkerton

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Problem 43

A peregrine falcon in a tight, circular turn can attain a centripetal acceleration 1.5 times the free-fall acceleration. If the falcon is flying at $20 \mathrm{m} / \mathrm{s},$ what is the radius of the turn?

Nicholas Mogoi

Numerade Educator

Problem 44

An airplane cruises at $880 \mathrm{km} / \mathrm{h}$ relative to the air. It is flying from Denver, Colorado, due west to Reno, Nevada, a distance of $1200 \mathrm{km},$ and will then return. There is a steady $90 \mathrm{km} / \mathrm{h}$ wind blowing to the east. What is the difference in flight time between the two legs of the trip?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 45

Anita is running to the right at $5 \mathrm{m} / \mathrm{s},$ as shown in Figure $\mathrm{P} 3.45 .$ Balls 1 and 2 are thrown toward her at $10 \mathrm{m} / \mathrm{s}$ by friends standing on the ground. According to Anita, what is the speed of each ball?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 46

In the 2016 Olympics in Rio, after the $50 \mathrm{m}$ freestyle competition, a problem with the pool was found. In lane 1 there was a gentle $1.2 \mathrm{cm} / \mathrm{s}$ current flowing in the direction that the swimmers were going, while in lane 8 there was a current of the same speed but directed opposite to the swimmers' direction. Suppose a swimmer could swim the $50 \mathrm{m}$ in $25.0 \mathrm{s}$ in the absence of any current. What would be her time in lane $1 ?$ In lane $8 ?$ How does the difference in these times compare to the actual 0.06 s difference in times between the gold medal winner and the fourthplace finisher?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 47

Anita is running to the right at $5 \mathrm{m} / \mathrm{s},$ as shown in Figure $\mathrm{P} 3.47 .$ Balls 1 and 2 are thrown toward her by friends standing on the ground. According to Anita, both balls are approaching her at $10 \mathrm{m} / \mathrm{s}$. According to her friends, with what speeds were the balls thrown?

Anand Jangid

Numerade Educator

Problem 48

Two children who are bored while waiting for their flight at the airport decide to race from one end of the $20-\mathrm{m}$ -long moving sidewalk to the other and back. Phillippe runs on the sidewalk at $2.0 \mathrm{m} / \mathrm{s}$ (relative to the sidewalk). Renee runs on the floor at $2.0 \mathrm{m} / \mathrm{s} .$ The sidewalk moves at $1.5 \mathrm{m} / \mathrm{s}$ relative to the floor. Both make the turn instantly with no loss of speed.
a. Who wins the race?
b. By how much time does the winner win?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 49

A boat takes $3.0 \mathrm{h}$ to travel $30 \mathrm{km}$ down a river, then $5.0 \mathrm{h}$ to return. How fast is the river flowing?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 50

Suppose $\vec{C}=\vec{A}+\vec{B}$ where vector $\vec{A}$ has components $A_{x}=5$ $A_{y}=2$ and vector $\vec{B}$ has components $B_{x}=-3, B_{y}=-5$
a. What are the $x$ - and $y$ -components of vector $\vec{C}$ ?
b. Draw a coordinate system and on it show vectors $\vec{A}, \vec{B},$ and $\vec{C}$.
c. What are the magnitude and direction of vector $\vec{C}$ ?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 51

Suppose $\vec{D}=\vec{A}-\vec{B}$ where vector $\vec{A}$ has components $A_{x}=5$, $A_{y}=2$ and vector $\vec{B}$ has components $B_{x}=-3, B_{y}=-5$
a. What are the $x$ - and $y$ -components of vector $\vec{D}$ ?
b. Draw a coordinate system and on it show vectors $\vec{A}, \vec{B}$ and $\vec{D}$.
c. What are the magnitude and direction of vector $\vec{D} ?$

William Dunkerton

William Dunkerton

Numerade Educator

Problem 52

Suppose $\vec{E}=2 \vec{A}+3 \vec{B}$ where vector $\vec{A}$ has components $A_{x}=5, A_{y}=2$ and vector $\vec{B}$ has components $B_{x}=-3, B_{y}=-5$
What are the $x$ - and $y$ -components of vector $\vec{E}$ ?
b. Draw a coordinate system and on it show vectors $\vec{A}, \vec{B}$, and $\vec{E}$.
c. What are the magnitude and direction of vector $\vec{E}$ ?

Brandy Heflin

Numerade Educator

Problem 53

For the three vectors shown in Figure $\mathrm{P} 3.53,$ the vector sum $\vec{D}=$ $\vec{A}+\vec{B}+\vec{C}$ has components $D_{x}=2$ and $D_{y}=0$
a. What are the $x$ - and $y$ -components of vector $\vec{B} ?$
b. Write $\vec{B}$ as a magnitude and a direction.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 54

Let $\vec{A}=\left(3.0 \mathrm{m}, 20^{\circ}\right.$ south of east $)$ $\vec{B}=(2.0 \mathrm{m},$ north $),$ and $\vec{C}=\left(5.0 \mathrm{m}, 70^{\circ}\right.$ south of west $)$
a. Draw and label $\vec{A}, \vec{B},$ and $\vec{C}$ with their tails at the origin. Use a coordinate system with the $x$ -axis to the east.
b. Write the $x$ - and $y$ -components of vectors $\vec{A}, \vec{B},$ and $\vec{C}$.
c. Find the magnitude and the direction of $\vec{D}=\vec{A}+\vec{B}+\vec{C}$.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 55

To get to his office from home, Greg walks 5 blocks north and then 3 blocks east. After work he meets some friends at a café; to get there he walks 2 blocks south and 5 blocks west. All blocks are 660 feet long. What is the straight-line distance from the café to his home?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 56

A pilot in a small plane encounters shifting winds. He flies $26.0 \mathrm{km}$ northeast, then $45.0 \mathrm{km}$ due north. From this point, he flies an additional distance in an unknown direction, only to find himself at a small airstrip that his map shows to be $70.0 \mathrm{km}$ directly north of his starting point. What were the length and direction of the third leg of his trip?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 57

In punting a football, the kicker tries to maximize both the distance of the kick and its "hang time"-the time that the ball is in the air. A kicker gets off a great punt with a hang time of $5.0 \mathrm{s}$ that lands 50 yards from the kicker.
a. What is the speed of the ball as it leaves the kicker's foot?
b. What is the angle of the ball's initial velocity?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 58

The bacterium Escherichia coli (or $E$. coli) is a single-celled organism that lives in the gut of healthy humans and animals. When grown in a uniform medium rich in salts and amino acids, these bacteria swim along zig-zag paths at a constant speed of $20 \mu \mathrm{m} / \mathrm{s} .$ Figure $\mathrm{P} 3.58$ shows the trajectory of an $E .$ coli as it moves from point A to point E. Each segment of the motion can be identified by two letters, such as segment $\mathrm{BC}$.
a. For each of the four segments in the bacterium's trajectory, calculate the $x$ - and $y$ -components of its displacement and of its velocity.
b. Calculate both the total distance traveled and the magnitude of the net displacement for the entire motion.
c. What are the magnitude and the direction of the bacterium's average velocity for the entire trip?

Supratim Pal

Numerade Educator

Problem 59

A skier gliding across the snow at $3.0 \mathrm{m} / \mathrm{s}$ suddenly starts down a $10^{\circ}$ incline, reaching a speed of $15 \mathrm{m} / \mathrm{s}$ at the bottom. Friction between the snow and her freshly waxed skis is negligible.
a. What is the length of the incline?
b. How long does it take her to reach the bottom?

Anand Jangid

Numerade Educator

Problem 60

As shown in Figure $\mathrm{P} 3.60,$ a skier speeds along a flat patch of snow, and then flies horizontally off the edge at $12.0 \mathrm{m} / \mathrm{s}$. He eventually lands on a straight, sloped section that is at an angle of $45^{\circ}$ below the horizontal. How long is he in the air?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 61

A physics student on Planet Exidor throws a ball, and it follows the parabolic trajectory shown in Figure $\mathrm{P} 3.61$ The ball's position is shown at $1.0 \mathrm{s}$ intervals until $t=3.0 \mathrm{s}$
At $t=1.0 \mathrm{s},$ the ball's velocity has components $v_{x}=2.0 \mathrm{m} / \mathrm{s}$ $v_{y}=2.0 \mathrm{m} / \mathrm{s}$
a. Determine the $x$ - and $y$ -components of the ball's velocity at $t=0.0 \mathrm{s}, 2.0 \mathrm{s},$ and $3.0 \mathrm{s}$
b. What is the value of $g$ on Planet Exidor?
c. What was the ball's launch angle?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 62

The archerfish uses a remarkable method for catching insects sitting on branches or leaves above the waterline. The fish rises to the surface and then shoots out a stream of water precisely aimed to knock the insect off its perch into the water, where the archerfish gobbles it up. Scientists have measured the speed of the water stream exiting the fish's mouth to be $3.7 \mathrm{m} / \mathrm{s}$. An archerfish spots an insect sitting $19 \mathrm{cm}$ above the waterline and a horizontal distance of $30 \mathrm{cm}$ away. The fish aims its stream at an angle of $39^{\circ}$ from the waterline. Does the stream hit its mark?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 63

In $1780,$ in what is now referred to as "Brady's Leap," Captain Sam Brady of the U.S. Continental Army escaped certain death from his enemies by running horizontally off the edge of the cliff above Ohio's Cuyahoga River, which is confined at that spot to a gorge. He landed safely on the far side of the river. It was reported that he leapt $22 \mathrm{ft}$ across while falling
$20 \mathrm{ft} .$ Tall tale, or possible?
a. What is the minimum speed with which he'd need to run off the edge of the cliff to make it safely to the far side of the river?
b. The world-record time for the $100 \mathrm{m}$ dash is approximately 10 s. Given this, is it reasonable to expect Brady to be able to run fast enough to achieve Brady's leap?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 64

The longest recorded pass in an NFL game traveled 83 yards in the air from the quarterback to the receiver. Assuming that the pass was thrown at the optimal $45^{\circ}$ angle, what was the speed at which the ball left the quarterback's hand?

Prabhat Tyagi

Numerade Educator

Problem 65

A spring-loaded gun, fired vertically, shoots a marble $6.0 \mathrm{m}$ straight up in the air. What is the marble's range if it is fired horizontally from $1.5 \mathrm{m}$ above the ground?

Nishant Kumar

Numerade Educator

Problem 66

Small-plane pilots regularly compete in "message drop" competitions, dropping heavy weights (for which air resistance can be ignored) from their low-flying planes and scoring points for having the weights land close to a target. A plane $60 \mathrm{m}$ above the ground is flying directly toward a target at $45 \mathrm{m} / \mathrm{s}$.
a. At what distance from the target should the pilot drop the weight?
b. The pilot looks down at the weight after she drops it. Where is the plane located at the instant the weight hits the ground-not yet over the target, directly over the target, or past the target?

Vishal Gupta

Numerade Educator

Problem 67

Paintball guns were originally developed to mark trees for logging. A forester aims his gun directly at a knothole in a tree that is $4.0 \mathrm{m}$ above the gun. The base of the tree is $20 \mathrm{m}$ away. The speed of the paintball as it leaves the gun is $50 \mathrm{m} / \mathrm{s}$. How far below the knothole does the paintball strike the tree?

Prabhu Ramji

Numerade Educator

Problem 68

Trained dolphins are capable of a vertical leap of $7.0 \mathrm{m}$ straight up from the surface of the water-an impressive feat. Suppose you could train a dolphin to launch itself out of the water at this same speed but at an angle. What maximum horizontal range could the dolphin achieve?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 69

A tennis player hits a ball $2.0 \mathrm{m}$ above the ground. The ball leaves his racquet with a speed of $20 \mathrm{m} / \mathrm{s}$ at an angle $5.0^{\circ}$ above the horizontal. The horizontal distance to the net is $7.0 \mathrm{m},$ and the net is $1.0 \mathrm{m}$ high. Does the ball clear the net? If so, by how much? If not, by how much does it miss?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 70

The shot put is a trackand-field event in which athletes throw a heavy ballthe shot- $-$ as far as possible. The best athletes can throw the shot as far as $23 \mathrm{m}$. Athletes who use the "glide" technique push the shot outward in a reasonably straight line, accelerating it over a distance of about $2.0 \mathrm{m} .$ What acceleration do they provide to the shot as they push on it? Assume that the shot is launched at an angle of $37^{\circ},$ a reasonable value for an excellent throw. You can assume that the shot lands at the same
height from which it is thrown; this simplifies the calculation considerably, and makes only a small difference in the final result.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 71

Water at the top of Horseshoe Falls (part of Niagara Falls) is moving horizontally at $9.0 \mathrm{m} / \mathrm{s}$ as it goes off the edge and plunges $53 \mathrm{m}$ to the pool below. If you ignore air resistance, at what angle is the falling water moving as it enters the pool?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 72

A supply plane needs to drop a package of food to scientists working on a glacier in Greenland. The plane flies $100 \mathrm{m}$ above the glacier at a speed of $150 \mathrm{m} / \mathrm{s}$. How far short of the target should it drop the package?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 73

A BMX bicycle rider takes off from a ramp at a point $1.8 \mathrm{m}$ above the ground. The ramp is angled at $40^{\circ}$ from the horizontal, and the rider's speed is $6.7 \mathrm{m} / \mathrm{s}$ when he leaves the ramp. How far from the end of the ramp does he land?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 74

Ships A and B leave port together. For the next two hours, ship $\mathrm{A}$ travels at $20 \mathrm{mph}$ in a direction $30^{\circ}$ west of north while ship $\mathrm{B}$ travels $20^{\circ}$ east of north at $25 \mathrm{mph}$.
a. What is the distance between the two ships two hours after they depart?
b. What is the speed of ship $A$ as seen by ship $B ?$

William Dunkerton

William Dunkerton

Numerade Educator

Problem 75

A flock of ducks is trying to migrate south for the winter, but they keep being blown off course by a wind blowing from the west at $12 \mathrm{m} / \mathrm{s}$. A wise elder duck finally realizes that the solution is to fly at an angle to the wind. If the ducks can fly at $16 \mathrm{m} / \mathrm{s}$ relative to the air, in what direction should they head in order to move directly south?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 76

A kayaker needs to paddle north across a $100-\mathrm{m}-$ wide harbor. The tide is going out, creating a tidal current flowing east at $2.0 \mathrm{m} / \mathrm{s}$. The kayaker can paddle with a speed of $3.0 \mathrm{m} / \mathrm{s}$
a. In which direction should he paddle in order to travel straight across the harbor?
b. How long will it take him to cross?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 77

A plane has an airspeed of 200 mph. The pilot wishes to reach a destination 600 mi due east, but a wind is blowing at $50 \mathrm{mph}$ in the direction $30^{\circ}$ north of east.
a. In what direction must the pilot head the plane in order to reach her destination?
b. How long will the trip take?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 78

The Gulf Stream off the east coast of the United States can flow at a rapid $3.6 \mathrm{m} / \mathrm{s}$ to the north. A ship in this current has a cruising speed of $10 \mathrm{m} / \mathrm{s}$. The captain would like to reach land at a point due west from the current position.
a. In what direction with respect to the water should the ship sail?
b. At this heading, what is the ship's speed with respect to land?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 79

A ball thrown horizontally at $25 \mathrm{m} / \mathrm{s}$ travels a horizontal distance of $50 \mathrm{m}$ before hitting the ground. From what height was the ball thrown?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 80

A sports car is advertised as capable of "reaching $60 \mathrm{mph}$ in 5 seconds flat, cornering at $0.85 g,$ and stopping from $70 \mathrm{mph}$ in only 168 feet." In which of those three situations is the magnitude of the car's acceleration the largest? In which is it the smallest?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 81

A Ford Mustang can accelerate from 0 to 60 mph in a time of 5.6 s. A Mini Cooper isn't capable of such a rapid start, but it can turn in a very small circle 34 ft in diameter. How fast would you need to drive the Mini Cooper in this tight circle to match the magnitude of the Mustang's acceleration?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 82

The "Screaming Swing" is a carnival ride that is -not surprisingly $-$ a giant swing. It's actually two swings moving in opposite directions. At the bottom of its arc, a rider in one swing is moving at $30 \mathrm{m} / \mathrm{s}$ with respect to the ground in a $50-\mathrm{m}$ -diameter circle. The rider in the other swing is moving in a similar circle at the same speed, but in the exact opposite direction.
a. What is the acceleration, in $\mathrm{m} / \mathrm{s}^{2}$ and in units of $g$, that riders experience?
b. At the bottom of the ride, as they pass each other, how fast do the riders move with respect to each other?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 83

On an otherwise straight stretch of road near Moffat, Colorado, the road suddenly turns. This bend in the road is a segment of a circle with radius $110 \mathrm{m}$. Drivers are cautioned to slow down to 40 mph as they navigate the curve.
a. If you heed the sign and slow to 40 mph, what will be your acceleration going around the curve at this constant speed? Give your answer in $\mathrm{m} / \mathrm{s}^{2}$ and in units of $g .$
b. At what speed would your acceleration be double that at the recommended speed?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 84

At the end of the first section of the motion, riders are moving at what approximate speed?
A. $3 \mathrm{m} / \mathrm{s}$
B. $6 \mathrm{m} / \mathrm{s}$
$\mathrm{C} .9 \mathrm{m} / \mathrm{s}$
D. $12 \mathrm{m} / \mathrm{s}$

William Dunkerton

William Dunkerton

Numerade Educator

Problem 85

Suppose the acceleration during the second section of the motion is too large to be comfortable for riders. What change could be made to decrease the acceleration during this section?
A. Reduce the radius of the circular segment.
B. Increase the radius of the circular segment.
C. Increase the angle of the ramp.
D. Increase the length of the ramp.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 86

What is the vertical component of the velocity of a rider as he or she hits the water?
A. $2.4 \mathrm{m} / \mathrm{s}$
B. $3.4 \mathrm{m} / \mathrm{s}$
C. $5.2 \mathrm{m} / \mathrm{s}$
D. $9.1 \mathrm{m} / \mathrm{s}$

William Dunkerton

William Dunkerton

Numerade Educator

Problem 87

Suppose the designers of the water slide want to adjust the height $h$ above the water so that riders land twice as far away from the bottom of the slide. What would be the necessary height above the water?
A. $1.2 \mathrm{m}$
B. $1.8 \mathrm{m}$
C. $2.4 \mathrm{m}$
D. $3.0 \mathrm{m}$

William Dunkerton

William Dunkerton

Numerade Educator

Problem 88

During which section of the motion is the magnitude of the acceleration experienced by a rider the greatest?
A. The first.
B. The second.
C. The third.
D. It is the same in all sections.

William Dunkerton

William Dunkerton

Numerade Educator