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Physics

James S. Walker

Chapter 3

Vectors in Physics

Educators


Problem 1

CE Suppose that each component of a certain vector is doubled.
(a) By what multiplicative factor does the magnitude of the vector
change? (b) By what multiplicative factor does the direction angle
of the vector change?

Andy C.
Numerade Educator

Problem 2

CE Rank the vectors in FlGURE $3-37$ in order of increasing
magnitude.

Andy C.
Numerade Educator

Problem 3

CE Rank the vectors in Figure $3-37$ in order of increasing value of
their $x$ component.

Andy C.
Numerade Educator

Problem 4

CE Rank the vectors in Figure $3-37$ in order of increasing value of
their $y$ component.

Andy C.
Numerade Educator

Problem 5

The press box at a baseball park is 44.5 ft above the ground. A
reporter in the press box looks at an angle of $13.4^{\circ}$ below the horizontal to see second base. What is the horizontal distance from the press box to second base?

Andy C.
Numerade Educator

Problem 6

You are driving up a long, inclined road. After 1.2 miles you
notice that signs along the roadside indicate that your elevation
has increased by 530 $\mathrm{ft}$ (a) What is the angle of the road above the
horizontal? (b) How far do you have to drive to gain an additional
150 ft of elevation?

Andy C.
Numerade Educator

Problem 7

A One-Percent Grade A road that rises 1 ft for every 100 ft traveled
horizontally is said to have a 1$\%$ grade. Portions of the Lewiston
grade, near Lewiston, Idaho, have a 6$\%$ grade. At what angle is this
road inclined above the horizontal?

Andy C.
Numerade Educator

Problem 8

You walk in a straight line for 95 $\mathrm{m}$ at an angle of $162^{\circ}$ above the positive $x$ axis. What are the $x$ and $y$ components of your displacement?

Andy C.
Numerade Educator

Problem 9

Find the $x$ and $y$ components of a position vector $\overrightarrow{\mathbf{r}}$ of magnitude $r=88 \mathrm{m},$ if its angle relative to the $x$ axis is (a) $32.0^{\circ}$ and
(b) $64.0^{\circ} .$

Andy C.
Numerade Educator

Problem 10

A vector has the components $A_{x}=22 \mathrm{mand} A_{y}=13 \mathrm{m}$ . (a) What
is the magnitude of this vector? (b) What angle does this vector
make with the positive $x$ axis?

Andy C.
Numerade Educator

Problem 11

A vector has the components $A_{x}=-36 \mathrm{m}$ and $A_{y}=43 \mathrm{m}$ .
(a) What is the magnitude of this vector? (b) What angle does this
vector make with the positive $x$ axis?

Andy C.
Numerade Educator

Problem 12

A baseball "diamond" (FIGURE $3-38$ ) is a square with sides 90 ft in
length. If the positive $x$ axis points from home plate to first base,
and the positive $y$ axis points from home plate to third base, find
the displacement vector of a base runner who has just hit (a) a
double, (b) a triple, or (c) a homerun.

Andy C.
Numerade Educator

Problem 13

A lighthouse that rises 49 ft above the surface of the water sits
on a rocky cliff that extends 19 ft from its base, as shown in FIGURE
$3-39 .$ A sailor on the deck of a ship sights the top of the lighthouse
at an angle of $30.0^{\circ}$ above the horizontal. If the sailor's eye level is
14 ft above the water, how far is the ship from the rocks?

Andy C.
Numerade Educator

Problem 14

$ \mathrm{H}_{2} \mathbf{0}$ A water molecule is shown schematically in FlGURE $3-40$ .
The distance from the center of the oxygen atom to the center of
a hydrogen atom is $0.96 \hat{\mathrm{A}},$ and the angle between the hydrogen
atoms is $104.5^{\circ} .$ Find the center-to-center distance between the
hydrogen atoms. $\left(1 \hat{\mathrm{A}}=10^{-10} \mathrm{m.}\right)$

Andy C.
Numerade Educator

Problem 15

Predict/Calculate The $x$ and $y$ components of a vector $\overrightarrow{\mathbf{r}}$ are
$r_{x}=14 \mathrm{m}$ and $r_{y}=-9.5 \mathrm{m},$ respectively. Find (a) the direction and
(b) the magnitude of the vector $\overrightarrow{\mathbf{r}}$ . (c) If both $r_{x}$ and $r_{y}$ are doubled, how do you predict your answers to parts (a) and (b) will change?
(d) Verify your prediction in part (c) by calculating the magnitude
and direction for this new case.

Andy C.
Numerade Educator

Problem 16

You drive $a$ car 680 ft to the east, then 340 ft to the north. (a) What
is the magnitude of your displacement? (b) Using a sketch, estimate the direction of your displacement. (c) Verify your estimate in part (b) with a numerical calculation of the direction.

Andy C.
Numerade Educator

Problem 17

Vector $\overrightarrow{\mathbf{A}}$ has a magnitude of 50 units and points in the positive
$x$ direction. A second vector, $\overline{\mathbf{B}},$ has a magnitude of 120 units and
points at an angle of $70^{\circ}$ below the $x$ axis. Which vector has (a) the
greater $x$ component, and (b) the greater $y$ component?

Andy C.
Numerade Educator

Problem 18

A treasure map directs you to start at a palm tree and walk due
north for 15.0 $\mathrm{m} .$ You are then to turn $90^{\circ}$ and walk 22.0 $\mathrm{m}$ ; then
turn $90^{\circ}$ again and walk 5.00 $\mathrm{m} .$ Give the distance from the palm
tree, and the direction relative to north, for each of the four possible locations of the treasure.

Andy C.
Numerade Educator

Problem 19

A whale comes to the surface to breathe and then dives at an
angle of $20.0^{\circ}$ below the horizontal (FIGURE $3-41 ) .$ If the whale
continues in a straight line for $215 \mathrm{m},$ (a) how deep is it, and
(b) how far has it traveled horizontally?

Andy C.
Numerade Educator

Problem 20

$\mathrm{CE}$ Consider the vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ shown in FlGuRE $3-42$ Which of the other four vectors in the figure $(\overrightarrow{\mathbf{c}}, \overrightarrow{\mathbf{D}}, \overrightarrow{\mathbf{E}},$ and $\overrightarrow{\mathbf{F}} )$ best represents the direction of $(\mathbf{a}) \overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}},(\mathbf{b}) \overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}},$ and $(\mathrm{c}) \overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}} ?$

Andy C.
Numerade Educator

Problem 21

CE Refer to Figure $3-42$ for the following questions: (a) Is the
magnitude of $\overrightarrow{\mathbf{A}}+\overline{\mathbf{D}}$ greater than, less than, or equal to the magnitude of $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{E}} ?$ (b) Is the magnitude of $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{E}}$ greater than, less than, or equal to the magnitude of $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{F}} ?$

Andy C.
Numerade Educator

Problem 22

A vector $\overrightarrow{\mathbf{A}}$ has a magnitude of 40.0 $\mathrm{m}$ and points in a direction
$20.0^{\circ}$ below the positive $x$ axis. A second vector, $\overrightarrow{\mathbf{B}},$ has a magnitude of 75.0 $\mathrm{m}$ and points in a direction $50.0^{\circ}$ above the positive $x$
axis. (a) Sketch the vectors $\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}},$ and $\overline{\mathbf{C}}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ . (b) Using the component method of vector addition, find the magnitude and
direction of the vector $\overrightarrow{\mathbf{C}} .$

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Problem 23

An air traffic controller observes two airplanes approaching
the airport. The displacement from the control tower to plane 1
is given by the vector $\mathbf{A},$ which has a magnitude of 220 $\mathrm{km}$ and
points in a direction $32^{\circ}$ north of west. The displacement from the
control tower to plane 2 is given by the vector $\overline{\mathbf{B}}$ , which has a magnitude of 140 $\mathrm{km}$ and points $65^{\circ}$ east of north. (a) Sketch the vectors $\overrightarrow{\mathbf{A}},-\overrightarrow{\mathbf{B}},$ and $\overrightarrow{\mathbf{D}}=\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ . Notice that $\overrightarrow{\mathbf{D}}$ is the displacement
from plane 2 to plane $1 .$ (b) Find the magnitude and direction of
the vector $\overline{\mathbf{D}} .$

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Problem 24

The initial velocity of a car, $\overrightarrow{\mathbf{v}}_{\mathrm{i}},$ is 45 $\mathrm{km} / \mathrm{h}$ in the positive $x$ direction. The final velocity of the car, $\overrightarrow{\mathbf{v}}_{\mathrm{f}},$ is 66 $\mathrm{km} / \mathrm{h}$ in a direction that points $75^{\circ}$ above the positive $x$ axis. (a) Sketch the vectors $-\overrightarrow{\mathbf{v}_{i}}, \overrightarrow{\mathbf{v}}_{i}$ and $\Delta \overrightarrow{\mathbf{v}}=\overrightarrow{\mathbf{v}}_{\mathrm{f}}-\overrightarrow{\mathbf{v}}_{\mathrm{i}}$ . (b) Find the magnitude and direction of the
change in velocity, $\Delta \overrightarrow{\mathbf{v}} .$

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Problem 25

Vector $\overrightarrow{\mathbf{A}}$ points in the positive $x$ direction and has a magnitude
of 75 $\mathrm{m} .$ The vector $\overrightarrow{\mathbf{C}}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ points in the positive $y$ direction and has a magnitude of 95 $\mathrm{m}$ . (a) Sketch $\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}},$ and $\overrightarrow{\mathbf{C}} .$ (b) Estimate
the magnitude and direction of the vector $\overline{\mathbf{B}}$ . (c) Verify your estimate in part (b) with a numerical calculation.

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Problem 26

Vector $\overrightarrow{\mathbf{A}}$ points in the negative $x$ direction and has a magnitude of 22 units. The vector $\overrightarrow{\mathbf{B}}$ points in the positive $y$ direction.
(a) Find the magnitude of $\overrightarrow{\mathbf{B}}$ if $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ has a magnitude of 37 units.
(b) Sketch $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ .

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Problem 27

Vector $\overrightarrow{\mathbf{A}}$ points in the negative $y$ direction and has a magnitude of 5 units. Vector $\overrightarrow{\mathbf{B}}$ has twice the magnitude and points in the positive $x$ direction. Find the direction and magnitude of $(\mathbf{a}) \overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}},(\mathbf{b}) \overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}},$ and $(\mathbf{c}) \overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}$.

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Problem 28

A basketball player runs down the court, following the path
indicated by the vectors $\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}},$ and $\overrightarrow{\mathbf{C}}$ in FlGURE $3-43 .$ The magnitudes of these three vectors are $A=10.0 \mathrm{m}, B=20.0 \mathrm{m},$ and $C=7.0 \mathrm{m} .$ Find the magnitude and direction of the net displacement of the player using (a) the graphical method and (b) the component method of vector addition. Compare your results.

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Problem 29

A particle undergoes a displacement $\Delta \overrightarrow{\mathbf{r}}$ of magnitude 54 $\mathrm{m}$ in
a direction $42^{\circ}$ below the $x$ axis. Express $\Delta \overrightarrow{\mathbf{r}}$ in terms of the unit
vectors $\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$

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Problem 30

A vector has a magnitude of 3.50 $\mathrm{m}$ and points in a direction that
is $145^{\circ}$ counterclockwise from the $x$ axis. Find the $x$ and $y$ components of this vector.

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Problem 31

A vector $\overrightarrow{\mathbf{A}}$ has a length of 6.1 $\mathrm{m}$ and points in the negative $x$
direction. Find (a) the $x$ component and $(\mathrm{b})$ the magnitude of the
vector $-3.7 \overline{\mathrm{A}}$

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Problem 32

The vector $-5.2 \overrightarrow{\mathbf{A}}$ has a magnitude of 34 $\mathrm{m}$ and points in the
positive $x$ direction. Find (a) the $x$ component and (b) the magnitude of the vector $\frac{\mathrm{A}}{\mathbf{A}} .$

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Problem 33

Find the direction and magnitude of the vectors.
(a) $\overrightarrow{\mathbf{A}}=(5.0 \mathrm{m}) \hat{\mathbf{x}}+(-2.0 \mathrm{m}) \hat{\mathbf{y}}$
(b) $\mathbf{B}=(-2.0 \mathrm{m}) \hat{\mathbf{x}}+(5.0 \mathrm{m}) \hat{\mathbf{y}},$ and (c) $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$

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Problem 34

Find the direction and magnitude of the vectors.
$\overrightarrow{\mathbf{A}}=(22 \mathrm{m}) \hat{\mathbf{x}}+(-14 \mathrm{m}) \hat{\mathbf{y}},$
(b) $\overrightarrow{\mathbf{B}}=(2.5 \mathrm{m}) \hat{\mathbf{x}}+(13 \mathrm{m}) \hat{\mathbf{y}},$ and $(\mathrm{c}) \overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$

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Problem 35

For the vectors given in Problem $34,$ express (a) $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ and
(b) $\overline{\mathbf{B}}-\overrightarrow{\mathbf{A}}$ in unit vector notation.

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Problem 36

Express each of the vectors in FIGURE $3-44$ in unit vector notation.

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Problem 37

Referring to the vectors in Figure $3-44,$ express the sum
$\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overline{\mathbf{C}}$ in unit vector notation.

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Problem 38

CE The blue curves shown in FIGURE $3-45$ display the constant-speed
motion of two different particles in the $x-y$ plane. For each of the
eight vectors in Figure $3-45,$ state whether it is (a) a position vector,
(b) a velocity vector, or (c) an acceleration vector for the particles.

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Problem 39

What are the direction and magnitude of your total displacement if you have traveled due west with a speed of 23 $\mathrm{m} / \mathrm{s}$ for 175 $\mathrm{s}$ ,
then due south at 12 $\mathrm{m} / \mathrm{s}$ for 285 $\mathrm{s}$ ?

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Problem 40

Predict/Calculate Moving the Knight Two of the allowed chess
moves for a knight are shown in FlGURE $3-46$ . (a) Is the magnitude of
displacement 1 greater than, less than, or equal to the magnitude
of displacement 2$?$ Explain. (b) Find the magnitude and direction
of the knight's displacement for each of the two moves. Assume
that the checkerboard squares are 3.5 $\mathrm{cm}$ on a side.

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Problem 41

To visit your favorite ice cream shop, you must travel 490 $\mathrm{m}$ west
on Main Street and then 950 $\mathrm{m}$ south on Division Street. (a) Find
the total distance you traveled. (b) Write your displacement vector
in unit vector notation, taking the $\hat{\mathbf{x}}$ direction to be east and the $\hat{\mathbf{y}}$
direction to be north. (c) Write the displacement vector required
to return to your starting point in unit vector notation.

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Problem 42

Referring to Problem $41,$ suppose you take 44 s to complete the
$490-\mathrm{m}$ displacement and 73 $\mathrm{s}$ to complete the $950-\mathrm{m}$ displacement. (a) What are the magnitude and direction of your average
velocity during this 117 -second period of time? (b) What is your average speed during the trip?

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Problem 43

You drive a car 1500 $\mathrm{ft}$ to the east, then 2500 $\mathrm{ft}$ to the north. If
the trip took 3.0 minutes, what were the direction and magnitude of your average velocity?

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Problem 44

Predict/Calculate A jogger runs with a speed of 3.25 $\mathrm{m} / \mathrm{s}$ in a
direction $30.0^{\circ}$ above the $x$ axis. (a) Find the $x$ and $y$ components
of the jogger's velocity. (b) How will the velocity components found in part (a) change if the jogger's speed is halved? (c) Calculate the components of the velocity for the case where the jogger's speed is halved.

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Problem 45

$\bullet$ You throw a ball upward with an initial speed of 4.5 $\mathrm{m} / \mathrm{s}$ . When
it returns to your hand 0.92 s later, it has the same speed in the
downward direction (assuming air resistance can be ignored).
What was the average acceleration vector of the ball?

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Problem 46

$\because$ Consider a skateboarder who starts from rest at the top of a
ramp that is inclined at an angle of $17.5^{\circ}$ to the horizontal. Assuming that the skateboarder's acceleration is $g \sin 17.5^{\circ},$ find his speed
when he reaches the bottom of the ramp in 3.25 $\mathrm{s}$ .

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Problem 47

$\cdots$ In a soccer game a midfielder kicks the ball from one touchline
directly toward the other with a velocity of $(14.0 \mathrm{m} / \mathrm{s}) \hat{\mathbf{x}}$ . After the
ball travels for 2.50 $\mathrm{s}$ , a striker makes an acrobatic kick that gives
the ball a new velocity in the $\hat{\mathbf{y}}$ direction, and after 0.900 s the ball passes the diving goalkeeper and goes into the net. If the total
displacement of the ball is 40.2 $\mathrm{m}$ at $29.5^{\circ}$ above the $x$ axis, with
what speed did the striker kick the ball?

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Problem 48

CE The accompanying photo shows a KC-10A Extender using a
boom to refuel an aircraft in flight. If the velocity of the KC-10A is
125 $\mathrm{m} / \mathrm{s}$ due east relative to the ground, what is the velocity of the
aircraft being refueled relative to (a) the ground, and (b) the KC-10A?

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Problem 49

$\cdot$ As an airplane taxies on the runway with a speed of 16.5 $\mathrm{m} / \mathrm{s}$ ,
a flight attendant walks toward the tail of the plane with a speed of 1.22 $\mathrm{m} / \mathrm{s} .$ What is the flight attendant's speed relative to the
ground?

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Problem 50

$\cdot$ Referring to part (a) of Example $3-11,$ find the time it takes for the
boat to reach the opposite shore if the river is 35 $\mathrm{m}$ wide.

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Problem 51

$\because$ A police car travels at 38.0 $\mathrm{m} / \mathrm{s}$ due east while in pursuit of a
vehicle that is traveling at 33.5 $\mathrm{m} / \mathrm{s}$ due east. (a) What is the velocity of the vehicle relative to the police car? (b) What is the velocity
of the police car relative to the vehicle?

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Problem 52

$\bullet$ Consider the river crossing problem in Example $3-11 .$ Suppose
we would like the boat to move directly across the river (in the
positive $x$ direction) with a speed of 5.0 $\mathrm{m} / \mathrm{s} .$ What is the corresponding speed and direction of the boat's velacity relative to the
water?

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Problem 53

$\because$ As you hurry to catch your flight at the local airport, you
encounter a moving walkway that is 61 $\mathrm{m}$ long and has a speed
of 2.2 $\mathrm{m} / \mathrm{s}$ relative to the ground. If it takes you 49 $\mathrm{s}$ to cover
61 $\mathrm{m}$ when walking on the ground, how much time will it take you to cover the same distance on the walkway? Assume that
you walk with the same speed on the walkway as you do on the
ground.

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Problem 54

$\cdots$ In Problem $53,$ how much time would it take you to cover the
$61-\mathrm{m}$ length of the walkway if, once you get on the walkway, you
immediately turn around and start walking in the opposite direction with a speed of 1.3 $\mathrm{m} / \mathrm{s}$ relative to the walkway?

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Problem 55

$\because$ Predict/Calculate The pilot of an airplane wishes to fly due
north, but there is a $65-\mathrm{km} / \mathrm{h}$ wind blowing toward the east. (a) In
what direction relative to north should the pilot head her plane if
its speed relative to the air is 340 $\mathrm{km} / \mathrm{h} ?$ (b) Draw a vector diagram that illustrates your result in part (a). (c) If the pilot decreases the
air speed of the plane, but still wants to head due north, should
the angle found in part (a) be increased or decreased?

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Problem 56

$\because$ A passenger walks from one side of a ferry to the other as it
approaches a dock. If the passenger's velocity is 1.50 $\mathrm{m} / \mathrm{s}$ due
north relative to the ferry, and 4.50 $\mathrm{m} / \mathrm{s}$ at an angle of $30.0^{\circ}$ west of north relative to the water, what are the direction and magnitude
of the ferry's velocity relative to the water?

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Problem 57

$\bullet$ You are riding on a Jet Ski at an angle of $35^{\circ}$ upstream on a river
flowing with a speed of 2.8 $\mathrm{m} / \mathrm{s}$ . If your velocity relative to the
ground is 9.5 $\mathrm{m} / \mathrm{s}$ at an angle of $20.0^{\circ}$ upstream, what is the speed
of the Jet Ski relative to the water? (Note: Angles are measured relative to the $x$ axis shown in Example $3-11 .$ )

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Problem 58

$\cdots \bullet$ Predict/Calculate In Problem $57,$ suppose the Jet Ski is
moving at a speed of 11 $\mathrm{m} / \mathrm{s}$ relative to the water. (a) At what
angle must you point the Jet Ski if your velocity relative to the
ground is to be perpendicular to the shore of the river? (b) If you
increase the speed of the Jet Ski relative to the water, does the angle in part (a) increase, decrease, or stay the same? Explain.
(c) Calculate the new angle if you increase the Jet Ski speed to
15 $\mathrm{m} / \mathrm{s} .$ (Note: Angles are measured relative to the $x$ axis shown
in Example $3-11 .$ )

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Problem 59

$\cdots \bullet$ Predict/Calculate Two people take identical Jet Skis across
a river, traveling at the same speed relative to the water. Jet Ski A
heads directly across the river and is carried downstream by the
current before reaching the opposite shore. Jet Ski B travels in a
direction that is $35^{\circ}$ upstream and arrives at the opposite shore
directly across from the starting point. (a) Which Jet Ski reaches
the opposite shore in the least amount of time? (b) Confirm your
answer to part (a) by finding the ratio of the time it takes for the
two Jet Skis to cross the river. (Note: Angles are measured relative to
the $x$ axis shown in Example $3-11 .$ )

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Problem 60

$\cdot$ CE Predict/Explain Consider the vectors $\overrightarrow{\mathbf{A}}=(1.2 \mathrm{m}) \hat{\mathbf{x}}$ and
$\overrightarrow{\mathbf{B}}=(-3.4 \mathrm{m}) \hat{\mathbf{x}}$ . (a) Is the magnitude of vector $\overrightarrow{\mathbf{A}}$ greater than, less than, or equal to the magnitude of vector $\overrightarrow{\mathbf{B}} ?$ (b) Choose the
best explanation from among the following:
I. The number 3.4 is greater than the number $1.2 .$
II. The component of $\overline{\mathbf{B}}$ is negative.
III. The vector $\overrightarrow{\mathbf{A}}$ points in the positive $x$ direction.

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Problem 61

$\cdot$ CE Predict/Explain Two vectors are defined as follows:
$\overrightarrow{\mathbf{A}}=(-3.8 \mathrm{m}) \hat{\mathbf{x}}$ and $\overrightarrow{\mathbf{B}}=(2.1 \mathrm{m}) \hat{\mathbf{y}}$ . (a) Is the magnitude of
$\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ greater than, less than, or equal to the magnitude of $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}} ?(\mathbf{b})$ Choose the best explanation from among the following:
I. A vector sum is always greater than a vector difference.
II. $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ and $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ produce similar triangles because $\overrightarrow{\mathbf{A}}$
is perpendicular to $\overline{\mathbf{B}}$ .
III. The vector $-\overrightarrow{\mathbf{B}}$ is in the same direction as $\overrightarrow{\mathbf{A}}$ .

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Problem 62

$\cdot$ To be compliant with regulations the inclination angle of a
wheelchair ramp must not exceed $4.76^{\circ} .$ If the wheelchair ramp
must have a vertical height of $1.22 \mathrm{m},$ what is its minimum horizontal length?

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Problem 63

$\cdot$ Find the direction and magnitude of the vector $2 \overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}},$ where
$\overrightarrow{\mathbf{A}}=(12.1 \mathrm{m}) \hat{\mathbf{x}}$ and $\overline{\mathbf{B}}=(-32.2 \mathrm{m}) \hat{\mathbf{y}}$

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Problem 64

CE The components of a vector $\overrightarrow{\mathbf{A}}$ satisfy $A_{x}<0$ and $A_{y}<0$ Is the direction angle of $\overrightarrow{\mathbf{A}}$ between $0^{\circ}$ and $90^{\circ},$ between $90^{\circ}$ and
$180^{\circ},$ between $180^{\circ}$ and $270^{\circ},$ or between $270^{\circ}$ and $360^{\circ} ?$

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Problem 65

CE The components of a vector $\overrightarrow{\mathbf{B}}$ satisfy $B_{x}>0$ and $B_{y}<0$. Is the direction angle of $\overrightarrow{\mathbf{B}}$ between $0^{\circ}$ and $90^{\circ},$ between $90^{\circ}$ and
$180^{\circ},$ between $180^{\circ}$ and $270^{\circ},$ or between $270^{\circ}$ and $360^{\circ} ?$

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Problem 66

It is given that $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}=(-51.4 \mathrm{m}) \hat{\mathbf{x}}, \overrightarrow{\mathbf{C}}=(62.2 \mathrm{m}) \hat{\mathbf{x}},$ and $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}=(13.8 \mathrm{m}) \hat{\mathbf{x}} .$ Find the vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$

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Problem 67

You pilot an airplane with the intent to fly 392 $\mathrm{km}$ in a direction $55.0^{\circ}$ south of east from the starting airport. Air traffic control requires you to first fly due south for 85.0 $\mathrm{km}$ . (a) After completing the $85.0-\mathrm{km}$ flight due south, how far are you from your destination? (b) In what direction should you point your airplane to complete the trip? Give your answer as an angle relative to due east.
(c) If your airplane has an average ground speed of 485 $\mathrm{km} / \mathrm{h}$ , what
estimated time of travel should you give to your passengers?

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Problem 68

Find the $x, y,$ and $z$ components of the vector $\overrightarrow{\mathbf{A}}$ shown in FIGURE $3-47,$ given that $A=65 \mathrm{m}$

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Problem 69

Observer 1 rides in a car and drops a ball from rest straight downward, relative to the interior of the car. The car moves horizontally with a constant speed of 3.50 $\mathrm{m} / \mathrm{s}$ relative to observer 2 standing on the sidewalk. (a) What is the speed of the ball 1.00 s after it is
released, as measured by observer 2$?$ (b) What is the direction of
travel of the ball 1.00 s after it is released, as measured relative to
the horizontal by observer 2 ?

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Problem 70

A person riding in a subway train drops a ball from rest straight
downward, relative to the interior of the train. The train is moving horizontally with a constant speed of 6.7 $\mathrm{m} / \mathrm{s} .$ A second person standing at rest on the subway platform observes the ball drop. From the point of view of the person on the platform, the ball is released at the position $x=0$ and $y=1.2 \mathrm{m} .$ Make a plot
of the position of the ball for the times $t=0,0.1 \mathrm{s}, 0.2 \mathrm{s}, 0.3$
$\mathrm{s},$ and 0.4 $\mathrm{s} .$ (Your plot is parabolic in shape, as we shall see in
Chapter 4.)

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Problem 71

A football is thrown horizontally with an initial velocity of
$(16.6 \mathrm{m} / \mathrm{s}) \hat{\mathbf{x}} .$ Ignoring air resistance, the average acceleration of the football over any period of time is $\left(-9.81 \mathrm{m} / \mathrm{s}^{2}\right) \hat{\mathbf{y}}$ . (a) Find the velocity vector of the ball 1.75 s after it is thrown. (b) Find the magnitude and direction of the velocity at this time.

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Problem 72

As a function of time, the velocity of the football described in
Problem 71 can be written as $\overrightarrow{\mathbf{v}}=(16.6 \mathrm{m} / \mathrm{s}) \hat{\mathbf{x}}-\left[\left(9.81 \mathrm{m} / \mathrm{s}^{2}\right) t\right] \hat{\mathbf{y}}$ . Calculate the average acceleration vector of the football for the
time periods (a) $t=0$ to $t=1.00 \mathrm{s},$ (b) $t=0$ to $t=2.50 \mathrm{s},$ and
(c) $t=0$ to $t=5.00 \mathrm{s}$

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Problem 73

Two airplanes taxi as they approach the terminal. Plane 1 taxies
with a speed of 12 $\mathrm{m} / \mathrm{s}$ due north. Plane 2 taxies with a speed of
7.5 $\mathrm{m} / \mathrm{sin}$ a direction $20^{\circ}$ north of west. (a) What are the direction and magnitude of the velocity of plane 1 relative to plane 2$?$ (b) What
are the direction and magnitude of the velocity of plane 2 relative
to plane 1$?$

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Problem 74

A shopper at the supermarket follows the path indicated by vectors $\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}}, \overrightarrow{\mathbf{B}},$ and $\overrightarrow{\mathbf{D}}$ in FlGURE $3-48 .$ Given that the vectors have the
magnitudes $A=51 \mathrm{ft}, B=45 \mathrm{ft}, C=35 \mathrm{ft},$ and $D=13 \mathrm{ft},$ find the total displacement of the shopper using (a) the graphical method
and (b) the component method of vector addition. Give the direction of the displacement relative to the direction of vector $\overline{\mathbf{A}}$ .

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Problem 75

BIO A food particle from your breakfast takes a circuitous path through your digestive system. Suppose its motion over a period of time can be represented by the four displacement vectors depicted
in FlauRE $3-49 .$ Let the vectors have magnitudes $A=8.0 \mathrm{cm}, B=$
$16 \mathrm{cm}, C=23 \mathrm{cm},$ and $D=5.6 \mathrm{cm} .$ (a) What is the total displacement of the food particle? Give its direction relative to the direction
of vector $\overline{\mathbf{A}}$ . (b) If the average speed of the particle is $0.010 \mathrm{mm} / \mathrm{s},$
what is its average velocity over the time interval of the four displacements?

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Problem 76

Initially, a particle is moving at 4.10 $\mathrm{m} / \mathrm{s}$ at an angle of $33.5^{\circ}$
above the horizontal. Two seconds later, its velocity is 6.05 $\mathrm{m} / \mathrm{s}$
at an angle of $59.0^{\circ}$ below the horizontal. What was the particle's
average acceleration during these 2.00 seconds?

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Problem 77

A passenger on a stopped bus notices that rain is falling vertically just outside the window. When the bus moves with constant velocity, the passenger observes that the falling raindrops are now
making an angle of 15^ \circ with respect to the vertical. (a) What is the ratio of the speed of the raindrops to the speed of the bus? (b) Find
the speed of the raindrops, given that the bus is moving with a
speed of 18 $\mathrm{m} / \mathrm{s} .$

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Problem 78

Predict/Calculate Suppose we orient the $x$ axis of a two-dimensional coordinate system along the beach at Waikiki, as shown in FIGURE $3-50 .$ Waves approaching the beach have a velocity relative to the shore given by $\overrightarrow{\mathbf{v}}_{\mathrm{ws}}=(-1.3 \mathrm{m} / \mathrm{s}) \hat{\mathbf{y}}$ . Surfers move more rapidly than the waves, but at an angle $\theta$ to the beach. The angle is chosen so that the surfers approach the shore with the
same speed as the waves along the $y$ direction. (a) If a surfer has
a speed of 7.2 $\mathrm{m} / \mathrm{s}$ relative to the water, what is his direction of
motion $\theta$ relative to the beach? (b) What his the surfer's velocity relative to the wave? (c) If the surfer's speed is increased, will the
angle in part (a) increase or decrease? Explain.

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Problem 79

Predict/Calculate The Longitude Problem In $1759,$ John Harrison
$(1693-1776)$ completed his fourth precision chronometer, the $\mathrm{H} 4,$ which eventually won the celebrated Longitude Prize. (For the human drama behind the Longitude Prize, see Longitude, by
Dava Sobel.) When the minute hand of the H4 indicates 10 minutes past the hour, it extends 3.0 $\mathrm{cm}$ in the horizontal direction. (a) How long is the H4's minute hand? (b) At 10 minutes past the hour, is the extension of the minute hand in the vertical direction
more than, less than, or equal to 3.0 $\mathrm{cm}$ ? Explain. (c) Calculate the
vertical extension of the minute hand at 10 minutes past the hour.

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Problem 80

Referring to Example $3-11,$ (a) what heading must the boat
have if it is to land directly across the river from its starting point?
(b) How much time is required for this trip if the river is 25.0 $\mathrm{m}$ wide? (c) Suppose the speed of the boat is increased, but it is still
desired to land directly across from the starting point. Should the
boat's heading be more upstream, more downstream, or the same
as in part (a)? Explain.

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Problem 81

Vector $\overrightarrow{\mathbf{A}}$ points in the negative $x$ direction. Vector $\overrightarrow{\mathbf{B}}$ points
at an angle of $30.0^{\circ}$ above the positive $x$ axis. Vector $\overrightarrow{\mathbf{C}}$ has a magnitude of 15 $\mathrm{m}$ and points in a direction $40.0^{\circ}$ below the positive
$x$ axis. Given that $\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}=0,$ find the magnitudes of $\overrightarrow{\mathbf{A}}$
and $\overrightarrow{\mathbf{B}} .$

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Problem 82

As two boats approach the marina, the velocity of boat 1 relative to boat 2 is 2.15 $\mathrm{m} / \mathrm{s}$ in a direction $47.0^{\circ}$ east of north. If boat
1 has a velocity that is 0.775 $\mathrm{m} / \mathrm{s}$ due north, what is the velocity
(magnitude and direction) of boat 2?

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Problem 83

BIO Motion Camouflage in Dragonflies
Dragonflies, whose ancestors were once the size of hawks, have
prowled the skies in search of small flying insects for over 250 million years. Faster and more maneuverable than any other insect, they even fold their front two legs in flight and tuck them behind their head to be as streamlined as possible. They also employ an
intriguing stalking strategy known as "motion camouflage" to approach their prey almost undetected. The basic idea of motion camouflage is for the dragonfly to move in such a way that the line of sight from the prey to the dragonfly is always in the same direction. Moving in this way, the dragonfly appears almost motionless to its prey, as if it were an object at infinity. Eventually the prey notices the dragonfly has grown in size and is therefore closer, but by that time it's too late for the prey to evade capture. A typical capture scenario is shown in FlGURE $3-51,$ where the prey moves in the positive $y$ direction with the constant speed $v_{p}=0.750 \mathrm{m} / \mathrm{s},$ and the dragonfly moves at an angle $\theta=48.5^{\circ}$ to the $x$ axis with the constant speed $v_{\mathrm{d}}$ . If the dragonfly chooses its speed correctly, the line of sight from the prey to the dragonfly will always be in the same direction-parallel to the $x$ axis in this case.
What speed must the dragonfly have if the line of sight, which
is parallel to the $x$ axis initially, is to remain parallel to the $x$ axis?
$$
\begin{array}{ll}{\text { A. } 0.562 \mathrm{m} / \mathrm{s}} & {\text { B. } 0.664 \mathrm{m} / \mathrm{s}} \\ {\text { C. } 1.00 \mathrm{m} / \mathrm{s}} & {\text { D. } 1.13 \mathrm{m} / \mathrm{s}}\end{array}
$$

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Problem 84

BIO Motion Camouflage in Dragonflies
Dragonflies, whose ancestors were once the size of hawks, have
prowled the skies in search of small flying insects for over 250 million years. Faster and more maneuverable than any other insect, they even fold their front two legs in flight and tuck them behind their head to be as streamlined as possible. They also employ an
intriguing stalking strategy known as "motion camouflage" to approach their prey almost undetected. The basic idea of motion camouflage is for the dragonfly to move in such a way that the line of sight from the prey to the dragonfly is always in the same direction. Moving in this way, the dragonfly appears almost motionless to its prey, as if it were an object at infinity. Eventually the prey notices the dragonfly has grown in size and is therefore closer, but by that time it's too late for the prey to evade capture. A typical capture scenario is shown in FlGURE $3-51,$ where the prey moves in the positive $y$ direction with the constant speed $v_{p}=0.750 \mathrm{m} / \mathrm{s},$ and the dragonfly moves at an angle $\theta=48.5^{\circ}$ to the $x$ axis with the constant speed $v_{\mathrm{d}}$ . If the dragonfly chooses its speed correctly, the line of sight from the prey to the dragonfly will always be in the same direction-parallel to the $x$ axis in this case.
Suppose the dragonfly now approaches its prey along a path
with $\theta>48.5^{\circ},$ but it still keeps the line of sight parallel to the
$x$ axis. Is the speed of the dragonfly in this new case greater than,
less than, or equal to its speed in Problem 83$?$

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Problem 85

BIO Motion Camouflage in Dragonflies
Dragonflies, whose ancestors were once the size of hawks, have
prowled the skies in search of small flying insects for over 250 million years. Faster and more maneuverable than any other insect, they even fold their front two legs in flight and tuck them behind their head to be as streamlined as possible. They also employ an
intriguing stalking strategy known as "motion camouflage" to approach their prey almost undetected. The basic idea of motion camouflage is for the dragonfly to move in such a way that the line of sight from the prey to the dragonfly is always in the same direction. Moving in this way, the dragonfly appears almost motionless to its prey, as if it were an object at infinity. Eventually the prey notices the dragonfly has grown in size and is therefore closer, but by that time it's too late for the prey to evade capture. A typical capture scenario is shown in FlGURE $3-51,$ where the prey moves in the positive $y$ direction with the constant speed $v_{p}=0.750 \mathrm{m} / \mathrm{s},$ and the dragonfly moves at an angle $\theta=48.5^{\circ}$ to the $x$ axis with the constant speed $v_{\mathrm{d}}$ . If the dragonfly chooses its speed correctly, the line of sight from the prey to the dragonfly will always be in the same direction-parallel to the $x$ axis in this case.
If the dragonfly approaches its prey with a speed of $0.950 \mathrm{m} / \mathrm{s},$
what angle $\theta$ is required to maintain a constant line of sight parallel to the $x$ axis?
$$
\begin{array}{ll}{\text { A. } 37.9^{\circ}} & {\text { B. } 38.3^{\circ}} \\ {\text { C. } 51.7^{\circ}} & {\text { D. } 52.1^{\circ}}\end{array}
$$

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Problem 86

BIO Motion Camouflage in Dragonflies
Dragonflies, whose ancestors were once the size of hawks, have
prowled the skies in search of small flying insects for over 250 million years. Faster and more maneuverable than any other insect, they even fold their front two legs in flight and tuck them behind their head to be as streamlined as possible. They also employ an
intriguing stalking strategy known as "motion camouflage" to approach their prey almost undetected. The basic idea of motion camouflage is for the dragonfly to move in such a way that the line of sight from the prey to the dragonfly is always in the same direction. Moving in this way, the dragonfly appears almost motionless to its prey, as if it were an object at infinity. Eventually the prey notices the dragonfly has grown in size and is therefore closer, but by that time it's too late for the prey to evade capture. A typical capture scenario is shown in FlGURE $3-51,$ where the prey moves in the positive $y$ direction with the constant speed $v_{p}=0.750 \mathrm{m} / \mathrm{s},$ and the dragonfly moves at an angle $\theta=48.5^{\circ}$ to the $x$ axis with the constant speed $v_{\mathrm{d}}$ . If the dragonfly chooses its speed correctly, the line of sight from the prey to the dragonfly will always be in the same direction-parallel to the $x$ axis in this case.
In a similar situation, the dragonfly is observed to fly at a constant speed, but its angle to the $x$ axis gradually increases from $20^{\circ}$ to $40^{\circ} .$ Which of the following is the best explanation for this behavior?
A. The prey is moving in the positive $y$ direction and maintaining a constant speed.
B. The prey is moving in the positive $y$ direction and increasing its speed.
C. The prey is moving in the positive $y$ direction and decreasing its speed.
D. The prey is moving in the positive $x$ direction and maintaining a constant speed.

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Problem 87

Referring to Example $3-11$ Suppose the speed of the boat relative
to the water is 7.0 $\mathrm{m} / \mathrm{s}$ . (a) At what angle to the $x$ axis must the
boat be headed if it is to land directly across the river from its starting position? (b) If the speed of the boat relative to the water is
increased, will the angle needed to go directly across the river
increase, decrease, or stay the same? Explain.

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Problem 88

Referring to Example $3-11$ Suppose that the boat has a speed of
6.7 $\mathrm{m} / \mathrm{s}$ relative to the water, and that the dock on the opposite
shore of the river is at the location $x=55 \mathrm{m}$ and $y=28 \mathrm{m}$ relative to the starting point of the boat. (a) At what angle relative to the
$x$ axis must the boat be pointed in order to reach the other dock?
(b) With the angle found in part (a), what is the speed of the boat
relative to the ground?

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