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Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 3

Vectors - all with Video Answers

Educators

+ 5 more educators

Chapter Questions

05:00

Problem 1

Two points in the $x y$ plane have Cartesian coordinates (2.00,-4.00) $\mathrm{m}$ and $(-3.00,3.00) \mathrm{m} .$ Determine
(a) the distance between these points and
(b) their polar coordinates.

Keshav Singh
Keshav Singh
Numerade Educator
03:47

Problem 2

Two points in a plane have polar coordinates $\left(2.50 \mathrm{~m}, 30.0^{\circ}\right)$ and $\left(3.80 \mathrm{~m}, 120.0^{\circ}\right) .$ Determine (a) the Cartesian coordinates of these points and (b) the distance between them.

Darren Wilson
Darren Wilson
Numerade Educator
08:22

Problem 3

The polar coordinates of a certain point are $(r=4.30 \mathrm{~cm}$, $\theta=214^{\circ}$ ). (a) Find its Cartesian coordinates $x$ and $y .$ Find the polar coordinates of the points with Cartesian coordinates
(b) $(-x, y)$
(c) $(-2 x,-2 y),$ and
(d) $(3 x,-3 y)$

Susan Hallstrom
Susan Hallstrom
Numerade Educator
03:11

Problem 4

Let the polar coordinates of the point $(x, y)$ be $(r, \theta)$. Determine the polar coordinates for the points
(a) $(-x, y)$,
(b) $(-2 x,-2 y),$ and
(c) $(3 x,-3 y)$.

Darren Wilson
Darren Wilson
Numerade Educator
01:12

Problem 5

Why is the following situation impossible? A skater glides along a circular path. She defines a certain point on the circle as her origin. Later on, she passes through a point at which the distance she has traveled along the path from the origin is smaller than the magnitude of her displacement vector from the origin.

Geoffrey Brandt
Geoffrey Brandt
Numerade Educator
01:12

Problem 6

Vector $\overrightarrow{\mathrm{A}}$ has a magnitude of 29 units and points in the positive $y$ direction. When vector $\overrightarrow{\mathbf{B}}$ is added to $\overrightarrow{\mathbf{A}},$ the resultant vector $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ points in the negative $y$ direction with a magnitude of 14 units. Find the magnitude and direction of $\mathbf{B}$.

Darren Wilson
Darren Wilson
Numerade Educator
04:00

Problem 7

$\begin{array}{llll}\text { A force } \overrightarrow{\mathbf{F}}_{1} & \text { of } & \text { magnitude } & 6.00\end{array}$ units acts on an object at the origin in a direction $\theta=30.0^{\circ}$ above the positive $x$ axis (Fig. $\mathrm{P} 3.7$ ). $\mathrm{A}$ second force $\overrightarrow{\mathbf{F}}_{2}$ of magnitude 5.00 units acts on the object in the direction of the positive $y$ axis. Find graphically the magnitude and direction of the resultant force $\overrightarrow{\mathbf{F}}_{1}+\overrightarrow{\mathbf{F}}_{2}$

Darren Wilson
Darren Wilson
Numerade Educator
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Problem 8

Three displacements are $\overrightarrow{\mathbf{A}}=200 \mathrm{~m}$ due south, $\overrightarrow{\mathbf{B}}=250 \mathrm{~m}$ due west, and $\overrightarrow{\mathbf{C}}=150 \mathrm{~m}$ at $30.0^{\circ}$ east of north. (a) Construct a separate diagram for each of the following possible ways of adding these vectors: $\overrightarrow{\mathbf{R}}_{1}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}} ; \overrightarrow{\mathbf{R}}_{2}=\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}+\overrightarrow{\mathbf{A}}$
$\overrightarrow{\mathbf{R}}_{3}=\overrightarrow{\mathbf{C}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{A}}$. (b) Explain what you can conclude from comparing the diagrams.

Gregory Devenport
Gregory Devenport
Numerade Educator
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Problem 9

The displacement vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ shown in Figure $\mathbf{P} 3.9$
both have magnitudes of $3.00 \mathrm{~m}$. The direction of vec-
tor $\overrightarrow{\mathbf{A}}$ is $\theta=30.0^{\circ}$. Find gra-
(b) $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$,
(c) $\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}},$ and
(d) $\overrightarrow{\mathbf{A}}-2 \overrightarrow{\mathbf{B}}$.
(Report all angles counterclockwise from the positive $x$ axis.)

Gregory Devenport
Gregory Devenport
Numerade Educator
06:52

Problem 10

A roller-coaster car moves $200 \mathrm{ft}$ horizontally and then rises 135 ft at an angle of $30.0^{\circ}$ above the horizontal. It next travels $135 \mathrm{ft}$ at an angle of $40.0^{\circ}$ downward. What is its displacement from its starting point? Use graphical techniques.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
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Problem 11

A minivan travels straight north in the right lane of a divided highway at $28.0 \mathrm{~m} / \mathrm{s}$. A camper passes the minivan and then changes from the left lane into the right lane. As it does so, the camper's path on the road is a straight displacement at $8.50^{\circ}$ east of north. To avoid cutting off the minivan, the north-south distance between the camper's back bumper and the minivan's front bumper should not decrease. (a) Can the camper be driven to satisfy this requirement? (b) Explain your answer.

Gregory Devenport
Gregory Devenport
Numerade Educator
01:28

Problem 12

A person walks $25.0^{\circ}$ north of east for $3.10 \mathrm{~km} .$ How far would she have to walk due north and due east to arrive at the same location?

Averell Hause
Averell Hause
Carnegie Mellon University
02:32

Problem 13

Your dog is running around the grass in your back yard. He undergoes successive displacements $3.50 \mathrm{~m}$ south, $8.20 \mathrm{~m}$ northeast, and $15.0 \mathrm{~m}$ west. What is the resultant displacement?

Geoffrey Brandt
Geoffrey Brandt
Numerade Educator
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Problem 14

Given the vectors $\overrightarrow{\mathbf{A}}=2.00 \hat{\mathbf{i}}+6.00 \hat{\mathbf{j}} \quad$ and $\quad \overrightarrow{\mathbf{B}}$
$3.00 \hat{\mathrm{i}}-2.00 \hat{\mathrm{j}}$
(a) draw the vector sum $\overrightarrow{\mathbf{C}}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$
and the vector difference $\overrightarrow{\mathbf{D}}=\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$.
(b) Calculate $\overrightarrow{\mathbf{C}}$ and $\overrightarrow{\mathbf{D}},$ in terms of unit vectors. (c) Calculate $\overrightarrow{\mathbf{C}}$ and $\overrightarrow{\mathbf{D}}$ in terms of polar coordinates, with angles measured with respect to the positive $x$ axis.

Gregory Devenport
Gregory Devenport
Numerade Educator
05:58

Problem 15

The helicopter view in Fig. P3.15 shows two people pulling on a stubborn mule. The person on the right pulls with a force $\overrightarrow{\mathbf{F}}_{1}$ of magnitude $120 \mathrm{~N}$ and direction of $\theta_{1}=60.0^{\circ}$ The person on the left pulls with a force $\overrightarrow{\mathbf{F}}_{2}$ of magnitude $80.0 \mathrm{~N}$ and direction of $\theta_{2}=75.0^{\circ} .$ Find (a) the single force that is equivalent to the two forces shown and (b) the force that a third person would have to exert on the mule to make the resultant force equal to zero. The forces are measured in units of newtons (symbolized $\mathrm{N}$ ).

Vishal Gupta
Vishal Gupta
Numerade Educator
02:06

Problem 16

A snow-covered ski slope makes an angle of $35.0^{\circ}$ with the horizontal. When a ski jumper plummets onto the hill, a parcel of splashed snow is thrown up to a maximum displacement of $1.50 \mathrm{~m}$ at $16.0^{\circ}$ from the vertical in the uphill direction as shown in Figure $\mathrm{P} 3.16$. Find the components of its maximum displacement (a) parallel to the surface and (b) perpendicular to the surface.

Anand Jangid
Anand Jangid
Numerade Educator
08:09

Problem 17

Consider the three displacement vectors $\overrightarrow{\mathbf{A}}=$ $(3 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}) \quad \mathbf{m}, \quad \overrightarrow{\mathbf{B}}=(\hat{\mathbf{i}}-4 \hat{\mathbf{j}}) \quad \mathrm{m},$ and $\overrightarrow{\mathbf{C}}=(-2 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}) \quad \mathbf{m}$
Use the component method to determine (a) the magnitude and direction of $\overrightarrow{\mathbf{D}}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}$ and
(b) the magnitude and direction of $\overrightarrow{\mathbf{E}}=-\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}$.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
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Problem 18

Vector $\overrightarrow{\mathbf{A}}$ has $x$ and $y$ components of $-8.70 \mathrm{~cm}$ and $\begin{array}{llllll}15.0 & \mathrm{~cm}, & \text { respectively; } & \text { vector } & \overrightarrow{\mathbf{B}} & \text { has } & x & \text { and } & y & \operatorname{com}-\end{array}$
$\begin{array}{llllll}\text { ponents of } 13.2 & \mathrm{~cm} & \text { and } & -6.60 & \mathrm{~cm}, & \text { respectively. }\end{array}$
If $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}+3 \overrightarrow{\mathbf{C}}=0,$ what are the components of $\overrightarrow{\mathbf{C}} ?$

Gregory Devenport
Gregory Devenport
Numerade Educator
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Problem 19

The vector $\overrightarrow{\mathbf{A}}$ has $x, y,$ and $z$ components of 8.00 12.0, and -4.00 units, respectively. (a) Write a vector expression for $\overrightarrow{\mathbf{A}}$ in unit-vector notation. (b) Obtain a unit-vector expression for a vector $\overrightarrow{\mathbf{B}}$ one-fourth the length of $\overrightarrow{\mathbf{A}}$ pointing in the same direction as $\overrightarrow{\mathbf{A}}$. (c) Obtain a unit-vector expression for a vector $\overrightarrow{\mathrm{C}}$ three times the length of $\overrightarrow{\mathbf{A}}$ pointing in the direction opposite the direction of $\overrightarrow{\mathbf{A}}$.

Gregory Devenport
Gregory Devenport
Numerade Educator
04:06

Problem 20

Given the displacement vectors $\overrightarrow{\mathbf{A}}=(3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \mathrm{m}$
and $\overrightarrow{\mathbf{B}}=(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-7 \hat{\mathbf{k}}) \quad \mathrm{m}, \quad$ find $\quad$ the $\quad$ magnitudes $\quad$ of the following vectors and express each in terms of its rectangular components. (a) $\overrightarrow{\mathbf{C}}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ (b) $\overrightarrow{\mathbf{D}}=$ $2 \overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$.

Darren Wilson
Darren Wilson
Numerade Educator
01:12

Problem 21

Vector $\overrightarrow{\mathbf{A}}$ has a negative $x$ component 3.00 units in length and a positive $y$ component 2.00 units in length.
(a) Determine an expression for $\overrightarrow{\mathbf{A}}$ in unit-vector notation. (b) Determine the magnitude and direction of $\overrightarrow{\mathbf{A}}$. (c) What vector $\overrightarrow{\mathbf{B}}$ when added to $\overrightarrow{\mathbf{A}}$ gives a resultant vector with no $x$ component and a negative $y$ component 4.00 units in length?

Anand Jangid
Anand Jangid
Numerade Educator
06:56

Problem 22

Three displacement vectors of a croquet ball are shown in Figure $\mathrm{P} 3.22$, where $|\overrightarrow{\mathbf{A}}|=20.0 \quad$ units $, \quad|\overrightarrow{\mathbf{B}}|=$ 40.0 units, and $|\overrightarrow{\mathbf{C}}|=30.0$ units. Find (a) the resultant in unit-vector notation and (b) the magnitude and direction of the resultant displacement.

Yaqub Khan
Yaqub Khan
Numerade Educator
02:23

Problem 23

(a) Taking $\overrightarrow{\mathbf{A}}=(6.00 \hat{\mathbf{i}}-8.00 \hat{\mathbf{j}})$ units, $\overrightarrow{\mathbf{B}}=(-8.00 \hat{\mathbf{i}}+3.00 \hat{\mathbf{j}})$ units, and $\overrightarrow{\mathbf{C}}=(26.0 \hat{\mathbf{i}}+19.0 \hat{\mathbf{j}})$ units, determine $a$ and $b$ such that $a \overrightarrow{\mathbf{A}}+b \overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}=0$.
(b) A student has learned that a single equation cannot be solved to determine values for more than one unknown in it. How would you explain to him that both $a$ and $b$ can be determined from the single equation used in part (a)?

Geoffrey Brandt
Geoffrey Brandt
Numerade Educator
05:41

Problem 24

Vector $\overrightarrow{\mathbf{B}}$ has $x, y,$ and $z$ components of $4.00,6.00,$ and 3.00 units, respectively. Calculate (a) the magnitude of $\overrightarrow{\mathbf{B}}$ and (b) the angle that $\overrightarrow{\mathbf{B}}$ makes with each coordinate axis.

Darren Wilson
Darren Wilson
Numerade Educator
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Problem 25

Use the component method to add the vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ shown in Figure $\mathrm{P} 3.9 .$ Both vectors have mag- nitudes of $3.00 \mathrm{~m}$ and vector $\overrightarrow{\mathrm{A}}$ makes an angle of $\theta=30.0^{\circ}$ with the $x$ axis. Express the resultant $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ in unit-vector notation.

Gregory Devenport
Gregory Devenport
Numerade Educator
02:22

Problem 26

A girl delivering newspapers covers her route by traveling 3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east. (a) What is her resultant displacement? (b) What is the total distance she travels?

Darren Wilson
Darren Wilson
Numerade Educator
11:04

Problem 27

A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude of $150 \mathrm{~cm}$ and makes an angle of $120^{\circ}$ with the positive $x$ axis. The resultant displacement has a magnitude of $140 \mathrm{~cm}$ and is directed at an angle of $35.0^{\circ}$ to the positive $x$ axis. Find the magnitude and direction of the second displacement.

Yaqub Khan
Yaqub Khan
Numerade Educator
06:12

Problem 28

Figure $\mathrm{P} 3.28$ illustrates typical proportions of male $(\mathrm{m})$ and female $(\mathrm{f})$ anatomies. The displacements $\overrightarrow{\mathrm{d}}_{1 \mathrm{~m}}$ and $\overrightarrow{\mathrm{d}}_{1 \mathrm{f}}$ from the soles of the feet to the navel have magnitudes of $104 \mathrm{~cm}$ and $84.0 \mathrm{~cm},$ respectively. The displacements $\overrightarrow{\mathbf{d}}_{2 \mathrm{~m}}$ and $\overrightarrow{\mathrm{d}}_{2 \mathrm{f}}$ from the navel to outstretched fingertips have magnitudes of $100 \mathrm{~cm}$ and $86.0 \mathrm{~cm},$ respectively. Find the vector sum of these displacements $\overrightarrow{\mathbf{d}}_{3}=\overrightarrow{\mathbf{d}}_{1}+\overrightarrow{\mathbf{d}}_{2}$ for both people.

Keshav Singh
Keshav Singh
Numerade Educator
04:16

Problem 29

Review. As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction $60.0^{\circ}$ north of west with a speed of $41.0 \mathrm{~km} / \mathrm{h}$.
(a) What is the unit-vector expression for the velocity of the hurricane? It maintains this velocity for $3.00 \mathrm{~h},$ at which time the course of the hurricane suddenly shifts due north, and its speed slows to a constant $25.0 \mathrm{~km} / \mathrm{h} .$ This new velocity is maintained for $1.50 \mathrm{~h}$.
(b) What is the unit-vector expression for the new velocity of the hurricane?
(c) What is the unit-vector expression for the displacement of the hurricane during the first $3.00 \mathrm{~h}$ ?
(d) What is the unit-vector expression for the displacement of the hurricane during the latter $1.50 \mathrm{~h}$ ?
(e) How far from Grand Bahama is the eye $4.50 \mathrm{~h}$ after it passes over the island?

Geoffrey Brandt
Geoffrey Brandt
Numerade Educator
08:15

Problem 30

In an assembly operation illustrated in Figure $\mathrm{P} 3.30, \mathrm{a}$ robot moves an object first straight upward and then also to the east, around an arc forming one-quarter of a circle of radius $4.80 \mathrm{~cm}$ that lies in an east-west vertical plane. The robot then moves the object upward and to the north, through one-quarter of a circle of radius $3.70 \mathrm{~cm}$ that lies in a north-south vertical plane. Find (a) the magnitude of the total displacement of the object and (b) the angle the total displacement makes with the vertical.

Yaqub Khan
Yaqub Khan
Numerade Educator
05:34

Problem 31

Review. You are standing on the ground at the origin of a coordinate system. An airplane flies over you with constant velocity parallel to the $x$ axis and at a fixed height of $7.60 \times$ $10^{3} \mathrm{~m}$. At time $t=0,$ the airplane is directly above you so that the vector leading from you to it is $\overrightarrow{\mathbf{P}}_{0}=7.60 \times 10^{3} \hat{\mathbf{j}} \mathrm{m}$ At $t=30.0 \mathrm{~s},$ the position vector leading from you to the airplane is $\overrightarrow{\mathbf{P}}_{30}=\left(8.04 \times 10^{3} \hat{\mathbf{i}}+7.60 \times 10^{3} \mathbf{j}\right) \mathrm{m}$ as suggested in
Figure $\mathrm{P} 3.31 .$ Determine the magnitude and orientation of the airplane's position vector at $t=45.0 \mathrm{~s}$.

Vishal Gupta
Vishal Gupta
Numerade Educator
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Problem 32

Why is the following situation impossible? A shopper pushing a cart through a market follows directions to the canned goods and moves through a displacement 8.00 ì m down one aisle. He then makes a $90.0^{\circ}$ turn and moves $3.00 \mathrm{~m}$ along the $y$ axis. He then makes another $90.0^{\circ}$ turn and moves $4.00 \mathrm{~m}$ along the $x$ axis. Every shopper who follows these directions correctly ends up $5.00 \mathrm{~m}$ from the starting point.

Gregory Devenport
Gregory Devenport
Numerade Educator
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Problem 33

In Figure P3.33, the line segment represents a path from the point with position vector $(5 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}) \mathrm{m}$ to the point with
location $(16 \hat{\mathbf{i}}+12 \hat{\mathbf{j}}) \quad \mathrm{m} .$ Point ( $\mathrm{A}$ is along this path, a fraction $f$ of the way to the destination.
(a) Find the position vector of point $@$ in terms of $f$. (b) Evaluate the expression from part (a) for $f=0 .$ (c) Explain whether the result in part (b) is reasonable. (d) Evaluate the expression for $f=1$. (e) Explain whether the result in part (d) is reasonable.

Gregory Devenport
Gregory Devenport
Numerade Educator
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Problem 34

You are spending the summer as an assistant learning how to navigate on a large ship carrying freight across Lake Erie. One day, you and your ship are to travel across the lake a distance of $200 \mathrm{~km}$ traveling due north from your origin port to your destination port. Just as you leave your origin port, the navigation electronics go down. The captain continues sailing, claiming he can depend on his years of experience on the water as a guide. The engineers work on the navigation system while the ship continues to sail, and winds and waves push it off course. Eventually, enough of the navigation system comes back up to tell you your location. The system tells you that your current position is $50.0 \mathrm{~km}$ north of the origin port and $25.0 \mathrm{~km}$ east of the port. The captain is a little embarrassed that his ship is so far off course and barks an order to you to tell him immediately what heading he should set from your current position to the destination port. Give him an appropriate heading angle.

Gregory Devenport
Gregory Devenport
Numerade Educator
07:43

Problem 35

A person going for a walk follows the path shown in Figure $\mathrm{P} 3.35 .$ The total trip consists of four straight-line paths. At the end of the walk, what is the person's resultant displacement measured from the starting point?

Sachin Rao
Sachin Rao
Numerade Educator
05:19

Problem 36

A ferry transports tourists between three islands. It sails from the first island to the second island, $4.76 \mathrm{~km}$ away, in a direction $37.0^{\circ}$ north of east. It then sails from the second island to the third island in a direction $69.0^{\circ}$ west of north. Finally it returns to the first island, sailing in a direction $28.0^{\circ}$ east of south. Calculate the distance between (a) the second and third islands and (b) the first and third islands.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:11

Problem 37

Two vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ have precisely equal magnitudes. For the magnitude of $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ to be 100 times larger than the magnitude of $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}},$ what must be the angle between them?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
06:34

Problem 38

Two vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ have precisely equal magnitudes. For the magnitude of $\vec{A}+\vec{B}$ to be larger than the magnitude of $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ by the factor $n$, what must be the angle between them?

Darren Wilson
Darren Wilson
Numerade Educator
05:09

Problem 39

Review. The biggest stuffed animal in the world is a snake $420 \mathrm{~m}$ long, constructed by Norwegian children. Suppose the snake is laid out in a park as shown in Figure $\quad$ P3.39, forming two straight sides of a $105^{\circ}$ angle, with one side $240 \mathrm{~m}$ long. Olaf and Inge run a race they invent. Inge runs directly from the tail of the snake to its head, and Olaf starts from the same place at the same moment but runs along the snake. (a) If both children run steadily at $12.0 \mathrm{~km} / \mathrm{h}$, Inge reaches the head of the snake how much earlier than Olaf? (b) If Inge runs the race again at a constant speed of $12.0 \mathrm{~km} / \mathrm{h},$ at what constant speed must Olaf run to reach the end of the snake at the same time as Inge?

Surjit Tewari
Surjit Tewari
Numerade Educator
05:28

Problem 40

Ecotourists use their global positioning system indicator to determine their location inside a botanical garden as latitude 0.00243 degree south of the equator, longitude 75.64238 degrees west. They wish to visit a tree at latitude 0.00162 degree north, longitude 75.64426 degrees west. (a) Determine the straight-line distance and the direction in which they can walk to reach the tree as follows. First model the Earth as a sphere of radius $6.37 \times 10^{6} \mathrm{~m}$ to determine the westward and northward displacement components required, in meters. Then model the Earth as a flat surface to complete the calculation. (b) Explain why it is possible to use these two geometrical models together to solve the problem.

Jacob Schulze
Jacob Schulze
Numerade Educator
01:16

Problem 41

A vector is given by $\overrightarrow{\mathbf{R}}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}} .$ Find (a) the magnitudes of the $x, y,$ and $z$ components; (b) the magnitude of $\overrightarrow{\mathbf{R}} ;$ and $(c)$ the angles between $\overrightarrow{\mathbf{R}}$ and the $x, y,$ and
$z$ axes.

Mayukh Banik
Mayukh Banik
Numerade Educator
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Problem 42

You are working as an assistant to an air-traffic controller at the local airport, from which small airplanes take off and land. Your job is to make sure that airplanes are not closer to each other than a minimum safe separation distance of $2.00 \mathrm{~km}$. You observe two small aircraft on your radar screen, out over the ocean surface. The first is at altitude $800 \mathrm{~m}$ above the surface, horizontal distance $19.2 \mathrm{~km}$, and $25.0^{\circ}$ south of west. The second aircraft is at altitude $1100 \mathrm{~m},$ horizontal distance $17.6 \mathrm{~km},$ and $20.0^{\circ}$ south of west. Your supervisor is concerned that the two aircraft are too close together and asks for a separation distance for the two airplanes. (Place the $x$ axis west, the $y$ axis south, and the $z$ axis vertical. $)$

Gregory Devenport
Gregory Devenport
Numerade Educator
00:57

Problem 43

Review. The instantaneous position of an object is specified by its position vector leading from a fixed origin to the location of the object, modeled as a particle. Suppose for a certain object the position vector is a function of time given by $\overrightarrow{\mathbf{r}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 t \hat{\mathbf{k}},$ where $\overrightarrow{\mathbf{r}}$ is in meters and $t$ is in seconds. (a) Evaluate $d \overrightarrow{\mathbf{r}} / d t$.
(b) What physical quantity does $d \overrightarrow{\mathbf{r}} / d t$ represent about the object?

Geoffrey Brandt
Geoffrey Brandt
Numerade Educator
04:36

Problem 44

Vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ have equal magnitudes of 5.00 . The sum of $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ is the vector $6.00 \hat{\mathbf{j}}$. Determine the angle between $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$.

Darren Wilson
Darren Wilson
Numerade Educator
01:03

Problem 45

A rectangular parallelepiped has dimensions $a, b,$ and $c$ as shown in Figure $\mathrm{P} 3.45$. (a) Obtain a vector expression for the face diagonal vector $\overrightarrow{\mathbf{R}}_{1} .$ (b) What is the magnitude of this vector? (c) Notice that $\overrightarrow{\mathbf{R}}_{1}, c \hat{\mathbf{k}},$ and $\overrightarrow{\mathbf{R}}_{2}$ make a right triangle. Obtain a vector expression for the body diagonal vector $\overrightarrow{\mathbf{R}}_{2}$.

Geoffrey Brandt
Geoffrey Brandt
Numerade Educator
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Problem 46

A pirate has buried his treasure on an island with five trees located at the points $(30.0 \mathrm{~m},-20.0 \mathrm{~m}),(60.0 \mathrm{~m}, 80.0 \mathrm{~m})$ $(-10.0 \mathrm{~m},-10.0 \mathrm{~m}),(40.0 \mathrm{~m},-30.0 \mathrm{~m}),$ and $(-70.0 \mathrm{~m}$$60.0 \mathrm{~m})$, all measured relative to some origin, as shown in Figure $\mathrm{P} 3.46 .$ His ship's log instructs you to start at tree $A$ and move toward tree $B$, but to cover only one-half the distance between $A$ and $B .$ Then move toward tree $C,$ covering one-third the distance between your current location and
C. Next move toward tree $D$, covering one-fourth the distance between where you are and $D .$ Finally move toward tree $E$, covering one-fifth the distance between you and $E$, stop, and dig. (a) Assume you have correctly determined the order in which the pirate labeled the trees as $A, B, C, D$, and $E$ as shown in the figure. What are the coordinates of the point where his treasure is buried? (b) What If? What if you do not really know the way the pirate labeled the trees? What would happen to the answer if you rearranged the order of the trees, for instance, to $B(30 \mathrm{~m},-20 \mathrm{~m}), A(60 \mathrm{~m},$ $80 \mathrm{~m}), E(-10 \mathrm{~m},-10 \mathrm{~m}), C(40 \mathrm{~m},-30 \mathrm{~m}),$ and $D(-70 \mathrm{~m}$ $60 \mathrm{~m}$ )? State reasoning to show that the answer does not depend on the order in which the trees are labeled.

Gregory Devenport
Gregory Devenport
Numerade Educator