64
Chapter 3
Vectors
Summary
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DEFINITIONS
Scalar quantities are those that have only a numerical value and no associated direction. Vector quantities have
both magnitude and direction and obey the laws of vector addition. The magnitude of a vector is always a positive
number.
CO N C E P T S A N D P R I N C I P L E S
When two or more vectors are added together, they
must all have the same units and all of them must be
S
the same type of quantity. We can add two vectors A
S
and B graphically. In this method (Active Fig. 3.6), the
S
S
S
S
resultant vector R ⫽ A ⫹ B runs from the tail of A to
S
the tip of B.
S
If a vector A has an x component Ax and a y component Ay, Sthe vector can be expressed in unit–vector
form as A ⫽ Ax î ⫹ Ay ĵ . In this notation, î is a unit
vector pointing in the positive x direction and ĵ is a
unit vector pointing in the positive y direction.
Because î and ĵ are unit vectors, 0 î 0 ⫽ 0 ĵ 0 ⫽ 1.
A second method of adding vectors involves components of the vectors. The x component Ax of the vector
S
S
A is equal to the projection of A along the x axis of a
coordinate system, where Ax ⫽ A cos u. The y compoS
S
nent Ay of A is the projection of A along the y axis,
where Ay ⫽ A sin u.
We can find the resultant of two or more vectors by
resolving all vectors into their x and y components,
adding their resultant x and y components, and then
using the Pythagorean theorem to find the magnitude
of the resultant vector. We can find the angle that the
resultant vector makes with respect to the x axis by
using a suitable trigonometric function.
Questions
䡺 denotes answer available in Student Solutions Manual/Study Guide; O denotes objective question
1. O Yes or no: Is each of the following quantities a vector?
(a) force (b) temperature (c) the volume of water in a
can (d) the ratings of a TV show (e) the height of a
building (f) the velocity of a sports car (g) the age of
the Universe
2. A book is moved once around the perimeter of a tabletop
with dimensions 1.0 m ⫻ 2.0 m. If the book ends up at its
initial position, what is its displacement? What is the distance traveled?
S
S
3. O Figure Q3.3 shows two vectors, D1 and D2. Which of the
S
S
possibilities (a) through (d) is the vector D2 ⫺ 2D1, or (e) is
it none of them?
4. O The cutting tool on a lathe is given two displacements,
one of magnitude 4 cm and one of magnitude 3 cm, in
each one of five situations (a) through (e) diagrammed
in Figure Q3.4. Rank these situations according to the
magnitude of the total displacement of the tool, putting
the situation with the greatest resultant magnitude first. If
the total displacement is the same size in two situations,
give those letters equal ranks.
(a)
(b)
(c)
(d)
(e)
Figure Q3.4
D1
D2
S
(a)
(b)
Figure Q3.3
(c)
(d)
5. O Let A represent a velocity vector pointing from the origin into the second quadrant. (a) Is its x component positive, negative, or zero? (b) Is its y component positive,
S
negative, or zero? Let B represent a velocity vector point-
Problems
ing from the origin into the fourth quadrant. (c) Is its x
component positive, negative, or zero? (d) Is its y component positive, negative, or zero? (e) Consider the vector
S
S
A ⫹ B. What, if anything, can you conclude about quadrants it must be in or cannot be in? (f) Now consider the
S
S
vector B ⫺ A. What, if anything, can you conclude about
quadrants it must be in or cannot be in?
6. O (i) What is the magnitude of the vector
110 î ⫺ 10k̂ 2 m>s? (a) 0 (b) 10 m/s (c) ⫺10 m/s
(d) 10 (e) ⫺10 (f) 14.1 m/s (g) undefined (ii) What
is the y component of this vector? (Choose from among
the same answers.)
7. O A submarine dives from the water surface at an angle
of 30° below the horizontal, following a straight path 50
m long. How far is the submarine then below the water
surface? (a) 50 m (b) sin 30° (c) cos 30° (d) tan 30°
(e) (50 m)/sin 30° (f) (50 m)/cos 30° (g) (50 m)/
tan 30° (h) (50 m)sin 30° (i) (50 m)cos 30°
(j) (50 m)tan 30° (k) (sin 30°)/50 m (l) (cos 30°)/50 m
(m) (tan 30°)/50 m (n) 30 m (o) 0 (p) none of these
answers
8. O (i) What is the x component of the vector shown in
Figure Q3.8? (a) 1 cm (b) 2 cm (c) 3 cm (d) 4 cm
(e) 6 cm (f) ⫺1 cm (g) ⫺2 cm (h) ⫺3 cm (i) ⫺4 cm
65
(j) ⫺6 cm (k) none of these answers (ii) What is the y
component of this vector? (Choose from among the same
answers.)
y, cm
2
⫺4 ⫺2
0
2
x, cm
⫺2
Figure Q3.8
S
9. O Vector A lies in the xy plane. (i) Both of its components
will be negative if it lies in which quadrant(s)? Choose all
that apply. (a) the first quadrant (b) the second quadrant (c) the third quadrant (d) the fourth quadrant
(ii) For what orientation(s) will its components have opposite signs? Choose from among the same possibilities.
S
10. If the component of vector A along the direction of vector
S
B is zero, what can you conclude about the two vectors?
11. Can the magnitude of a vector have a negative value?
Explain.
12. Is it possible to add a vector quantity to a scalar quantity?
Explain.
Problems
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with additional quizzing and conceptual questions.
1, 2, 3 denotes straightforward, intermediate, challenging; 䡺 denotes full solution available in Student Solutions Manual/Study
Guide ; 䊱 denotes coached solution with hints available at www.thomsonedu.com; 䡵 denotes developing symbolic reasoning;
䢇 denotes asking for qualitative reasoning;
denotes computer useful in solving problem
Section 3.1 Coordinate Systems
1. 䊱 The polar coordinates of a point are r ⫽ 5.50 m and
u ⫽ 240°. What are the Cartesian coordinates of this point?
2. Two points in a plane have polar coordinates (2.50 m,
30.0°) and (3.80 m, 120.0°). Determine (a) the Cartesian
coordinates of these points and (b) the distance between
them.
3. A fly lands on one wall of a room. The lower left-hand corner of the wall is selected as the origin of a two-dimensional
Cartesian coordinate system. If the fly is located at the
point having coordinates (2.00, 1.00) m, (a) how far is it
from the corner of the room? (b) What is its location in
polar coordinates?
4. The rectangular coordinates of a point are given by (2, y),
and its polar coordinates are (r, 30°). Determine y and r.
5. Let the polar coordinates of the point (x, y) be (r, u).
Determine the polar coordinates for the points (a) (⫺x, y),
(b) (⫺2x, ⫺2y), and (c) (3x, ⫺3y).
2 = intermediate;
3 = challenging;
䡺 = SSM/SG;
䊱
Section 3.2 Vector and Scalar Quantities
Section 3.3 Some Properties of Vectors
6. A plane flies from base camp to lake A, 280 km away in
the direction 20.0° north of east. After dropping off supplies it flies to lake B, which is 190 km at 30.0° west of
north from lake A. Graphically determine the distance
and direction from lake B to the base camp.
7. A surveyor measures the distance across a straight river by
the following method: starting directly across from a tree
on the opposite bank, she walks 100 m along the riverbank to establish a baseline. Then she sights across to the
tree. The angle from her baseline to the tree is 35.0°.
How wide is the river?
S
8. A force F1 of magnitude 6.00 units acts on an object at
the origin in a direction 30.0° above the positive x axis. A
S
second force F2 of magnitude 5.00 units acts on the object
in the direction of the positive y axis. Graphically find the
S
S
magnitude and direction of the resultant force F1 ⫹ F2.
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66
Chapter 3
Vectors
䊱
A skater glides along a circular path of radius 5.00 m. If
he coasts around one half of the circle, find (a) the magnitude of the displacement vector and (b) how far he
skated. (c) What is the magnitude of the displacement if
he skates all the way around the circle?
10. Arbitrarily define the “instantaneous vector height” of a
person as the displacement vector from the point halfway
between his or her feet to the top of the head. Make an
order-of-magnitude estimate of the total vector height of
all the people in a city of population 100 000 (a) at
10 o’clock on a Tuesday morning and (b) at 5 o’clock on
a Saturday morning. Explain your reasoning.
S
S
11. 䊱 Each of the displacement vectors A and B shown in
Figure P3.11 has a magnitude of 3.00 m. Graphically find
S
S
S
S
S
S
S
S
(a) A ⫹ B, (b) A ⫺ B, (c) B ⫺ A, and (d) A ⫺ 2B. Report
all angles counterclockwise from the positive x axis.
9.
18.
19.
20.
21.
y
B
22.
3.00 m
A
0m
3.0
30.0⬚
x
O
Figure P3.11
23.
Problems 11 and 32.
S
S
12. 䢇 Three displacementsS are A ⫽ 200 m due south, B ⫽
250 m due west, and C ⫽ 150 m at 30.0° east of north.
Construct a separate diagram for each Sof the
following
S
S
S
possible
ways
of
adding these vectors: R1 ⫽ A ⫹ B ⫹ C;
S
S
S
S
S
S
S
S
R2 ⫽ B ⫹ C ⫹ A; R3 ⫽ C ⫹ B ⫹ A. Explain what you can
conclude from comparing the diagrams.
13. A roller-coaster car moves 200 ft horizontally and then
rises 135 ft at an angle of 30.0° above the horizontal. It
next travels 135 ft at an angle of 40.0° downward. What is
its displacement from its starting point? Use graphical
techniques.
14. 䢇 A shopper pushing a cart through a store moves 40.0 m
down one aisle, then makes a 90.0° turn and moves 15.0 m.
He then makes another 90.0° turn and moves 20.0 m.
(a) How far is the shopper away from his original position? (b) What angle does his total displacement make
with his original direction? Notice that we have not specified whether the shopper turned right or left. Explain
how many answers are possible for parts (a) and (b) and
give the possible answers.
Section 3.4 Components of a Vector and Unit Vectors
15. 䊱 A vector has an x component of ⫺25.0 units and a y
component of 40.0 units. Find the magnitude and direction of this vector.
16. A person walks 25.0° north of east for 3.10 km. How far
would she have to walk due north and due east to arrive
at the same location?
17. 䢇 A minivan travels straight north in the right lane of a
divided highway at 28.0 m/s. A camper passes the minivan and then changes from the left into the right lane. As
it does so, the camper’s path on the road is a straight displacement at 8.50° east of north. To avoid cutting off the
minivan, the north–south distance between the camper’s
2 = intermediate;
3 = challenging;
䡺 = SSM/SG;
䊱
24.
25.
26.
rear bumper and the minivan’s front bumper should not
decrease. Can the camper be driven to satisfy this requirement? Explain your answer.
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?
Obtain expressions in component form for the position
vectors having the following polar coordinates: (a) 12.8 m,
150° (b) 3.30 cm, 60.0° (c) 22.0 in., 215°
A displacement vector lying in the xy plane has a magnitude of 50.0 m and is directed at an angle of 120° to the
positive x axis. What are the rectangular components of
this vector?
While exploring a cave, a spelunker starts at the entrance
and moves the following distances. She goes 75.0 m
north, 250 m east, 125 m at an angle 30.0° north of east,
and 150 m south. Find her resultant displacement from
the cave entrance.
A map suggests that Atlanta is 730 miles in a direction of
5.00° north of east from Dallas. The same map shows that
Chicago is 560 miles in a direction of 21.0° west of north
from Atlanta. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago.
A man pushing a mop across a floor causes it to undergo
two displacements. The first has a magnitude of 150 cm
and makes an angle of 120° with the positive x axis. The
resultant displacement has a magnitude of 140 cm and is
directed at an angle of 35.0° to the positive x axis. Find
the magnitude and direction of the second displacement.
S
S
Given the vectors A ⫽ 2.00 î ⫹ 6.00 ĵ and B ⫽ 3.00 î ⫺
S
S
S
2.00 ĵ , (a) draw the vector sum C ⫽ A ⫹ B and the vector
S
S
S
S
S
difference D ⫽ A ⫺ B. (b) Calculate C and D, first in
terms of unit vectors and then in terms of polar coordinates, with angles measured with respect to the ⫹x axis.
S
S
Consider the two vectors A ⫽ 3 î ⫺ 2ĵ and B ⫽ ⫺ î ⫺ 4 ĵ .
S
S
S
S
S
S
S
S
Calculate (a) A ⫹ B, (b) A ⫺ B, (c) 0 A ⫹ B 0 , (d) 0 A ⫺ B 0 ,
S
S
S
S
and (e) the directions of A ⫹ B and A ⫺ B.
A snow-covered ski slope makes an angle of 35.0° with the
horizontal. When a ski jumper plummets onto the hill, a
parcel of splashed snow projects to a maximum position
of 5.00 m at 20.0° from the vertical in the uphill direction
as shown in Figure P3.26. Find the components of its
maximum position (a) parallel to the surface and (b) perpendicular to the surface.
20.0°
35.0°
Figure P3.26
27. A particle undergoes the following consecutive displacements: 3.50 m south, 8.20 m northeast, and 15.0 m west.
What is the resultant displacement?
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Problems
28. In a game of American football, a quarterback takes the
ball from the line of scrimmage, runs backward a distance
of 10.0 yards, and then runs sideways parallel to the line
of scrimmage for 15.0 yards. At this point, he throws a forward pass 50.0 yards straight downfield perpendicular to
the line of scrimmage. What is the magnitude of the football’s resultant displacement?
29. A novice golfer on the green takes three strokes to sink
the ball. The successive displacements of the ball are 4.00 m
to the north, 2.00 m northeast, and 1.00 m at 30.0° west
of south. Starting at the same initial point, an expert
golfer could make the hole in what single displacement?
S
30. Vector A has x and y components of ⫺8.70 cm and 15.0 cm,
S
respectively; vector B has x and y components of 13.2 cm
S
S
S
and ⫺6.60 cm, respectively. If A ⫺ B ⫹ 3C ⫽ 0, what are
S
the components of C?
31. The helicopter view in Figure P3.31 shows two people
pulling on a stubborn mule. 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 N).
Figure P3.36
37.
38.
y
F1 ⫽
120 N
F2 ⫽
80.0 N
75.0⬚
60.0⬚
x
39.
Figure P3.31
S
S
32. Use the component method to add the vectors A and B
S
S
shown in Figure P3.11. Express the resultant A ⫹ B in
unit–vector notation.
S
33. Vector B has x, y, and z components of 4.00, 6.00, and
S
3.00 units, respectively. Calculate the magnitude of B and
S
the angles B makes with the coordinate axes.
S
34. Consider
the three displacement
vectors A ⫽ 13 î ⫺ 3 ĵ 2 m,
S
S
B ⫽ 1 î ⫺ 4 ĵ 2 m, and C ⫽ 1⫺2 î ⫹ 5 ĵ 2 m. Use the component method to determine (a) the magnitude and
S
S
S
S
direction of the vector D ⫽ A ⫹ B ⫹ C and (b) the magS
S
S
S
nitude and direction of E ⫽ ⫺A ⫺ B ⫹ C.
S
35. GivenS the displacement vectors A ⫽ 13 î ⫺ 4 ĵ ⫹ 4k̂ 2 m
and B ⫽ 12 î ⫹ 3 ĵ ⫺ 7k̂ 2 m, find the magnitudes of the
S
S
S
S
S
S
vectors (a) C ⫽ A ⫹ B and (b) D ⫽ 2A ⫺ B, also expressing each in terms of its rectangular components.
36. In an assembly operation illustrated in Figure P3.36, 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 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 cm
that lies in a north–south vertical plane. Find (a) the mag2 = intermediate;
3 = challenging;
䡺 = SSM/SG;
䊱
67
40.
41.
42.
43.
nitude of the total displacement of the object and (b) the
angle the total displacement makes with the vertical.
S
The vector A has x, y, and z components of 8.00, 12.0, and
–4.00 units, respectively. (a) Write a vector expression for
S
A in unit–vector notation. (b) Obtain a unit–vector expresS
S
sion for a vector B one-fourth the length of A pointing in
S
the same direction as A. (c) Obtain a unit–vector expresS
S
sion for a vector C three times the length of A pointing in
S
the direction opposite the direction of A.
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 ⫻
103 m. At time t ⫽ 0 the airplane is directly above you
so
S
that the vector leading from you to it is P0 ⫽
17.60 ⫻ 103 m 2 ĵ . At t ⫽ 30.0 sSthe position vector leading from you to the airplane is P30 ⫽ 18.04 ⫻ 103 m 2 î ⫹
17.60 ⫻ 103 m 2 ĵ . Determine the magnitude and orientation of the airplane’s position vector at t ⫽ 45.0 s.
A radar station locates a sinking ship at range 17.3 km
and bearing 136° clockwise from north. From the same
station, a rescue plane is at horizontal range 19.6 km,
153° clockwise from north, with elevation 2.20 km.
(a) Write the position vector for the ship relative to the
plane, letting î represent east, ĵ north, and k̂ up. (b) How
far apart are the plane and ship?
S
(a) Vector E has magnitude 17.0 cm and is directed 27.0°
counterclockwise from the ⫹x axis. Express it in unit–vector
S
notation. (b) Vector F has magnitude 17.0 cm and is
directed 27.0° counterclockwise from the ⫹y axis. Express
S
it in unit–vector notation. (c) Vector G has magnitude
17.0 cm and is directed 27.0° clockwise from the ⫺y axis.
Express it in unit–vector notation.
S
Vector A has a negative x component 3.00 units in
length and a positive y component 2.00 units in length.
S
(a) Determine an expression for A in unit–vector notaS
tion. (b) Determine the magnitude and direction of A.
S
S
(c) What vector B when added to A gives a resultant vector with no x component and a negative y component
4.00 units in length?
As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 60.0° north of west with a
speed of 41.0 km/h. Three hours later the course of the
hurricane suddenly shifts due north, and its speed slows
to 25.0 km/h. How far from Grand Bahama is the eye
4.50 h after it passes over the island?
䊱 Three displacement vectors of a croquet ball are shown
S
S
in Figure P3.43, where 0 A 0 ⫽ 20.0 units, 0 B 0 ⫽ 40.0 units,
S
and 0 C 0 ⫽ 30.0 units. Find (a) the resultant in unit–vector
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68
Chapter 3
Vectors
notation and (b) the magnitude and direction of the
resultant displacement.
y
B
A
47.
45.0⬚
O
x
45.0⬚
C
48.
Figure P3.43
S
S
44. 䢇 (a) Taking A ⫽S 16.00 î ⫺ 8.00 ĵ 2 units, B ⫽ 1⫺8.00 î ⫹
3.00 ĵ 2 units, and C ⫽
126.0
î ⫹S 19.0 ĵ 2 units, determine a
S
S
and b such that a A ⫹ b B ⫹ 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)?
45. 䢇 Are we there yet? In Figure P3.45, the line segment represents a path from the point with position vector
15 î ⫹ 3 ĵ 2 m to the point with location 116 î ⫹ 12 ĵ 2 m.
Point A is along this path, a fraction f of the way to the
destination. (a) Find the position vector of point A in
terms of f. (b) Evaluate the expression from part (a) in
the case f ⫽ 0. Explain whether the result is reasonable.
(c) Evaluate the expression for f ⫽ 1. Explain whether the
result is reasonable.
y
49.
50.
map of the successive displacements. (b) What total distance did she travel? (c) Compute the magnitude and
direction of her total displacement. The logical structure
of this problem and of several problems in later chapters
was suggested by Alan Van Heuvelen and David Maloney,
American Journal of Physics 67(3) 252–256, March 1999.
S
S
Two vectors A and B have precisely equal magnitudes. For
S
S
the magnitude of A ⫹ B to be 100 times larger than the
S
S
magnitude of A ⫺ B, what must be the angle between
them?
S
S
Two vectors A and B have precisely equal magnitudes. For
S
S
the magnitude of A ⫹ B to be larger than the magnitude
S
S
of A ⫺ B by the factor n, what must be the angle between
them?
An air-traffic controller observes two aircraft on his radar
screen. The first is at altitude 800 m, horizontal distance
19.2 km, and 25.0° south of west. The second is at altitude
1 100 m, horizontal distance 17.6 km, and 20.0° south of
west. What is the distance between the two aircraft? (Place
the x axis west, the y axis south, and the z axis vertical.)
The biggest stuffed animal in the world is a snake 420 m
long, constructed by Norwegian children. Suppose the
snake is laid out in a park as shown in Figure P3.50, forming two straight sides of a 105° angle, with one side 240 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. If both children run steadily at 12.0 km/h,
Inge reaches the head of the snake how much earlier
than Olaf?
(16, 12)
A
(5, 3)
O
x
Figure P3.45 Point A is a fraction f of the distance from the initial point (5, 3) to the final point (16, 12).
Additional Problems
46. On December 1, 1955, Rosa Parks (1913–2005), an icon of
the early civil rights movement, stayed seated in her bus
seat when a white man demanded it. Police in Montgomery, Alabama, arrested her. On December 5, blacks
began refusing to use all city buses. Under the leadership
of the Montgomery Improvement Association, an efficient
system of alternative transportation sprang up immediately, providing blacks with approximately 35 000 essential
trips per day through volunteers, private taxis, carpooling,
and ride sharing. The buses remained empty until they
were integrated under court order on December 21, 1956.
In picking up her riders, suppose a driver in downtown
Montgomery traverses four successive displacements represented by the expression
1⫺6.30b2 î ⫺ 14.00b cos 40° 2 î ⫺ 14.00b sin 40°2 ĵ
⫹ 13.00b cos 50°2 î ⫺ 13.00b sin 50°2 ĵ ⫺ 15.00b2 ĵ
Here b represents one city block, a convenient unit of distance of uniform size; î is east; and ĵ is north. (a) Draw a
2 = intermediate;
3 = challenging;
䡺 = SSM/SG;
䊱
Figure P3.50
51. A ferryboat transports tourists among three islands. It sails
from the first island to the second island, 4.76 km away, in
a direction 37.0° north of east. It then sails from the second island to the third island in a direction 69.0° west of
north. Finally, it returns to the first island, sailing in a
direction 28.0° east of south. Calculate the distance
between (a) the second and third islands and (b) the first
and third islands.
S
52. A vector is given by R ⫽ 2 î ⫹ ĵ ⫹ 3k̂ . Find (a) the magnitudes of the x, y, and z components, (b) the magnitude of
S
S
R, and (c) the angles between R and the x, y, and z axes.
53. A jet airliner, moving initially at 300 mi/h to the east,
suddenly enters a region where the wind is blowing at
100 mi/h toward the direction 30.0° north of east. What
are the new speed and direction of the aircraft relative to
the ground?
S
54. 䢇 Let A ⫽ 60.0 cm at 270° measured from the horizontal.
S
Let B ⫽ 80.0 cm at some angle u. (a) Find the magnitude
S
S
of A ⫹ B as a function of u. (b) From the answer to part
S
S
(a), for what value of u does 0 A ⫹ B 0 take on its maximum
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Problems
value? What is this maximum value? (c) From the answer
S
S
to part (a), for what value of u does 0 A ⫹ B 0 take on its
minimum value? What is this minimum value? (d) Without reference to the answer to part (a), argue that the
answers to each of parts (b) and (c) do or do not make
sense.
55. After a ball rolls off the edge of a horizontal table at time
t ⫽ 0, its velocity as a function of time is given by
S
v ⫽ 1.2 î m>s ⫺ 9.8t ĵ m>s2
The ball’s displacement away from the edge of the table,
during the time interval of 0.380 s during which it is in
flight, is given by
0.380 s
S
¢r ⫽
冮 v dt
S
0
To perform the integral, you can use the calculus theorem
冮 1A ⫹ Bf 1x2 2 dx ⫽ 冮 A dx ⫹ B 冮 f 1x 2 dx
You can think of the units and unit vectors as constants,
represented by A and B. Do the integration to calculate
the displacement of the ball.
56. 䢇 Find the sum of these four vector forces: 12.0 N to the
right at 35.0° above the horizontal, 31.0 N to the left at
55.0° above the horizontal, 8.40 N to the left at 35.0°
below the horizontal, and 24.0 N to the right at 55.0°
below the horizontal. Follow these steps. Guided by a
sketch of this situation, explain how you can simplify the
calculations by making a particular choice for the directions of the x and y axes. What is your choice? Then add
the vectors by the component method.
57. A person going for a walk follows the path shown in Figure P3.57. 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?
y
Start 100 m
x
300 m
End
200 m
60.0⬚
30.0⬚
150 m
Figure P3.57
58. 䢇 The instantaneous position of an object is specified by
its position vector Sr 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,
S
given by r ⫽ 4 î ⫹ 3 ĵ ⫺ 2t k̂ , where r is in meters and t is
in seconds. Evaluate drS>dt. What does it represent about
the object?
59. 䢇 Long John Silver, a pirate, has buried his treasure on
an island with five trees, located at the points (30.0 m,
⫺20.0 m), (60.0 m, 80.0 m), (⫺10.0 m, ⫺10.0 m), (40.0 m,
⫺30.0 m), and (⫺70.0 m, 60.0 m), all measured relative
2 = intermediate;
3 = challenging;
䡺 = SSM/SG;
䊱
69
to some origin as shown in Figure P3.59. 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 m, ⫺20 m), A(60 m,
80 m), E(⫺10 m, –10 m), C(40 m, ⫺30 m), and
D(⫺70 m, 60 m)? State reasoning to show the answer
does not depend on the order in which the trees are
labeled.
B
E
y
x
C
A
D
Figure P3.59
60. 䢇 Consider a game in which N children position themselves at equal distances around the circumference of a
circle. At the center of the circle is a rubber tire. Each
child holds a rope attached to the tire and, at a signal,
pulls on his or her rope. All children exert forces of the
same magnitude F. In the case N ⫽ 2, it is easy to see that
the net force on the tire will be zero because the two
oppositely directed force vectors add to zero. Similarly, if
N ⫽ 4, 6, or any even integer, the resultant force on the
tire must be zero because the forces exerted by each pair
of oppositely positioned children will cancel. When an
odd number of children are around the circle, it is not as
obvious whether the total force on the central tire will be
zero. (a) Calculate the net force on the tire in the case
N ⫽ 3 by adding the components of the three force vectors. Choose the x axis to lie along one of the ropes.
(b) What If? State reasoning that will determine the net
force for the general case where N is any integer, odd or
even, greater than one. Proceed as follows. Assume the
total force is not zero. Then it must point in some particular direction. Let every child move one position clockwise. Give a reason that the total force must then have a
direction turned clockwise by 360°/N. Argue that the
total force must nevertheless be the same as before.
Explain what the contradiction proves about the magnitude
of the force. This problem illustrates a widely useful technique of proving a result “by symmetry,” by using a bit
of the mathematics of group theory. The particular situation is actually encountered in physics and chemistry
= ThomsonNOW;
䡵 = symbolic reasoning;
䢇 = qualitative reasoning
70
Chapter 3
Vectors
when an array of electric charges (ions) exerts electric
forces on an atom at a central position in a molecule or
in a crystal.
S
S
61. Vectors
A and
B have equal magnitudes of 5.00. The sum
S
S
of A and B is the vector 6.00 ĵ . Determine the angle
S
S
between A and B.
62. A rectangular parallelepiped has dimensions a, b, and c as
shown in Figure P3.62. (a)
Obtain a vector expression for
S
the face diagonal vector R1. What is the magnitude of this
vector? (b)SObtain a vectorSexpression Sfor the body diagonal vector R2. Notice that R1, c k̂ , and R2 make a right triS
angle. Prove that the magnitude of R2 is 1 a 2 ⫹ b 2 ⫹ c 2.
z
a
b
O
x
R2
c
R1
y
Figure P3.62
Answers to Quick Quizzes
3.1 Scalars: (a), (d), (e). None of these quantities has a direction. Vectors: (b), (c). For these quantities, the direction
is necessary to specify the quantity completely.
3.2 (c). The resultant has its maximum magnitude A ⫹ B ⫽
S
12 ⫹ 8 ⫽ 20 units when vector A is oriented in the same
S
direction as vector B. The resultant vector has its miniS
mum magnitude A ⫺ B ⫽ 12 ⫺ 8 ⫽ 4 units when vector A
S
is oriented in the direction opposite vector B.
3.3 (b) and (c). To add to zero, the vectors must point in
opposite directions and have the same magnitude.
2 = intermediate;
3 = challenging;
䡺 = SSM/SG;
䊱
3.4 (b). From the Pythagorean theorem, the magnitude of a
vector is always larger than the absolute value of each
component, unless there is only one nonzero component,
in which case the magnitude of the vector is equal to the
absolute value of that component.
S
3.5 (c). The magnitude of C is 5 units, the same as the z component. Answer (b) is not correct because the magnitude
of any vector is always a positive number, whereas the y
S
component of B is negative.
= ThomsonNOW;
䡵 = symbolic reasoning;
䢇 = qualitative reasoning