COMMON CORE EDITION
COURSE
Preparation for Calculus
Christian R. Hirsch
James T. Fey
Harold L. Schoen
Eric W. Hart
Ann E. Watkins
with
Beth E. Ritsema
Rebecca K. Walker
Robin Marcus
CP_SE_C4_TP.indd 1
Brin A. Keller
Arthur F. Coxford
12/16/13 12:24 PM
U N I T
2
Vectors and Motion
M
otion is a pervasive aspect of
our lives. You walk and travel by
bike, car, bus, subway, or perhaps
even by boat from one location
to another. You watch the paths of balls
thrown or hit in the air and of space shuttles
launched into orbit. Each of these motions
involves both direction and distance. Vectors
provide a powerful way for mathematically
representing and analyzing motion.
In this unit, you will learn how to use vectors
and vector operations to solve problems
about navigation and force. You will extend
and further connect your understanding
of geometry, trigonometry, and algebra to
establish properties of vector operations. You
will also create and use parametric equations
to model linear and nonlinear motion.
The key ideas will be developed through work
on problems in three lessons.
101_CP4_SE_U2_UO_665790.indd 101
L ES SO N S
©Royalty-Free/Corbis
1 Modeling Linear
Motion
Develop skill in using
vectors, equality of vectors,
scalar multiplication, vector
sums, and component
analysis to model and
analyze situations involving
magnitude and direction.
2 Vectors and
Parametric
Equations
3 Modeling Nonlinear
Motion
Use parametric equations
to model nonlinear motion,
Represent and analyze
including the motion of
vectors and vector operations
projectiles and circular and
using coordinates. Use
elliptical orbits.
position vectors to develop
parametric equations to
model linear motion.
11/28/13 4:08 PM
LES S O N
1
Modeling Linear Motion
In this unit, you will learn to use an
important tool for modeling motion—
vectors. Vectors are useful in situations
that involve magnitude (such as
distance) and direction. These are
important descriptors of motion. The
simplest motion is linear—movement
along a straight line.
Linear motion is used to plan and guide
hiking routes and courses of boats
and ships. Think about how you might
describe or represent a planned route
on a map. Also think about conditions
that might affect a planned route
and how you might incorporate that
information in the planning process.
Getty Images/Digital Vision
Each day you confront motion in nearly
everything you do. You may walk, ride a
bicycle, or ride in a car or bus to school.
You may take a subway train to meet
friends at a shopping mall. You see aircraft
fly overhead and you see the position of
the Sun in the sky move, from morning
when it rises in the east to evening when
it sets in the west. You might run in a race
or throw, kick, or hit a ball.
102 UNIT 2 Vectors and Motion
102_110_CP4_SE_U2_L1_665790.indd 102
11/28/13 4:08 PM
THINK ABOUT THIS SITUATION
Suppose you wanted to map out a route that involved sailing 3 km west from Bayview Harbor
to Presque Island, then 6 km south to Rudy Point, and then 5 km southeast to Pleasant Bay.
a
How could you represent the planned route geometrically?
b How could you represent a direct sailing route from Bayview Harbor to Pleasant Bay?
c
How could you estimate the length of the route in Part b? How would you describe its
direction?
d How would a northeast water current affect the path along which you would steer the
boat to maintain the route in Part b?
In this lesson, you will learn how to represent vectors geometrically, how to
scale vectors, and how to combine vectors by addition in the context of solving
applied problems.
INVESTIGATION
1
Navigation: What Direction and How Far?
Vectors and vector operations are used extensively in navigation on water and
in the air. As you work on the problems of this investigation, look for answers to
these questions:
How can vectors be represented geometrically with directed line segments?
How can vectors and scalar multiples of vectors be used to model navigation routes?
Charting a Boat’s Course
Imagine that you are
navigating a boat along
the small portion of the
Massachusetts coast shown
in the nautical chart at the
right. Note that within the
chart itself, there are several
aids to navigation such as
buoys, landmarks, and scales.
The buoys are painted red
or green and may have a red
or green flashing light. A
circle (on land) with a dot at
its center indicates an easily
recognized landmark such as
a stone tower or a tank.
01'
72° 00'
SG
“1A”
59'
58'
57'
56'
55'
Oak Island
Sunken
Ledge
03'
5 nm
02'
4 nm
01'
3 nm
00'
2 nm
59'
1 nm
STK
Hog
Island
C“1”
Fl R 4 sec
“2”
Tank
Bell “3”
Fl G 4 sec
“6”
Fl R 4 sec
42° 00'
Cupola
“SH”
Fl 6 sec
GONG
SM “2”
Fl R 2.5 sec
Priv Maint D Stone Harbor
Launch
SG “3”
Center
Great
Point
“GP”
Fl G 2.5 sec
Bell
Stone
Tower
41° 58'
71° 55'
0 nm
Adapted from Frank J. Larkin. Basic Coastal
Navigation. Sheridan House Inc. 1998.
Lesson 1
102_110_CP4_SE_U2_L1_665790.indd 103
Modeling Linear Motion 103
11/28/13 4:09 PM
1 As a class, examine a copy of the nautical chart.
a. At the right of the chart is a nautical mile (nm) scale. Use this scale to find
the distance from the “SH” buoy to the “GP” buoy. Measure between the
centers of the circles that mark the buoys.
b. There are other scales at the top and along the right edge of the chart.
What do you think these scales represent? Share your ideas with
classmates.
c. What other scale on this chart can be used to measure nautical miles?
What does a nautical mile represent based on this scale? Share your
ideas with your classmates.
d. A nautical mile is 6,076.1033 feet. How does a nautical mile compare to a
statute mile (regular mile)?
Coastal water nautical charts are designed so that the top is due north and the
right side is due east. You can use your knowledge of directed angles measured
counterclockwise from the horizontal (due east) to describe the direction of a
craft. Thus, you can say due east is 0°, due north is 90°, due west is 180°, and
due south is 270°.
2 The course of a boat starting at Buoy 6 and moving 30° north of east is shown
in the chart below. Use a copy of the nautical chart to complete this problem.
Using a ruler made
North
01'
72° 00'
59'
58'
57'
56'
55'
from the nautical mile
03'
5 nm
Oak Island
SG
STK
scale, measure distances
Sunken
“1A”
a. Mark and label a
point P on a copy
of the chart to
represent a boat that
is 3 nautical miles
from the “3” bell
and is headed at an
angle of 290°. What
buoy is nearest to P?
Ledge
Hog
Island
C“1”
Fl R 4 sec
“2”
Tank
Bell “3”
Fl G 4 sec
02'
4 nm
01'
3 nm
30°
“6”
Fl R 4 sec
East
1
n m.
to the nearest _
10
Measure angles to the
nearest degree using a
protractor.
42° 00'
00'
2 nm
59'
1 nm
Cupola
“SH”
Fl 6 sec
GONG
SM “2”
Fl R 2.5 sec
Priv Maint D Stone Harbor
Launch
SG “3”
Center
Great
Point
“GP”
Fl G 2.5 sec
Bell
Stone
Tower
41° 58'
71° 55'
0 nm
104 UNIT 2 Vectors and Motion
102_110_CP4_SE_U2_L1_665790.indd 104
11/28/13 5:41 PM
b. Draw an arrow from the “SH” buoy to the “6” buoy. What is the direction
in degrees? What is the distance in nautical miles?
c. What are the direction and distance of the path from the “6” buoy to the
center of the mouth of the channel at Stone Harbor?
d. Why are arrows particularly useful representations for nautical paths?
3 The arrows that indicate boating routes are directed line
segments. They have both a magnitude (length) and a
direction. Thus, an arrow is a geometric representation
of a vector—a quantity with magnitude and direction.
A vector with a length of 1" and direction of 45° is shown
at the right.
1"
45°
E
a. Accurately draw arrows representing vectors with the following
characteristics.
i. Length: 2 nm; direction: 70° (Use your nautical ruler and a protractor.)
ii. Length: 5 cm; direction: 110°
iii. Magnitude: 7 cm; direction: 300°
b. Draw an arrow for each vector described. State what length you chose to
represent 1 knot and what length you chose to represent 1 mph.
i. A boat with a speed of 2 knots (nautical miles per hour) at a direction
of 180°
ii. Speed of 40 mph at a direction of 240°
iii. Force of a 15 mph wind blowing from the northeast
c. Compare the arrows you drew in Parts a and b with your classmates.
Resolve any differences.
Denoting Vectors Vectors can be denoted in various ways. One way is to use
italicized letters with arrows over them, such as a
or v
. When the initial point, or
⎯⎯⎯ can be used.
tail, A and terminal point, or head, B are labeled, the notation AB
head or
terminal point
tail or
initial point
B
AB
a
A
Since a vector v
is determined by its magnitude r, and its direction θ, v
can also
be represented as a pair, [r, θ], called the polar form or polar representation
of the vector.
r
[r, θ]
θ
E
Lesson 1
102_110_CP4_SE_U2_L1_665790.indd 105
Modeling Linear Motion 105
12/12/13 10:41 AM
4 Since arrows representing vectors, as in Problem 3, can be drawn anywhere
in a plane, it is important to know that two arrows drawn using the same
scale represent equal vectors when they have the same magnitude and the
same direction.
Explain why the following method provides a
geometric test for the equality of the vectors
⎯⎯⎯ and RS
⎯⎯⎯.
PQ
S
Q
Step 1. Connectt he heads Q and S and connect
the tails P and R.
⎯⎯⎯ = RS
⎯⎯⎯.
Step 2. If PQSR is a parallelogram, then PQ
R
P
Scalar Multiples of a Vector In the problems that follow, use either a ruler,
protractor and graph paper, or interactive geometry software with vector drawing
and analysis capabilities. It may also be helpful to use the “Stone Harbor” custom
app in CPMP-Tools. When the instructions ask you to make an “accurate drawing”
or an “accurate sketch” of a vector, you can do so with geometry software or
carefully draw an arrow on graph paper using a ruler and protractor to measure. If,
however, the instructions are simply “sketch” or “draw” a vector, you may make a
freehand sketch of an arrow that approximates the characteristics of an accurately
drawn vector in order to guide your thinking. Note that “draw a vector” actually
means “draw a geometric representation of the vector” (an arrow).
5 A fishing boat leaves the mouth of the Stone Harbor channel trolling on a
heading due north at a speed of 1.5 knots (nautical miles per hour).
a. On your copy of the nautical chart, sketch the vector v
representing the
distance and direction traveled from the middle of the channel opening
during the first hour.
c. Sketch and label a vector that locates the fishing boat at the end of
20 minutes and another that locates it at the end of 2.5 hours.
d. Now sketch another vector that has the same length as 2v
but is not equal
to 2v
,
and
another
vector
that
is
equal
to
2
v
.
Compare
your
vectors with
those of your classmates.
e. In general, how would you sketch a vector that was a positive number
k times a given vector? How are the lengths and directions of these
two vectors related?
Goodshoot - JupiterImages France/Alamy
b. Use the vector v
in Part a to determine the vector for a 2-hour trip at the
same speed and in the same direction. Sketch this vector. Label it 2v
.
106 UNIT 2 Vectors and Motion
102_110_CP4_SE_U2_L1_665790.indd 106
11/28/13 4:09 PM
6 Suppose another boat begins a trip at the same point at the mouth of the
channel at Stone Harbor headed at a direction of 20° and at a speed of 2 knots.
a. Sketch the vector v
showing the approximate
position at the end of the
first hour.
b. Suppose the boat returns
to the harbor along the
same route at the same
speed. Sketch the return
vector and give its
magnitude and direction.
c. The word “opposites” can be used to denote the vectors in Parts a and b.
How is the word “opposite” descriptive of the relationship between the
two vectors?
d. Sketch a vector opposite to the vector v
in Part a with initial point at the
“3” bell. Give its magnitude and direction.
When a vector a
is multiplied by a
real number k, the number is called
a scalar and the product, ka
, is a
scalar multiple of a
.
(In
a
similar
⎯⎯⎯ is a scalar multiple of
manner, kAB
⎯⎯⎯.) When k > 0, ka
the vector AB
is the
vector whose length is k times the length
of a
and has the same direction as a
as
shown at the right.
a
k a, k > 1
k a, 0 < k <
7 For vector a shown above, the opposite of vector
a
, denoted -a
, is shown at the right. The scalar
multiple ka
when
k < 0 is shown at the far right.
a. Compare the relationship between a
and ka
when k > 0 to the relationship between -a
and
ka
when k < 0.
-a
k a, k > 0
b. Suppose a
= [10, 50°] and v
= [8, 20°]. Write each of the following vectors
in polar form [r, θ], where r is the vector’s magnitude and its direction θ
satisfies 0° ≤ θ < 360°.
i. 5a
ii. 0.2v
iii. -a
iv. -3v
Janet S. Robbins
c. Suppose k < 0 and a
= [r, θ]. Write ka
in polar form.
Lesson 1
102_110_CP4_SE_U2_L1_665790.indd 107
Modeling Linear Motion 107
11/28/13 4:09 PM
SUMMARIZE THE MATHEMATICS
In this investigation, you explored how vectors—quantities with magnitude and direction—
can be represented geometrically by arrows.
a
Describe how you know when two arrows represent equal vectors.
b How are a vector and a scalar multiple of that vector similar? How are they different?
c
⎯⎯⎯ and BA
⎯⎯⎯ alike? How are they different? What is another way to
How are vectors AB
⎯⎯⎯ using A and B ?
write BA
d What is always true about the magnitudes and directions of two opposite vectors?
Be prepared to explain your ideas to the class.
CHECK YOUR UNDERSTANDING
Escanaba
Charlevoix
Wausau
Green Bay
Manitowoc
Fond du Lac
Lake Michig
an
b. Find the magnitude and
direction of -v
. Draw -v
beginning at Charlevoix,
Michigan.
Michigan
Milwaukee
Cadillac
Ludington
Grand Haven
Grand Rapids
c. Sketch 0.5v
from Milwaukee,
Wisconsin. Find its magnitude
and direction.
d. Are the vectors representing
the route from Charlevoix to
Escanaba and the route from
South Haven to Milwaukee
approximately the same?
Explain.
Michigan
a. Draw the vector for the ferry
route from Manitowoc to
Ludington. Label it v
. Measure
its magnitude and direction.
N
Wisconsin
Daily ferries shuttle people and cars
between Manitowoc, Wisconsin,
and Ludington, Michigan. Use a
copy of this map of Lake Michigan
to complete the following tasks.
South Haven
Chicago
Illinois
South Bend
Gary
Indiana
Kankakee
Champaign
West Lafayette
50
100 miles
Photodisc/Getty Images
0
108 UNIT 2 Vectors and Motion
102_110_CP4_SE_U2_L1_665790.indd 108
11/28/13 4:09 PM
INVESTIGATION
2
Changing Course
In the previous investigation, you used vectors to model straight-line paths.
But hiking paths or the course of a ship often involve changes in direction.
As you complete the problems in this investigation, look for answers to the
following question:
How can vectors be used to model routes when there
is a change of course during the trip?
1 Vector Sums Roberta, the skipper of the fishing boat High Hopes, leaves
the mouth of the Stone Harbor channel at a speed of 6 knots at 25°. She travels
for 20 minutes, then turns so that she is heading in a direction of 100° at the
same speed and travels for 30 minutes before deciding to drop anchor and
begin fishing.
a. Using the “Stone Harbor” custom
app or a copy of the nautical chart,
draw a vector diagram showing
the paths taken and the position
of the High Hopes at the end of 50
minutes. What is the magnitude of
each of these two vectors?
b. Suppose the fish are biting and
Roberta wants to inform Clarissa,
the skipper of the Salmon King, of
where she is located so Clarissa
can join her. Draw a vector
representing the path Clarissa
should take from the mouth of
Stone Harbor channel directly to
the High Hopes. What direction
should Roberta advise her to take?
How far will she need to travel?
c. The vector representing the path that Clarissa should travel to the good
fishing spot is called the sum or resultant of the two vectors that describe
the route taken by the High Hopes. How are the initial and terminal points
of the resultant vector in Part b related to the two vectors that represent the
route taken by the High Hopes?
d. Suppose Roberta had left the harbor at a speed of 6 knots in a direction of
100° for 30 minutes, and then turned to a direction of 25° and traveled for
20 minutes at the same speed. Draw a sketch of Roberta’s path and the
resultant vector. What are the direction and the magnitude of the
resultant vector?
e. Now sketch two additional two-leg routes to this good fishing spot.
Lesson 1
102_110_CP4_SE_U2_L1_665790.indd 109
Modeling Linear Motion 109
11/28/13 4:09 PM
2 Based on your work in Problem 1, decide if each of the following
generalizations is true or false. In each case, explain your reasoning.
a. The vector sum of any two given vectors is unique.
b. If a vector is the sum of two given vectors, it cannot be the sum of two
other vectors.
c. If v
= [r, θ 1]an d w
= [s, θ 2], then v
+w
= [r + s, θ 1 + θ 2].
3 For the following vectors, the magnitude is in centimeters and the given angle
= [4, 30°], measure is the vector’s direction: a
c = [4, 350°],
= [5, 70°], b
= [3, 250°]. Make accurate drawings of each vector sum and measure
and d
to find the magnitude (to the nearest 0.1 cm) and direction (to the nearest 5°)
for each resultant vector.
a. a
+ b
b. a
+ d
+ c. a
c
+ b
4 Now investigate some general properties of vector addition. Begin by
as arrows that have different directions
sketching any two vectors a
and b
but no points in common.
to find a
. Do the same
a. Draw a diagram showing how to place a
and b
+ b
+ a
for b
. What do you notice about the two vector sums? Compare your
observations to those of others and resolve any differences.
b. To which property of real number operations is this similar?
so their initial points are at this
c. Choose a point in the plane. Place a
and b
and b
+ a
point. Then draw a single diagram showing how to find a
+ b
.
What shape is formed? Prove your conjecture.
5 On a sheet of plain paper or graph paper, make an accurate drawing
of vector u
with magnitude 4 cm and direction 200° and vector v
with
magnitude 5 cm and direction 70°.
a. Without drawing or measuring, find the magnitude and direction of
as many of the following vectors as possible. Explain your reasoning in
each case.
i. 2u
iii. u
+ v
v. 3u
+ 3v
vii. 2v
+ (-2u
)
ii. v
+ u
iv. 3(u
+ v
)
vi. -2v
viii. -2v
+ (-2u
)
b. For the remaining vectors in Part a, find the magnitude and direction by
measuring. Use as few drawings as possible. Look for possible connections
between pairs of vectors that might reduce your work.
c. What general rule is suggested by parts iv and v in Part a? Test your
conjecture.
d. The sum of two vectors is always a vector. Describe the resultant vector
for u
+ (-u
).
110 UNIT 2 Vectors and Motion
102_110_CP4_SE_U2_L1_665790.indd 110
11/28/13 4:09 PM
6 Horizontal and Vertical Components The chart below shows a vector v with
magnitude 1.3 nm and direction 0° and a vector w
with magnitude 1.8 nm and
direction 90° that represent one route to a good fishing area.
01'
72° 00'
SG
“1A”
59'
58'
57'
56'
55'
Oak Island
Sunken
Ledge
03
STK
Hog
Island
C“1”
Tank
Fl R 4 sec
“2”
Bell “3”
Fl G 4 sec
02'
01'
“6”
Fl R 4 sec
42° 00'
w
“SH”
Fl 6 sec
GONG
SM “2”
Fl R 2.5 sec
Priv Maint D Stone Harbor
Launch
SG “3”
Center
Stone
Tower
v
00
Cupola
Great
Point
59
“GP”
Fl G 2.5 sec
Bell
41° 58'
71° 55'
a. Calculate (do not measure) the magnitude of the resultant vector v
+w
.
b. Use trigonometric ratios to compute the direction of the direct route v
+w
to the good fishing spot.
c. Starting at the harbor, is it possible to find another pair of vectors with
directions 0° and 90° that have the same vector sum as in Part a? Explain
your reasoning.
7 Now investigate further how a vector can be thought of in terms of the sum of
horizontal and vertical vectors called its components.
a. Suppose a vector represents a 2-nautical mile route with a direction of 78°.
Use trigonometric ratios to compute the lengths of the east (0°) and north
(90°) legs of a route to the same location.
b. Suppose a vector v
represents a 2-nm route with a direction of 125°.
Make a sketch of the vector v
and include the west and north vectors
that would give the resultant vector v
. Compute the magnitudes of the
west and north vectors.
c. Now think more generally. How would you compute the magnitudes of
the horizontal and vertical components of the vectors described below?
Compare your methods with those of your classmates and resolve any
differences.
i. Any 2-nm vector with a direction θ between 180° and 270°
ii. Any 5-nm vector with a direction θ between 270° and 360°
iii. Any 10-nm vector that points due north or due south
iv. Any 10-nm vector that points due east or due west
Lesson 1
111_119_CP4_SE_U2_L1_665790.indd 111
Modeling Linear Motion 111
11/28/13 7:00 PM
SUMMARIZE THE MATHEMATICS
In this investigation, you explored the geometry of addition of vectors.
a
Describe geometrically how you can find the resultant, or sum, of two vectors.
b Any nonzero vector can be represented as the sum of a horizontal vector and a vertical
vector. Illustrate and explain how this can be done for a given vector.
c
⎯⎯⎯ and CB
⎯⎯⎯ are the horizontal and vertical components,
In the vector diagram below, AC
⎯⎯⎯.
respectively, of AB
⎯⎯⎯ and CB
⎯⎯⎯,
i. If you know the magnitudes of AC
how would you calculate the magnitude and
⎯⎯⎯?
direction of AB
⎯⎯⎯,
ii. If you know the magnitude and direction of AB
⎯⎯⎯
how would you calculate the magnitudes of AC
⎯⎯⎯?
and CB
B
C
A
Be prepared to share your ideas and reasoning with the class.
CHECK YOUR UNDERSTANDING
Use what you have learned about adding vectors and horizontal and vertical
components of a vector to compute (not measure) answers to the questions below.
Check that your answers are reasonable by measuring.
a. Suppose Clarissa wants to fish in the
secluded bay behind Great Point,
as shown in the chart. The vector
[3.1 nm, 10°] represents a direct route
to the bay. Since this route crosses
land, Clarissa decides to head east
and then due north to the fishing
spot. How many nautical miles
should she travel east before turning
north? How far north from there is
the fishing spot?
b. Suppose Roberta needs to travel
from her location south of Hog
Island to the west side of Oak Island
before nightfall. The west and north
vectors for one route are shown on
the chart. If Roberta decides to take
a direct route (across the rocky area)
rather than the west/north route,
how many nautical miles can she
shave off the trip? In what direction
should she head?
01'
72° 00'
SG
“1A”
59'
58'
57'
56'
55'
Oak Island
Sunken
Ledge
C“1”
03'
STK
Hog
Island
0.8 nm
Fl R 4 sec
“2”
1.9 nm
Tank
Bell “3”
Fl G 4 sec
02'
01'
“6”
Fl R 4 sec
42°
00
Cupola
“SH”
Fl 6 sec
GONG
SM “2”
Fl R 2.5 sec
Priv Maint D Stone Harbor
Launch
SG “3”
Center
[3.1 nm, 10°]
Great
Point
59'
“GP”
Fl G 2.5 sec
Bell
Stone
Tower
41° 58'
71° 55'
112 UNIT 2 Vectors and Motion
111_119_CP4_SE_U2_L1_665790.indd 112
11/28/13 4:09 PM
INVESTIGATION
3
Go with the Flow
The vector models you have been
using for navigation assume that the
force moving a boat is the only one
acting on the craft. When this is the
case, the craft moves in a straight
line in the direction of the force. In
reality though, two (or more) forces
often act simultaneously on an object.
For example, currents in the ocean
are forces that move the boats in the
direction of the current. Sailing ships
without motors use water currents to
help them enter and leave port. The
wind, too, is a force that affects the
path that a boat or an airplane follows.
A fundamental principle of physics is
that the effect of two forces acting on a
body is the sum of the forces.
As you work on problems in this
investigation, look for answers to this
question:
How can vectors be used to analyze the effect of two or more
forces simultaneously acting on an object?
1 Suppose a boat leaves port P headed in a
direction of 60° with the automatic pilot set
for 10 knots. On this particular day, there is a
4-knot ocean current with a direction of 30°.
The vector diagram at the right shows the
effect of the current on the position of the
boat at the end of one hour.
30°
E
a. Assuming a scale of 1 cm = 2 nm, verify
the accuracy of the diagram.
b. The sum of the original course and current
vectors gives the position of the boat in
one hour. Determine how far the boat will
actually travel in one hour:
60°
P
E
i. using the scale diagram.
Steve Mason/Getty Images
ii. using the Law of Cosines. (Hint: The obtuse angle of the triangle
is 150°. Why?)
c. At what speed and in what direction will the boat actually travel during
the first hour? Will it continue to travel similarly during the next hour if
all conditions remain the same? Explain.
Lesson 1
111_119_CP4_SE_U2_L1_665790.indd 113
Modeling Linear Motion 113
11/28/13 4:09 PM