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Chapter 23
Linear Motion
The simplest example of a parametrized curve arises when studying the
motion of an object along a straight line in the plane. We will start by
studying this kind of motion when the starting and ending locations are
known.
23.1 Motion of a Bug
P=(2,5)
Q=(6,3)
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-axis
-axis
-axis
(a) A bug walking from
to .
for x−motion
casts shadow
light on bug
for y−motion
casts shadow
light on bug
3
5
Q=(6,3)
R=(x ,y )
1 1
2 2
P=(2,5)
S=(x ,y )
shadow
on y−axis
2 6
shadow on x−axis
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-axis
-axis
(b) How to view motion in
the coordinates.
Figure 23.1: Visualizing the
model for the bug problem.
Example 23.1.1. A bug is spotted at in the -
plane. The bug walks in a straight line from to
at a constant speed . It takes the bug 5 seconds to reach
. Assume the units of our coordinate system are feet.
What is the speed of the bug along the line connecting
and ? Compute the horizontal and vertical speeds of the
bug and show they are both constant.
Solution. A standard technique in motion problems is to
analyze the and -motion separately. This means we
look at the projection of the bug location onto the and
-axis separately, studying how each projection moves.
We can think of these projections as “shadows” cast by a
flashlight onto the two axes:
For the -motion, we study the “shadow” on the -axis
which starts at “2” and moves toward “6” on the -axis.
For the -motion, we study the “shadow” on the -axis
which starts at “5” and moves toward “3” along the -axis.
In general, speed is computed by dividing distance by
time elapsed, so
dist !"
ft
sec
ft
sec
feet
sec.
(23.1)
This is the speed of the bug along the line connecting and .
315
316 CHAPTER 23. LINEAR MOTION
The hard part of this problem is to show that the speed of the hori-
zontal and vertical shadows are also constant. This might seem obvious
when you first think about it. In order to actually show it, let’s take two
intermediary positions and &
along the bugs path.
We are going to relate the horizontal speed
of the bug between and
, the vertical speed
of the bug between and
and the speed of
the bug from to . Actually, because there are positive or negative di-
rections for the and axes, we will allow horizontal and vertical “speed”
to be a quantity, with the obvious meaning. If it takes seconds for
the bug to travel from to , then is the elapsed time for the horizontal
motion from to
and also the elapsed time for the vertical motion from
to
. The horizontal speed
is the directed distance
$ di-
vided by the time elapsed , whereas the vertical speed
is the directed
distance
$ divided by the time elapsed.
We want to show that
and
are both constants! To do this, we
have these three equations:
$
$
distance )
$ ' (
$ '
Now, square each side of the three equations and combine them to con-
clude:
(
. We can multiply through by
and that gives
us the key equation:
(
(23.2)
On the other hand, the ratio of the vertical and horizontal speed gives
$
“slope of line connecting and ”
$
$
(23.3)
Solving for
in terms of
we can write
$
(23.4)
Since
is positive,
'
( '
'
(! $
#"
%$
$
'&
318 CHAPTER 23. LINEAR MOTION
Then the speed
in the -direction and the speed
in the -direction are
both constant. We also have two useful formulas:
slope of line of travel,
when the line is non-
vertical.
This fact is established using the same reasoning as in Example 23.1.1.
Let’s make a few comments. To begin with, if the line of travel is either
vertical or horizontal, then either
or
and Fact 23.2.1 isn’t
really saying anything of interest.
The formulas in Fact 23.2.1 only work for linear motion.
CAUTION
!!!
!!!
ements
-axis
-axis
-axis
P Q
P
Q
no y−motion
no x−motion
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-axis
-axis
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-axis
Figure 23.3: Horizontal or vertical motion.
For any other line of travel, we can use the reasoning used in Exam-
ple 23.1.1. Pay attention that the horizontal speed
and the vertical
speed
are both directed quantities; i.e. these can be positive or nega-
tive. The sign of
will indicate the direction of motion: If
is positive,
then the horizontal motion is to the right and if
is negative, then the
horizontal motion is to the left. Similarly, the sign of
tells us if the
vertical motion is upward or downward.
Returning to Figure 23.2, to describe the -motion, two pieces of in-
formation are needed: the starting location (in the -direction) and the
constant speed
in the -direction. So,
-coordinate of the object at time
beginning -coordinate (
distance traveled in
the -direction in
time units
23.2. GENERAL SETUP 319
If we are not given the horizontal velocity directly, rather the time
required to travel from to , then we could compute
using the fact
that the object starts at " and travels to "
:
directed horizontal distance traveled
time required to travel this distance
ending -coordinate $ starting -coordinate
time required to travel this distance
To describe the -motion in Figure 23.2, we proceed similarly. We will
denote by
the constant vertical speed of the object, then after time
units the object has traveled
units. So,
-coordinate of object at time
beginning -coordinate (
distance traveled in
the -direction in
time units
In summary,
Important Fact 23.2.2 (Linear motion). Suppose an object begins at a
point and moves at a constant speed along a line connecting
to another point
. Then the motion of this object will trace out
a line segment which is parametrized by the equations:
circular region
radius 1 ft.
location E where
bug enters region
Q=(6,3)
P=(2,5)
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-axis
-axis
Figure 23.4: A bug crosses a
circular boundary.
Example 23.2.3. Return to the linear motion problem stud-
ied in Example 23.1.1 and 23.1.2. However, now assume
that the point * is located at the center of a circular
region of radius 1 ft. When and where does the bug enter
this circular region?
Solution. The parametric equations for the linear motion
of the bug are given by:
The equation of the boundary of the circular region centered at is given
by
