Chapter 2
Motion in One Dimension
2.1
The Important Stuff
2.1.1
Position, Time and Displacement
We begin our study of motion by considering objects which are very small in comparison to
the size of their movement through space. When we can deal with an object in this way we
refer to it as a
particle
. In this chapter we deal with the case where a particle moves along
a straight line.
The particle’s location is specified by its
coordinate
, which will be denoted by
x
or
y
.
As the particle moves, its coordinate changes with the time,
t
. The change in position from
x
1
to
x
2
of the particle is the
displacement
Δ
x
, with Δ
x
=
x
2

x
1
.
2.1.2
Average Velocity and Average Speed
When a particle has a displacement Δ
x
in a change of time Δ
t
, its
average velocity
for
that time interval
is
v
=
Δ
x
Δ
t
=
x
2

x
1
t
2

t
1
(2.1)
The
average speed
of the particle is absolute value of the average velocity and is given
by
s
=
Distance travelled
Δ
t
(2.2)
In general, the value of the average velocity for a moving particle depends on the initial
and final times for which we have found the displacements.
2.1.3
Instantaneous Velocity and Speed
We can answer the question “how fast is a particle moving at a particular time
t
?” by finding
the
instantaneous velocity
. This is the limiting case of the average velocity when the time
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CHAPTER 2.
MOTION IN ONE DIMENSION
interval Δ
t
include the time
t
and is as small as we can imagine:
v
= lim
Δ
t
→
0
Δ
x
Δ
t
=
dx
dt
(2.3)
The
instantaneous speed
is the absolute value (magnitude) of the instantaneous ve
locity.
If we make a plot of
x
vs.
t
for a moving particle the instantaneous velocity is the slope
of the tangent to the curve at any point.
2.1.4
Acceleration
When a particle’s velocity changes, then we way that the particle undergoes an
acceleration
.
If a particle’s velocity changes from
v
1
to
v
2
during the time interval
t
1
to
t
2
then we
define the
average acceleration
as
v
=
Δ
x
Δ
t
=
x
2

x
1
t
2

t
1
(2.4)
As with velocity it is usually more important to think about the
instantaneous accel
eration
, given by
a
= lim
Δ
t
→
0
Δ
v
Δ
t
=
dv
dt
(2.5)
If the acceleration
a
is positive it means that the velocity is instantaneously
increasing
;
if
a
is negative, then
v
is instantaneously
decreasing
. Oftentimes we will encounter the word
deceleration
in a problem. This word is used when the sense of the acceleration is opposite
that of the instantaneous velocity (the motion). Then the
magnitude
of acceleration is given,
with its direction being understood.
2.1.5
Constant Acceleration
A very useful
special case
of accelerated motion is the one where the acceleration
a
is constant.
For this case, one can show that the following are true:
v
=
v
0
+
at
(2.6)
x
=
x
0
+
v
0
t
+
1
2
at
2
(2.7)
v
2
=
v
2
0
+ 2
a
(
x

x
0
)
(2.8)
x
=
x
0
+
1
2
(
v
0
+
v
)
t
(2.9)
In these equations, we mean that the particle has position
x
0
and velocity
v
0
at time
t
= 0;
it has position
x
and velocity
v
at time
t
.
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 Spring '11
 camus

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