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with her, and he proceeds to chat her up. At t
3
, he realizes that his advances
are getting him nowhere, and he begins to pace back and forth along the
track dejectedly until time t
4
. At that point, he decides to take a nap.
Meanwhile, the tortoise has been making slow and steady progress, and at
time t
5
, he catches up with the sleeping hare. The tortoise plods along and
crosses the tape at t
6
. Quickly thereafter, the hare, perhaps awakened by
the sound of the crowd celebrating the tortoise's victory, wakes up at time
t
7
and hurries in a frenzy to the finish. At t
8
, the hare crosses the finish
line, where he is humiliated by all his peers, and the cute girl bunny, too.
To measure the average velocity of either animal during any time in-
terval, we divide the animal's displacement by the duration of the interval.
We'll be focusing on the hare, and we'll denote the position of the hare as
x, or more explicitly as x(t), to emphasize the fact that the hare's position
varies as a function of time. It is a common convention to use the capital
Greek letter delta (“∆”) as a prefix to mean “amount of change in.” For
example, ∆x would mean “the change in the hare's position,” which is a
displacement of the hare. Likewise ∆t means “the change in the current
time,” or simply, “elapsed time between two points.” Using this notation,
the average velocity of the hare from t
a
to t
b
is given by the equation
average velocity =
displacement
elapsed time
=
∆x
∆t
=
x(t
b
) − x(t
a
)
Definition of average
velocity
.
t
b
− t
a
This is the definition of average velocity. No matter what specific units we
use, velocity always describes the ratio of a length divided by a time, or to
use the notation discussed in
Section 11.2,
velocity is a quantity with units
L/T.
If we draw a straight line through any two points on the graph of the
hare's position, then the slope of that line measures the average velocity of
the hare over the interval between the two points. For example, consider
the average velocity of the hare as he decelerates from time t
1
to t
2
, as
shown in
Figure 11.2.
The slope of the line is the ratio ∆x/∆t. This slope
is also equal to the tangent of the angle marked α, although for now the
values ∆x and ∆t are the ones we will have at our fingertips, so we won't
need to do any trig.
Returning to
Figure 11.1
,
notice that the hare's average velocity from
t
2
to t
3
is negative. This is because velocity is defined as the ratio of net
displacement over time. Compare this to speed, which is the total distance
divided by time and cannot be negative. The sign of displacement and
velocity are sensitive to the direction of travel, whereas distance and speed
are intrinsically nonnegative. We've already spoken about these distinctions
way back in Section 2.2. Of course it's obvious that the average velocity
is negative between t
2
and t
3
, since the hare was going backwards during
the entire interval. But average velocity can also be negative on an interval
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