61
We assume that our speedometer tells us what is happening to the speed
of our car at every instant, but how can we define speed mathematically in a
case like this. We can’t just define it as the slope of the curvy graph, because
a curve doesn’t have a single well-defined slope as does a line. A mathemati-
cal definition that corresponded to the speedometer reading would have to
be one that attached a different velocity value to a single point on the curve,
i.e. a single instant in time, rather than to the entire graph. If we wish to
define the speed at one instant such as the one marked with a dot, the best
way to proceed is illustrated in (d), where we have drawn the line through
that point called the tangent line, the line that “hugs the curve.” We can
then adopt the following definition of velocity:
definition of velocity
The velocity of an object at any given moment is the slope of
the tangent line through the relevant point on itsx-t graph.
One interpretation of this definition is that the velocity tells us how many
meters the object would have traveled in one second, if it had continued
moving at the same speed for at least one second. To some people the
graphical nature of this definition seems “inaccurate” or “not mathemati-
cal.” The equation
v=
.
x/
.
t by itself, however, is only valid if the velocity is
constant, and so cannot serve as a general definition.
Example
Question: What is the velocity at the point shown with a dot on
the graph.
Solution: First we draw the tangent line through that point. To
find the slope of the tangent line, we need to pick two points on
it. Theoretically, the slope should come out the same regardless
of which two points we picked, but in practical terms we’ll be able
to measure more accurately if we pick two points fairly far apart,
such as the two white diamonds. To save work, we pick points
that are directly above labeled points on the t axis, so that
.
t=4.0
s is easy to read off. One diamond lines up with x
˜
17.5 m, the
other with x
˜
26.5 m, so
.
x=9.0 m. The velocity is
.
x/
.
t=2.2 m/s.
Conventions about graphing
The placement of t on the horizontal axis and x on the upright axis may
seem like an arbitrary convention, or may even have disturbed you, since
your algebra teacher always told you that x goes on the horizontal axis and y
goes on the upright axis. There is a reason for doing it this way, however.
In example (e), we have an object that reverses its direction of motion twice.
It can only be in one place at any given time, but there can be more than
one time when it is at a given place. For instance, this object passed
through x=17 m on three separate occasions, but there is no way it could
have been in more than one place at t=5.0 s. Resurrecting some terminol-
ogy you learned in your trigonometry course, we say that x is a function of
t, but t is not a function of x. In situations such as this, there is a useful
convention that the graph should be oriented so that any vertical line passes
through the curve at only one point. Putting the x axis across the page and
t upright would have violated this convention. To people who are used to
interpreting graphs, a graph that violates this convention is as annoying as
0
5
10
15
20
25
30
0246810
t (s)
x
(m)
Example: finding the velocity at the
point indicated with the dot.
0
5
10
15
20
25
30
0246810
t (s)
x
(m)
.
t=4.0 s
.
x
0
5
10
15
20
25
30
0246810
t (s)
x
(m)
.
t
.
x
(d) The velocity at any given moment
is defined as the slope of the tangent
line through the relevant point on the
graph.
(e) Reversing the direction of
motion.
Section 2.3Graphs of Motion; Velocity.