69
2.6Graphs of Velocity Versus Time
Since changes in velocity play such a prominent role in physics, we need
a better way to look at changes in velocity than by laboriously drawing
tangent lines on x-versus-t graphs. A good method is to draw a graph of
velocity versus time. The examples on the left show the x-t and v-t graphs
that might be produced by a car starting from a traffic light, speeding up,
cruising for a while at constant speed, and finally slowing down for a stop
sign. If you have an air freshener hanging from your rear-view mirror, then
you will see an effect on the air freshener during the beginning and ending
periods when the velocity is changing, but it will not be tilted during the
period of constant velocity represented by the flat plateau in the middle of
the v-t graph.
Students often mix up the things being represented on these two types
of graphs. For instance, many students looking at the top graph say that
the car is speeding up the whole time, since “the graph is becoming greater.”
What is getting greater throughout the graph is x, not v.
Similarly, many students would look at the bottom graph and think it
showed the car backing up, because “it’s going backwards at the end.” But
what is decreasing at the end is v, not x. Having both the x-t and v-t graphs
in front of you like this is often convenient, because one graph may be
easier to interpret than the other for a particular purpose. Stacking them
like this means that corresponding points on the two graphs’ time axes are
lined up with each other vertically. However, one thing that is a little
counterintuitive about the arrangement is that in a situation like this one
involving a car, one is tempted to visualize the landscape stretching along
the horizontal axis of one of the graphs. The horizontal axes, however,
represent time, not position. The correct way to visualize the landscape is
by mentally rotating the horizon 90 degrees counterclockwise and imagin-
ing it stretching along the upright axis of the x-t graph, which is the only
axis that represents different positions in space.
2.7
.
Applications of Calculus
The integral symbol,
.
, in the heading for this section indicates that it is
meant to be read by students in calculus-based physics. Students in an
algebra-based physics course should skip these sections. The calculus-related
sections in this book are meant to be usable by students who are taking
calculus concurrently, so at this early point in the physics course I do not
assume you know any calculus yet. This section is therefore not much more
than a quick preview of calculus, to help you relate what you’re learning in
the two courses.
Newton was the first person to figure out the tangent-line definition of
velocity for cases where the x-t graph is nonlinear. Before Newton, nobody
had conceptualized the description of motion in terms of x-t and v-t graphs.
In addition to the graphical techniques discussed in this chapter, Newton
also invented a set of symbolic techniques called calculus. If you have an
equation for x in terms of t, calculus allows you, for instance, to find an
equation for v in terms of t. In calculus terms, we say that the function v(t)
0
2
048
t (s)
0
20
x
(m)
v
(m/s)
Section 2.6Graphs of Velocity Versus Time