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