42 Analytic geometry in two dimensions
will be completely familiar and meaningful to us. Meanwhile, we make a
preliminary study of ways in which they are related to experiments
involving falling bodies. Chapters 3 and 4 will give much less informa-
tion about the physics experiments but much more information about
the mathematics.
-3 Suppose we have, as in Figure 1.62, a vertical s axis with
-2 the positive s axis below the origin. For laboratory experi--1 ments, we can place one meter stick above another and
° 1 place minus signs in front of the numbers on the upper2 stick. We can suppose that a body is, at time t = 0,
3 falling or just being dropped so that it travels past the
s markings on our meter sticks with increasing rapidity as
Figure 1.62 time passes. On the other hand, we can suppose that thebody is rising at time t = 0 so that it rises for a while
before it begins its descent. We may suppose that distances are meas-
ured in centimeters, so that s = 20 when the body is 20 centimeters
below the origin, and that t is measured in seconds, so that t = 0.5 when a
timing device shows a half-second after our time origin or zero-hour.
Anyone who tosses a body upward and observes the ensuing motion
must realize that it is not an easy matter to use an ordinaryclock to
obtain accurate data giving the coordinate s of the body at various times
t. While solid information about such matters must be obtained from
physicists, we can all recognize the possibility of getting useful data with
the aid of apparatus so arranged that at each of the times t = 0, t = 0.01,
t = 0.02, t = 0.03, an electric spark jumps from a pointer onthe
falling body to burn a tiny hole in a long strip of
paper attached to the meter sticks. When enough
reasonably accurate information has been obtained
(12,82) in one way or another, we can use it to plot points
(t,s) in a is plane and obtain a graph more or less
like that shown in Figure 1.63. For each t within
the domain for which measurements are made,
0 " tl tz t the s coordinate of the point P(t,s) on our graph is
Figure 1.63 a more or less good approximation to the coordi-
nate or displacement of the body at time t.
Some information should be in hand when we undertake to use our data
and graph to obtain information about our falling body. Without pre-
tending to have precise ideas yet, we can start with the rough idea that
forces and velocities and accelerations exist and that these things are
vectors or are represented by vectors. The reason our falling body
plummets toward the center of the earth, with speed increasing when it is
headed downward, is that the earth exerts a gravitational force upon it.
The magnitude of this force is the weight of the body. Since we find no
perceptible change in the weight of a body when we raise or lower it a few