Calculus: Analytic Geometry and Calculus, with Vectors

44 Analytic geometry in two dimensions

average scalar velocity. After appropriate preliminary topics have been
studied, Chapter 3 will tell precisely how the velocity V and the scalar
velocity v at time t are defined. It turns out that the scalar velocity v

and a numberd,called the derivative of s with respect to t, are equal to

each other and, moreover, that the quotient in (1.64) is nearly equal to

v and dt whenever ti = t and t2 is nearly equal to t but t2 s t. To obtain

an estimate of the scalar velocity v at a particular time t from experimental

data, which may be presented in a graph, it therefore suffices to calculate

and use the average scalar velocity over a short time interval beginning

or ending at t. Use of experimental data for this purpose is rendered

difficult by the fact that, when tl and t2 are nearly equal, small relative

errors in measurements can produce huge errors in estimates of the value

Figure 1.65

of the quotient (s2 - sl)/(t2 - t1). It is a truly

remarkable fact that when reasonably accurate

data are collected and intelligently used, it is pos-

sible to estimate v for various values of t and to

find that the points (t,v) in a tv plane come so close

to lying on a line that all of the deviations can be

attributed to errors in measurement and calcula-

tion. Thus our experimental work leads to the con-

clusion that, as in Figure 1.65, the graph of v versus t

is either a part of a line or a very close approxima-

tion to a part of a line.

The scalar acceleration a of our falling body is defined in terms of the

scalar velocity v in the same way that the scalar velocity v is defined in

terms of the scalar displacement s. Thus, in addition to the basic

formula (1.64), we have the basic formula

(1.66)

t2 - tl

= average scalar acceleration

in which vl and v2 are the scalar velocities at times tl and t2. The scalar

acceleration a at time t and the derivative dt are equal to each other and,

moreover, the quotient in (1.66) is nearly equal to a and todtwhenever

t1 = t and t2 is nearly equal to t but t2 0 t. On the basis of the assump-

tion that the graph of v versus t is a part of a line as in Figure 1.65, the

average scalar acceleration is the slope ml of the part of the line. The

hypothesis that each average scalar acceleration is the constantm1 leads

to the conclusion that, at each time t, the scalar acceleration is ml; that

is, a = ml. Calculations from reasonably accurate data show that ml

is about 980 when centimeters and secondsare used and about 32 when