Calculus: Analytic Geometry and Calculus, with Vectors

1.6 Introduction to velocity and acceleration 45

feet and seconds are used. This number ml is the gravitational constant

g to which wehave referred.
The simplest reasonable conclusion that can be drawn from data
involving falling bodies is the following. To each place on the surface of
the earth there corresponds a positive constant g, the scalar acceleration
of gravity at that place, such that when a body moves on a vertical line
near the surfaceof the earth with no appreciable external force other than
the gravitational force exerted upon it, reasonable answers to problems
can be based uponthe assumption that the body is accelerated toward the
center of the earth and that the scalar acceleration is g. Another similar
but more lengthy conclusion involves the idea that the graph of v versus t
is a line and that reasonable results are obtainable from the formula
v = gt + vo, where vo is aparticular constant that depends upon choice of
the time-origin used when studying a particular flight. Finally, it is
possible to use the data and quite primitive mathematics to reach the
more abstruse conclusion that there exist constants g, vo, and so, the latter
two of which depend upon the time-origin and the space-origin used in the
study of a particular flight, such that reasonable results are obtainable from
(1.611). A campaign to reach this conclusion can start with the observa-
tion that the graph in Figure 1.63 does look like a part of a parabola.
Mathematicians do not, except when they are behaving like physicists,
actually perform physical experiments. Mathematicians cannot, unless
they have physical laws or other information upon which proofs can be
based, prove the formulas that are useful in mechanical dynamics and
thermodynamics and hydrodynamics and aerodynamics and electro-
dynamics and economics and psychology and genetics and chemistry and
cosmology. But mathematicians can, when they are given a few weeks,
learn enough about derivatives and other things to enable them to start
with given information and produce more information with astonishing
ease. One who knows the content of Chapter 3 can start with the first
of the three formulas

(1.671) s =

2

gt2 + vot + so

(1.672) v = dt= gt + vo

d's

(1673) a=7=-t dt2_ g

and produce the other two as fast as he can write. All he needs to do is

apply standard rules for writing derivatives. The problems at the end of

this section provide preliminary ideas about this matter. It is much

more significant that one who knows the contentof Chapter 4 can start

with the last of the formulas and produce the other two as fast as he can

write. All he needs to do is apply standard rules for writing integrals.