Calculus: Analytic Geometry and Calculus, with Vectors

1.6 Introduction to velocity and acceleration 47

toward the center. The length of the vector r is always R, and the length
of the vector a is always OR. This shows that the magnitude of the
acceleration is always w2R. These results are important in physics and
engineering. Physics books that do not make effective use of good
mathematics do not derive their results so efficiently.

Problems 1.69

1 Supposing that g, eo, and so are constants and that

(1) s =-gt2 +Sot + so

at each time t, use notation like that in (1.64), so that s = sl when t

s = s2 when t = t2, to obtain the formula

(2) s2 - Sl = 'lEg (t2 - ti) + vo(t2 - tl)+

and hence

(3)

s2-sl

= -'g(t2 + ti) + so

t2 - tl

= toand

when to 0 to. Remark: Even though we have not yet encountered procedures by

which such statements are made precise, we can temporarily accept without

question the statement that the right side of (3) must be near gt + so whenever

tl and t2 are both near t and hence that

(4)

2

(1)

V=gt+Vo.

Supposing that g and 9o are constants such that

V = gt + no

at each time t, use notation like that in (1.66), so that v = vi when t = tl and

9 = 92 when t = t2, to obtain the formula

(2)

and hence

(3) to - tl

when t2 0 t1. Remark: A remark similar to that of the preceding problem is

applicable here; the scalar acceleration a at time t is g.

3 We should now be well aware of the fact that Problems 9.29 will appear at

the end of Chapter 9, Section 2. While the trick is not used in this book, we can

use the numbers 9.2908 and 9.2922 to identify problems 8 and 22 of Problems9.29.

Now comes the problem. Write a single number to identify formula 15 of Prob-

lem 4 at the end of Section 6 of Chapter 12. Ins.: 12.690415. Persons who feel

that this trick is complicated should think about the matter to capture some of the

spirit of members of a research staff of a data processing department of IBM

(International Business Machine Corporation) who find that such tricks keep

them in business.

92-ti1 = g(t2 - tl)

• g