240

THEORY AND PRACTICE

TRACT 37-

1-jib ball, when w — 1-^., or \/w = 1'06, this time 9'6\/w, is10'176 seconds.

54. . Corol. 3. For any other size of ball, as that of d di-ameter, instead of the one of 2 inches diameter, employedabove, we must take, in the theorem, \d*a instead of a, and.then the whole time in this general case will be X V--

— x 9'6fw = ^-fw. But w, the ball’s weight,

is as d 3 , the cube of the diameter, therefore is as ^ -or as «/d; that is, the whole time, for the ascent of differentballs, is as f d, the square-root of the diameter. Thus, fora 24 pound ball, having its diameter almost 5'6, the ratio isto -v/5-6, or 1 to ^2-8 = 1-6733 ; then 1-6733 X 10-176= 17 seconds nearly, the whole time of the 24 pound ball’sascent, when projected with an infinite velocity.

PROBLEM VII.

55. To determine the Time of a Ball’s ascending to its great-est Height, using the same Formula of Besistance as inProb. 4.

Now, as in that problem, x —

(z + jO z<i

2g X (mu 2

Qgmd 9

nv) d* + w

X ( ^ + + ^.r ; then dividing by v, or z + p, = t =; the general fluent of which is t =

+

x arc to tang - or& 9

Zgmqdr

v p and radius 1; or, by correction, t =

2gmqd Cj

x (arc to tang.

9

\ — p

arc to tang. -—-). But, when

V — p

the first velocity v is great, the arc to this latter tangentmay be omitted, as equal to nothing, or as of no effect, sincethe value of p is 115^, and when the small velocity v is about100 or 200, the resistance, by the formula -00003026-z/ 1 —'OQlv, here used, comes out either nothing, ora small quan-tity negative. The latter aim being rejected then, there re-mains only x arc to tangent to radius 1.