TRACT 37.

OF GUNNERV.

25i

PROBLEM XX.

78 . To determine the Circumstances of the Motion of a Ballprojected horizontally in the Air; abstracted from its Ver-tical Descent by its Gravitation, or supposing the Ball void ofGravity.

Patting d for the diameter, and a) the weight of the ball,v the velocity of projection, and v the velocity of the ballafter having moved through the space .r. 1 hen, by corol. 1

to prob. 2, if the velocity is considerable, such as usual inpractice, the resistance of the ball, moving with the velocity•v, is (niv 1 — nv)d l , and therefore”'' “ d* is the retardiveforce f; hence the common formula vv = 2gfx, is — vv =

■dr, and theref. x

32* x

— V

— v

, the fluent of which is obviously

me—n 32 iPm

n

v — —m

X — hyp. log. of v — ; and by the correction by

32 md

n

v —..—

m

the first velocity v, it becomes x = pypggi x h. l°g-

H

V — -

m

x h. Ioet . -—the general formula for the distance

passed over in terms of the velocity, where q = ~ —

•00175

•000007565

79. But now to reduce the general coefficient Ml— to asimple number in terms of d the diameter only; the weight”00 zz *523 6d 3 x 4*3 2\d } = f d 3 , where 4’3 is the weight

in ounces of 1 cubic inch of cast iron, or w = f f d l in poundsweight. Therefore ~ — A z d, and -f-- = 58Ud; then the

theorem becomes x = 581^(/ x h. log. ^— l; or, multiplyingthe coefficient by 2-30258 the hyp. log. of iO, to reduce thehyp. log. to the common log. then it is x — 1338rfx com. log.

°f tl )e diameter d being in inches.