TRACT 37.

OF GUNNERY.

245

PROBLEM IX.

65. To determine the Time of a Ball's ascending, consideringthe decreasing Density of the Air as in the last Prob.

The fluxion of the time is t = But the general equa-tions of the fluxions of the space x and velocity v, in the lastprob. W as ~x = 33 x Q —; there*, f = „ x — X

But x, which

l ,* X tv c + x — o

—r— ; hence t or — = — x — x —-.

av*-rzc v 32 av 2 + w

is always small in respect of c, is nearly = b as determined

c + ^ _... . i_ j „ c + i

in the last problem; theref.-- may be substituted for —

c + b

without sensible error; and then t becomes = — X X

— V

av* + w % Now, this fluxion being to that in prob. 6 , in theconstant ratio of c — b to c + b, their fluents will be alsoin the same constant ratio. But, by the last prob. cc= 55000, and b — 2930 for the first example in prob. 5;therefore c — b— 52070, and c + b = 57930, also the timein problem 6 was 9"*5; therefore, as 52070 : 57930 : : 9 ,,- 5 :]0' ,- 57 for the time in this case, being l"’07 more than the for-mer, or nearly the 9th part more; which is nearly the double,or as the square of the difference, in the last prob. in theheight ascended.

problem x.

66. To determine the circumstances of Space, Time, and Ve-locity, of a Ball descending through the Atmosphere by itsown Weight.

It is here meant that the balls are at least as heavy as castiron, and therefore their loss of weight in the air insensible ;and that their motion commences by their own gravity from astate of rest. The first object of enquiry may be, the utmostdegree of velocity any such ball acquires by thus descending.Now it is manifest that the ball’s motion is commenced, anduniformly increased, by its own weight, which is its constant