TRACT 37.

OF GUNNERY.

239

X y

50. But — x v or — denotes the time of the body’s

2 g 32 _ ■>

whole ascent in free space, projected up with the same ve-locity v. Therefore the whole time of ascent in free space,is to the whole time of ascent in the atmosphere, commenc-ing with the same velocity, as the tangent of v, is to its arc,the radius being */—.

51. Now, as in the last prob. v = 2000, w = 1-J-, a =

’° 0002 ^ = -T = 42900 ’ “ d ^*7= 207 >

an( l - = 9’66184 the tangent, to which corresponds the

\/T

arc of 34° 6', whose length is 1-4678 : then — x 207 x1 4678 _-—- — 9 - 5, the whole time of ascent.

52. Corol. 1 . The time of freely ascending or descend-ing through the same height 2800 feet, nearly the mean be-tween the two former results, that is, without the air’s re-sistance, would be f —— = 5 t 3 = 13"i; and the time offreely ascending, till all the velocity is lost, commencing

with the same velocity 2000, would be = -■$- = 62"! =1' 2'f, being in proportion to 9"~ the time as above in the at-mosphere, as 9’6618 the tangent of 84°6', to 1-4678 its arc.But the time of ascending freely through the space 2930only, commencing with the same velocity 2000, would beonly l"£.

53. Corol. 2. Since the time of ascent is in general equal

to — f arc tang, to radius 1, nearly, the great-

est possible time, when v is infinite, or the tangent v x

is infinite, or when the arc is a quadrant, will be nearly

r—X the quadrant to the radius 1, that is, —*/—

2 g “ ^ . 4g v a,

where c = 3T416, the circumference of the circle to the di-ameter 1. Now, with the given numbers, viz, a = -000026-!,c = 3-1416, g = 64, the greatest time ~\f — becomes 9'6X V10, which therefore is the greatest time nearly. For the