TRACT 37.

OF GUNNERY.

239

50. But -j- xvor-r denotes the time of the body’s

^s 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 beins: */ —.

b y a

51. Now, as in the last prob. v = 2000, rw = l-§-, a —

*0000265. = ™ Hence — = 42900, and */— = 207,

» yooouuu a v a 7

and - Y = 9*66184 the tangent, to which corresponds the

a/T

arc of 84° 6’, whose length is T4678 ; then ~ x 207 X1*4678 =: — 7 - X J, 4678 = 9 "-5, the whole time of ascent.

52. Corol. 1. The time of f reely 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 = V = 13*1-; and the time of

freely ascending,"till all the velocity is lost, commencing. l » » v 2000 _ ro ff i —.

with the same velocity 2000 , would be — = t —■

T 2"-§., being in proportion to 9 "a the time as above in the at-mosphere, as 9*6618 the tangent of 84°6', to T4678 its arc.But the time of ascending freely through the space 2930only, commenciivg with the same velocity 2000 , would beonly l"^.

53 . Corol. 2. Since the time of ascent is in general equalt0 «/ -7 arc tang. V-J*, to radius 1 , nearly, the great-est possible time, when V is infinite, or the tangent v xV - 2 - is infinite, or when the arc is a quadrant, will be nearly

1 £ XtJ

x the quadrant to the radius 1 , that is, —a/ ~ twhere c = 3 - l416, the circumference of the circle to the di-ameter 1 . Now, with the given numbers, viz, <2 = "000026-5,c = 3"1416, g = 64, the greatest time^v/ a becomes 9*6X V 10 , which therefore is the greatest time nearly. For the