TRACT 38.

PROBLEMS.

333

Put

r = cs the radius of the earth,a = ca the dist. fallen from,x — cp any variable distance,

■o = the velocity at p,t = time of falling there, andg •— 16^, half the veloc. or force at s,f — the force at the point p.

Then we have the three following equations, viz.x z : r 1 : : 1 ;f = — the force at p, when the force of gra-vity is considered as 1;tv = — x , because x decreases; and

vv =

= - 9 ~sp

2 gr^xx* *

The fluents of the last equation give v z = But whenx = a, the velocity v = 0; therefore, by correction, v z =

a general

4er 2 4ff7 3 . . a — x

—-— = 4<gr z X —— ;

. , 4gr

or v = \S( —

expression for the velocity at any point p.

When x = r, this gives v = \Z(4gr x -—) for the great-est velocity, or the velocity when the body strikes theearth.

When a is very great in respect of r, the last velocity be-comes (1 — -^) X v'&gr very nearly, or nearly \Z4gr only,which is accurately the greatest velocity by falling from aninfinite height. And this, when r =. 3965 miles, is 6-9506miles per second. Also, the velocity acquired in falling fromthe distance of the sun, or 12000 diameters of the earth,is 6-9505 miles per second. And the velocity acquired infalling from the distance of the moon, or 30 diameters, is6-8927 miles per second.

Again, to find the time; since tv = — x, thereforet = —— •/— x Xx ; the correct fluent of which

gives t = x [a/ ax — xx + arc to diameter a and

vers, a x) ; or the time of failing to any point F =