• I 



THEORV Ot THE MOTION OF ROCKETS. 083 



has just lost all its motion , which is nearly equal to 2? 



mile«. 



But if the height to which it will farther rise be de- j-igj^i^t ^f ^j^^ 



manded on the true principle, that gravity varies inversely real ascent ac- 



as the square of the distance from the Earth's centre ; Then, t"ue jaVc/ 



this reUrda* 

 tion. 



Putting r — CL the rad. of the Earth 



a ~ CD the diftance of the point to which 

 the rocket has already ascended from 

 the centre C io 



a; :r C I any variable distance from C I 



V = velocity at I 

 and c zz velocity at D =: 2836-9895 ft. 



' Then x* : 7* : : 1 : — ,- the retardive force of gravity at I 



when that of the surface L is considered as unitj\ 



2 fif r A' 

 Hence — v-i; n 2g/x 'zz ■ " .^ - (the negative sign being 



-J^ — , which, when x zz a, and v — c, \s c — -^ — 



used because the velocity decreases) whose fluent is r* — 



IS c - ~^^-; 

 a 



therefore the fluent corrected will be v"^ zz c* + ' ^ ^' ' 



a X 



So that when v zz o, we shall have c* -\ ^^^ — ~ zz 



ax 



4:0 "" r^ 



0, and a- — J^ 5 :=: (taking the Earth's radius 



4gr — a c 



at 3979 miles) 21 145 1 43-65521 feet, the whole height of the 

 rocket from the centre of the Earth ; and consequently 

 21145143-65521 — r zz 1360-23-65521 feet is the whole 

 height from the surface. Whence also the height to whicli 

 the rocket rises from the point where the impelling force of 

 the composition ceases or is destroyed is 132007-67221 feet. 



Hence it appears, that, in consequence of the diminu- 

 tion of the force of retardation from gravity upwards ac- 

 *;ording to the inverse square of the distance from the 



Earth's 



