THEORY OF TME MOTION #P ROCKETS. 



the time of its burnings and the weight and dimensions of 

 the rocket ; to find the height to which it will rise in the at- 

 mosphere if thrown perpendicularly , and aho the velocity 

 acquired at the end of that time ; the law of resistance being 

 supposed to vary directly as the square of the velocity; and 

 the lamincE of the composition to fire uniformly and to burn 

 parallel to the rocket's base. 



Put w =: weight of the case of the rocket and head 



c = weight of the whole quantity of matter with 



which it is filled 

 a zz time in which the same i$ consuming itself uni- 

 formly 

 n zz 230 ozs. 

 s zz 1000 



d — diameter of the rocket's base -B 



x z= P D the space the rocket describes in the 



time t 



V zz the acquired velocity in that time 

 Rz: the resistance of the air to the rocket when mov- 

 ing with a velocity of 6 feet per second 



Rw* 



Then 6* : «* : : R : -rj- the resistance at D; and conse- 

 quently* wed* — (m ) Tj^- (see Prop, 1) will be 



the motive force of the rocket at D in this case ; and 



i — — -^ 1 the accelerative force. Therefore 



[am'^ct)b^ 



^. {*ned*6*— Ri;*) 2^a/* ^ . 

 .= 2g/. = l~ZTl)b^ 2g.;orput- 



ting <2ag'Ksned b'^zzhy ^ag^—k, am 6* z: /and c6* 



, ,. , . ht — kv^i . J , . 



:=/!, we shall have a; — — 2 ^/ ; and / v — p t v 



I — — p t 



=z ht — A; i;* /' — 2 ^ / f ' + 2gp tt'; and further, putting 

 h-^^gl zzq to render the expression as simple as possi- 

 ble. It will be /«w — pt*v — qi ■\- kv"^ t -\- ^gptt'zzo; 

 where r may be determined in terms of t as follows ; 



Assume v = A f + B <* + C t^ + D <♦ + E <* -f &c. : 

 then making t'z^ 1 ; we have v = A -f 2 B * -f 3 C <* + 



4D 



167 



