8 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. Jl 



material P is expelled downward with a constant velocity, c. It is 

 further supposed that the casing, K, drops away continuously as the 

 propellant material P burns, so that the base of the rocket always 

 remains plane. It will be seen that this approximates to the case of 

 a rocket in which the casing and firing chamber of a primary rocket 

 are discarded after the magazine has been exhausted of cartridges, 

 as well as to the case in which cartridge shells are ejected as fast as 

 the cartridges are fired. 

 Let us call 



M = the initial mass of the rocket, 

 m = the mass that has been ejected up to the time, t, 

 v = the velocity of the rocket, at time t, 

 c = the velocity of ejection of the mass expelled, 

 R = the force, in absolute units, due to air resistance, 

 g=the acceleration of gravity, 

 dm = the mass expelled during the time dt, 

 k = the constant fraction of the mass dm that consists of casing 

 K, expelled with zero velocity relative to the remainder of 

 the rocket, and 

 dv = the increment of velocity given the remaining mass of the 

 rocket. 

 The differential equation for this ideal rocket will be the analytical 

 statement of Newton's Third Law, obtained by equating the momen- 

 tum at a time t to that at the time t + dt, plus the impulse of the forces 

 of air resistance and gravity, 



(M-m)v=:dm(i-k) (v-c) +vkdm 



+ (M-m-dm)(v + dv) + [R + g(M-m)]dt. 



If we neglect terms of the second order, this equation reduces to 



c(i-k)dmi=(M-m)dv+[R + g(M-m)]dt. (i) 



A check upon the correctness of this equation may be had from 

 the analytical expression for the Co'nservation of Energy, obtained by 

 equating the heat energy evolved by the burning o'f the mass of 

 propellant, dm(i— k), to'the additional kinetic energy of the system 

 produced by this mass plus the work done against gravity and air 

 resistance during the time dt. The equation thus derived is found to 

 be identical with equation (i). 



REDUCTION OF EQUATION TO THE SIMPLEST FOR^I 

 In the most general case, it will be found that R and g are most 

 simply expressed when in terms of v and s. In particular, the 



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