NO. 2 METHOD OF REACtlING EXTREME ALTITUDES 9 



quantity R, the air resistance of the rocket at time t, depends not only 

 upon the density of the air and the velocity of the rocket, but also 

 upon the cross section, S, at the time t. The cross section, S, should 

 obviously be as small as possible ; and this condition will be satisfied 

 at all times, provided it is the following function of the mass of the 

 rocket (M — m), 



S = A(M-m)S (2) 



where A is a constant of proportionality. This condition is evidently 

 satisfied by the ideal rocket, figure i. Equation (2) expresses the 

 fact that the shape of the rocket apparatus is at all times similar to 

 the shape at the start ; or, expressed differently, S must vary as the 

 square of the linear dimensions, whereas the mass (M — m) varies as 

 the cube. Provision that this condition may approximately be ful- 

 filled is contained in the principle of primary and secondary rockets. 



The resistance, R, may be taken as independent of the length of 

 the rocket by neglecting " skin friction." For velocities exceeding 

 that of sound this is entirely permissible, provided the cross section 

 is greatest at the head of the apparatus, as shown in United States 

 Patent No. 1,102,653. 



The quantities R, g, and v, are evidently expressible most simply 

 in terms of the altitude s, provided the cross-section S is also so 

 expressed, giving, in place of equation (i) 



c(i-k)dm=(M-m)dv + ~-^[R(s)+g(s)(M-m)]ds. (3) 



v(^s J 



RIGOROUS SOLUTION FOR MINIMUM M AT PRESENT 

 IMPOSSIBLE 



The success of the method depends entirely upon the possibility of 

 using an initial mass, M, of explosive material that is not imprac- 

 ticably large. It amounts to the same thing, of course, if we say that 

 the mass ejected up to the time t (i. e., m) must be a minimum, con- 

 ditions for the existence of a minimum being involved in the integra- 

 tion of the equation of motion. 



That a minimum mass, m, exists when a required mass is to be 

 given an assigned upward velocity at a given altitude is evident 

 intuitively from the following consideration : If , at any intermediate 

 altitude, the velocity of ascent be very great, the air resistance R 

 (depending upon the square of the velocity) will also be great. On 

 the other hand, if the velocity of ascent be very small, force will be 

 required to overcome gravity for a long period of time. In both 

 cases the mass necessary to be expelled will be excessively large. 



