NO. 2 METHOD OF REACHING EXTREME ALTITUDES 37 



PART III. CALCULATIONS BASED ON THEORY 

 AND EXPERIMENT 



APPLICATION OF APPROXIMATE METHOD 

 As already explained this method consists in employing the 

 equations 



T? / _3±S_ t \ _-^g_ t 



M: 



and 



a + i 





M = ec"('-k), (7) 



to obtain a minimum M in each interval, where 



M = the initial mass, for the interval, when the final mass is one 



pound, and , 

 R = the air resistance in poundals over the cross-section S, at 

 the altitude of the rocket. If we call, P, the air resistance 



per unit cross-section, we shall have for R, PS — where 



Po 



p is the density at the altitude of the rocket, and po is the 

 density at sea-level. 

 a=the acceleration in ft. per second ^ taken constant through- 

 out the interval, 

 g— -the acceleration of gravity, 

 t = the time of ascent through the interval, and 

 c(i — k) =:what will be called the " effective velocity," for the reason 

 that the problem would remain unchanged if the rocket 

 were considered to be composed entirely of propellant 

 material, ejected with the velocity, c(i— k). It will be 

 remembered that c actually stands for the true velocity 

 of ejection of the propellant, and k for the fraction of 

 the entire mass that consists of material other than 

 propellant. The effective velocity is taken constant 

 throughout any one calculation. 

 The altitude is divided into intervals short enough to justify the 

 quantities involved in the above equations being taken as constants. 

 The equations are then used to find the minimum value of M for each 

 interval — ^the mean values of R and g, in the interval, being employed 

 ■ — and the " total initial mass " required to raise a final mass of one 

 pound to a desired altitude is then obtained as the product of 

 these M's. 



