NO. 2 METHOD OF REACHING EXTREME ALTITUDES 4I 



The value o'f at may be had from the equation 



Vi = Vo + at, (9) 



and t is at once obtained from the relation 



s = Vot + -jat^ ; i. e., 



whence, of course, a is at once known. 



a+g ^ at_ 



The calculation of ecd-io and e^u-k) call for no comment; and 

 R is obtained as P, the mean ordinate between Vq and v^ from the 



curves as already explained, multiplied by S and ^ . 



The value of M, the initial mass, for the interval, necessary in 

 order that the final mass in the interval shall be one pound, is then 

 obtained from equation (7) ; and finally, the ratio of equations (6) 



to (7) / i. e., — \ is calculated. This is the ratio of the initial 



(■ 



gCd-k) / 



mass necessary, including losses due to both R and g, to the mass 

 necessary to give the one pound the same velocity, Vi, without over- 

 coming R and g ; and the entire calculation must be repeated until a 

 minimum value of this ratio is obtained — when the corresponding 

 mass, M, will be the minimum mass for the interval in question. Each 

 minimum M is marked in the tables by an asterisk. 



This process is carried out for each interval beginning with the 

 first. 



It should be noticed that, although P and the density are not really 

 constant in any interval, the result obtained by taking the mean of the 

 quantities must nevertheless give results close to the truth, owing 

 to the fact that P increases during the ascent, whereas the density 

 decreases. 



EXPLANATION OF TABLES V AND VI 



It should first be explained why no minimum M has been calculated 

 for the intervals s^ and Sg. Although the minima for the preceding 

 intervals are clearly defined, a trial will show that a minimum M can 

 occur, for s^ and Sg, only for extremely high velocities, v^ ; although 

 for S-, a secondary minimum occurs for Vi = 8,000 ft./sec. Even for 

 Vi = 30,000 ft./sec. the minimum has not yet been attained for this 

 interval, although the acceleration required to produce this velocity 



