NO. 2 MKTHOD OF REACHING EXTREME ALTITUDES 47 



over the distance h, we have by the Principle of Work and Energy, 



-.T 2 



h= 



2(Sf + p)' 



The values of p are small, owing to small atmospheric density, being 

 1.59 poundals for the h beyond Sg ; 0.28 beyond s. (a = 5o) ; and 

 0.465 beyond s- (a= 150) . For Sg the low density makes this quantity 

 negligible. 



The altitudes obtained by adding to the interval the corresponding 

 h, are called the " Greatest altitude attained " in table VII. 



Obviously if the start is made at a high elevation, the " total initial 

 mass " required to reach a given height will be less than for a start 

 at sea-level, due not only to the fact that the apparatus is not raised 

 through so great a height, but also to the fact that the denser part 

 of the atmosphere is avoided. Table VI gives minimum masses, M, 

 calculated for a start with zero velocity from the beginning of interval 

 s, (/'. e., 15,000 ft.), the effective velocity being 7,000 ft. /sec. as in 

 table V. 



It happens that the velocity v^ for minimum M in the interval Sg 

 of table VI is the same as the v^ for the same interval in table \' . 

 The calculations that have been made for the intervals beyond Sg 

 apply therefore to the present case, and the only difference between 

 the two cases is that the masses required to reach s- will be greater, 

 for the start at sea-level, than for the start at 15,000 ft. 



The calculations, beginning at 15,000 ft. have been carried out in 

 table VII for all but the low^est " effective velocity " ; and it will be 

 observed that the start from a high elevation becomes important onlv 

 for the lower " effective velocities." 



The most striking as well as the most important conclusion to be 

 drawn from table VII is the small " total initial mass " required to 

 raise one pound to very great altitudes when the " effective velocity " 

 is 7,000 ft. /sec, the mass for the height of 437 miles (2,310,000 ft.) 

 for example, being but 12.33 lbs., starting from sea-level. Even for 

 an " effective velocity " of 3,500 ft. /sec, which allows of considerable 

 inefficiency in the rocket apparatus, the mass is sufficiently moderate 

 to render the method perfectly practicable, for in this case an altitude 

 of over 230 miles from sea-level, practically the limit of the earth's 

 atmosphere, requires under 90 lbs.'"; and an altitude of 118 miles, 

 close under the geocoronium sphere, only 38 lbs. For a start at 

 15,000 ft., the masses are of course, less, namely 49.3 lbs. and 

 20.9 lbs., respectively." 



