NO. 2 METHOD OF REACHING EXTREME ALTITUDES 6l 



But V is the maximum velocity during the motion and hence 

 V = w.s, where s is the maximum displacement, and w is a constant, 

 having the usual significance ; also 



P= — . 



m,g=(M+^' 



Hence 



3 / s- 

 But by the Conservation of Linear Momentum, 



/M+^^W = moV. 



Hence 



giving, for the kinetic energy per gram of mass expelled, 



(M+^^)(m,g) 



= -^ "^—^ s^ 



2 2m^- 



From this it is possible to obtain the efficiency, by dividing by the 



heat value of the powder, in ergs ; and also the velocity in kilometers 



per second by multiplying by two, extracting the square root, and 



dividing by lO^. 



CORRECTION OF THE DISPLACEMENT, s, FOR FRICTION 

 The displacement, s, in the preceding calculation is assumed to be 

 the corrected displacement. This is obtained from the upward dis- 

 placement s^. and the downward displacement s,, as 



^=^-it 



APPENDIX C 

 THEORY OF DIRECT-LIFT IMPULSE-METER 

 The theory of the direct-lift impulse-meter is as follows : 

 Calling I, the momentum of the gas that strikes the end of the 

 aluminium cylinder, 



mc=the mass of the aluminium cylinder, 

 Vc = the initial upward velocity of the cylinder, 

 Ac = the area of cross-section of the cylinder, 

 Ag = the maximum area of cross-section of the suspended system 

 comprising the gun, lead weight, and holders, 

 and s=the displacement of the aluminium c)dinder, as obtained 

 5 



