Viewing page 31 of 212

This transcription has been completed. Contact us with corrections.

7.

the analytical expression for the conservation of energy, obtained by equating the heat energy evolved by the burning of the mass propellant, dm(1-k), to the additional kinetic energy of the system produced by this mass plus the work done against gravity and air resistance during the time [[underline]] dt [[/underline]].  The equation thus derived is found to be identical with equation (1).

[[underline]] REDUCTION OF EQUATION TO THE SIMPLEST FORM [[/underline]].

In the most general case, it will be found that [[underline]] R [[/underline]] and [[underline]] g [[/underline]] are most simply expressed when in terms of [[underline]] v [[/underline]] and [[underline]] s [[/underline]].  In particular, the quantity [[underline]] R [[/underline]], the air resistance of the rocket at time [[underline]] t [[/underline]], depends not only upon the density of the air and the velocity of the rocket, but also upon the cross-section, [[underline]] S [[/underline]], at the time [[underline]] t [[/underline]]. The cross-section, [[underline]] S [[/underline]] should obviously be as small as possible; and this condition will be satisfied [[underline]] at all times [[/underline]], provided it is the following function of the mass of the rocket, (M-m),

^[[S = [[underline]] A [[/underline]](M-m)^2/3]]  (2)

where [[underline]] A [[/underline]] is a constant of proportionality.  This condition is evidently satisfied by the ideal rocket, Fig. 1. 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, [[underline]] S [[/underline]] 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 fulfilled is contained in the principle of primary and secondary rockets.  

The resistance, [[underline]] R [[/underline]], may be taken as independent of the length of the rocket by neglecting the "skin fraction".  For velocities exceeding that of sound, this is entirely permissible, provided the

Transcription Notes:
Reviewer 1 notes: my only changes were to modify all single brackets to double brackets as found in instruction guidelines, and to indicate handwriting ^[[example]]. I don't see in instructions how to indicate underlining in parts of handwritten text, and it's important in this case to be sure that the A in the handwritten text is underlined... so I guessed. Would be happy to learn how it should be done.