Viewing page 157 of 212

This transcription has been completed. Contact us with corrections.

58.

in the last few intervals, in which the rocket has become so small that the condition of one-square-inch-per-pound is approximately satisfied.

Before leaving the subject of air resistance, attention should be called to the fact that the velocities (Table V) do not exceed that for which air resistance has been studied by Mallock until [[underlined]] s[[subscript]] 7 [[/subscript]] [[/underlined]], for a = 150 ft/sec.[[superscript]] 2 [[/superscript]] and in s[[subscript]] 8 [[/subscript]], for a = 50 ft/sec.[[superscript]] 2 [[/superscript]]; and furthermore, that the velocities do not become much in excess until the densities have become almost negligible.

[[underlined]] CHECK ON APPROXIMATE METHOD OF CALCULATION.  [[/underlined]] 

A simple calculation, involving only the most elementary formulae instead of equations (6) and (7) will show that the "total initial masses" in Table VII cannot be far from the truth.

Consider, for simplicity, a rocket of the form shown in Fig. 1, and suppose that one third of the mass of the rocket is fired downward, with a velocity of 7,000 ft/sec, at the first shot; one third of the remaining mass, at the second shot; and so on, for successive shots. From the principle of the Conservation of Momentum it will be evident that the mass that remains is given an [[underlined]] additional [[/underlined]] upward velocity of 3,500 ft/sec. [[underlined]] after each shot [[/underlined]]. 

Thus, after the 4th shot, the mass that remains is 16/81, or practically 1/5, of the initial mass; and the velocity is 14,000 ft/sec. The velocity is sufficient, [[underlined]] if we neglect air resistance [[/underlined]], to raise the part of the rocket that remains to an altitude of [[underlined]] 580 miles [[/underlined]] (by the familiar relation, v[[superscript]] 2 [[/superscript]] = 2 g h). Although the range would be much reduced if air resistance were considered, it should nevertheless be remembered that the values in Table VII