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77.

Appendix E.

[[underline each individual word]] Check on Approximate Method of Calculation. for Small Charges Fired in Rapid Succession [[/underline each individual word]].

Consider a rocket weighing 10 lbs., having 2 lbs. of propelling material, fired [[strikethrough]] one [[/strikethrough]] two ounces at a time, eight times per second, with a velocity of 6,000 ft/sec. -- much less than the highest velocity attained in the experiments, either in air or [[underline each indivdiaul word]] in vacuo [[/underline each individual word]].

Let us suppose that, for simplicity, the rocket is directed upward and that each shot takes place instantly (a supposition not far from the truth); the velocity remaining constant between successive shots.

After the first shot, the mass, 9 7/8 lbs., has an upward velocity [[underline]] v [[/underline]] [[subscript]] o [[/subscript]] due to the downward velocity of the 1/8 lb. expelled. This velocity, [[underlined]] v [[/underlined]] [[subscript]] o [[/subscript]], is at once found by the Conservation of Momentum. But it is decreased by gravity until, at the end of 1/8 second, it is reduced to 

v [[superscript]] 1 [[/superscript]] [[subscript]] o [[/subscript]] = v [[subscript]] o [[/subscript]] - g t,
the space passed over during this time being

s = v [[subscript]] o [[/subscript]] t - 1/2 g t [[superscript]] 2 [[/superscript]].

We have then, v [[superscript]] 1 [[/superscript]] [[subscript]] o [[/subscript]] = 71.8 ft/sec., and [[underline]] s [[/underline]] = 9.23 ft.

At the beginning of the second interval of 1/8 sec., an additional velocity is given the remaining mass, of 76.8 ft/sec., and the final velocity and space passed over may be found in the same way. By completing the calculations for the remaining intervals, we shall have