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64.

If a rocket such as has been discussed were projected to the upper end of internal [[underlined]] s[[subscript]] 8 [[/subscript]] [[/underlined]], either with an acceleration of 50 or 150 ft/sec[[superscript]]2[[/superscript], and [[underlined]] this acceleration were maintained to a sufficient distance beyond [[underlined]] s [[subscript]] 8 [[/subscript]] [[/underlined]], until the parabolic velocity were attained, the mass finally remaining would certainly never return.

If we designate as the upper end of [[underlined]] s[[subscript]] 9 [[/subscript]] [[/underlined]] the height at which the velocity of ascent becomes the "parabolic" velocity, it will be evident that this height will be different for the two accelerations chosen; inasmuch as the "parabolic" velocity decreases with increasing distance from the center of the earth.

If we call [[underlined]] u [[/underlined]] = the "parabolic" velocity at a distance [[underlined]] H [[/underlined]] above the surface of the earth.

[[underlined]] v [[/underlined]][[subscript]] 1 [[/subscript]] = the velocity acquired at the upper end of internal [[underlined]] s [[/underlined]][[subscript]] 8 [[/subscript]],

[[underlined]] s [[/underlined]][[subscript]] o [[/subscript]] = the height of the upper end of [[underlined]] s [[/underlined]][[subscript]] 8 [[/subscript]] above sea-level,

we have, taking the radius of the earth as 20,900,000 feet,

u = v[[subscript]] 1 [[/subscript]] + at, (11)
[[underlined]] H [[/underlined]] = s[[subscript]] o [[/subscript]] + v[[subscript]] 1 [[/subscript]]t + 1/2 at[[superscript]] 2 [[/superscript]], (12)

and also the equation relating "parabolic" velocity to distance from the center of the earth

36,700/u = √20,900,000 + [[underlined]] H [[/underlined]] / 20,900,000   (13)

On putting the values of [[underlined]] u [[/underlined]] and [[underlined]] H [[/underlined]], from (11) and (12), in (13), we have

20,900,000 x 36,700 = (v[[subscript]] 1 [[/subscript]] + at) √21,400,000 + v.t + 1/2 at[[subscript]] 1 [[/subscript]][[superscript]] 2 [[/superscript]]  (14)

Equation (14) is a bi-quadratic in [[underlined]] t [[/underlined]], from which [[underlined]] t [[/underlined]] may easily be obtained (by trial and error). The values of [[underlined]] t [[/underlined]], for the two accelerations chosen, given in Table V, enables [[underlined]] u [[/underlined]] and the initial masses for [[underlined]] s [[/underlined]][[subscript]] 9 [[/subscript]], to be at once obtained.