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6.

constant velocity, [[underline]] c [[/underline]]. It is further supposed that the casing, [[underline]] K  [[/underline]], drops away continuously as the propellant material [[underline]] P [[/underline]] burns, so that the base of the rocket always remains plane. It will be seen that this approximates to the case of a rocket in which the casing and firing chamber of a primary rocket are discarded after the magazine has been exhausted of cartridges, as well as to the case in which cartridge shells are ejected as fast as the cartridges are fired.

Let us call

M = the initial mass of the rocket,

m - the mass that has been ejected up to the time, [[underline]] t [[/underline]],

v = the velocity of ejection of the mass expelled,

R = the force, in absolute units, due to air resistance,

g = the acceleration of gravity,

dm = the mass expelled during the time [[underline]] dt [[/underline]],

k = the constant fraction of the mass [[underline]] dm [[/underline]] that consists of casing [[underline]] k [[/underline]], expelled with zero velocity relative to the remainder of the rocket, and

dv = the increment of velocity given the remaining mass of the rocket.

The differential equation for this ideal rocket will be the analytical statement of Newton's Third Law, obtained by equating the momentum at a time [[underline]] t [[/underline]] to that at the time [[underline]] t [[/underline]] ^[[+]] [[underline]] dt [[/underline]], plus the impulse of the forces of air resistance and gravity,

^[[(M-m)v = dm (1-K) (v-c) + [[strikethrough]] c [[/strikethrough]] w dm + (M-m-dm)(v+dv) + {R + g (M-m)} dt.

If we neglect terms of the second order, this equation reduces to

^[[c (1-k) dm = (M-m0 dr + {R+g (M-m)} dt.

(1)

A check upon the correctness of this equation may be had from