A rocket of initial total mass mi leaves a space station in deep space, far from all sources of gravitation. The rocket ejects burn fuel at a constant speed u << c relative to the rocket.
(a) Derive an expression for the rocket relative to the space station when the total mass of the rocket plus remaining fuel is mf.
(b) Some time after the rocket has left the space station, a small quantity of the ejected fuel remains stationary in the frame of the space station. What limit does this observation place upon the fraction of the original total mass that was burnt as fuel?
(c) The rocket ejects fuel at a rate proportional to its remaining total mass m, so that (dm/dt) = -αm. Derive an expression for the distance x between the space station and the stationary burnt fuel. Obtain a value for x given that u = 1000 ms-1, and that m is reduced to half its starting value after 10 minutes.
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