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We show the traditional rocket problem, where the ejecta velocity is assumedconstant, can be reduced to an integral quadrature of which the completelynon-relativistic equation of Tsiolkovsky, as well as the fully relativisticequation derived by Ackeret, are limiting cases. By expanding this quadraturein series, it is shown explicitly how relativistic corrections to the massratio equation as the rocket transitions from the Newtonian to the relativisticregime can be represented as products of exponential functions of the rocketvelocity, ejecta velocity, and the speed of light. We find that even low ordercorrection products approximate the traditional relativistic equation to a highaccuracy in flight regimes up to $0.5c$ while retaining a clear distinctionbetween the non-relativistic base-case and relativistic corrections. Wefurthermore use the results developed to consider the case where the rocket isnot moving relativistically but the ejecta stream is, and where the ejectastream is massless.

A General Quadrature Solution for Relativistic, Non-relativistic, and Weakly-Relativistic Rocket Equations