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Concept# Tsiolkovsky rocket equation

Summary

The classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity can thereby move due to the conservation of momentum.
It is credited to the Russian scientist Konstantin Tsiolkovsky (Константи́н Эдуа́рдович Циолко́вский) who independently derived it and published it in 1903, although it had been independently derived and published by the British mathematician William Moore in 1810, and later published in a separate book in 1813. American Robert Goddard also developed it independently in 1912, and German Hermann Oberth derived it independently about 1920.
The maximum change of velocity of the vehicle, (with no external forces acting) is:
where:
is the effective exhaust velocity;
is the specific impulse in dimension of time;
is standard gravity;
is the natural logarithm function;
is the initial total mass, including propellant, a.k.a. wet mass;
is the final total mass without propellant, a.k.a. dry mass.
Given the effective exhaust velocity determined by the rocket motor's design, the desired delta-v (e.g. orbital speed or escape velocity), and a given dry mass , the equation can be solved for the required propellant mass :
The necessary wet mass grows exponentially with the desired delta-v.
The equation is named after Russian scientist Konstantin Tsiolkovsky who independently derived it and published it in his 1903 work.
The equation had been derived earlier by the British mathematician William Moore in 1810, and later published in a separate book in 1813.
American Robert Goddard independently developed the equation in 1912 when he began his research to improve rocket engines for possible space flight. German engineer Hermann Oberth independently derived the equation about 1920 as he studied the feasibility of space travel.

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