Tsiolkovsky rocket equation

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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 Konstantin Tsiolkovsky, who independently derived it and published it in 1903,^[1][2] although it had been independently derived and published by William Moore in 1810,^[3] and later published in a separate book in 1813.^[4] Robert Goddard also developed it independently in 1912, and Hermann Oberth derived it independently about 1920.

The maximum change of velocity of the vehicle, ${\displaystyle \Delta v}$ (with no external forces acting) is:

$${\displaystyle \Delta v=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}=I_{\text{sp}}g_{0}\ln {\frac {m_{0}}{m_{f}}},}$$

• ${\displaystyle v_{\text{e}}=I_{\text{sp}}g_{0}}$ is the effective exhaust velocity;
  • ${\displaystyle I_{\text{sp}}}$ is the specific impulse in dimension of time;
  • ${\displaystyle g_{0}}$ is standard gravity;
• ${\displaystyle \ln }$ is the natural logarithm function;
• ${\displaystyle m_{0}}$ is the initial total mass, including propellant, a.k.a. wet mass;
• ${\displaystyle m_{f}}$ 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 ${\displaystyle m_{f}}$, the equation can be solved for the required propellant mass ${\displaystyle m_{0}-m_{f}}$:

$${\displaystyle m_{0}=m_{f}e^{\Delta v/v_{\text{e}}}.}$$

The necessary wet mass grows exponentially with the desired delta-v.