Tsiolkovsky Rocket Equation - Stages

Stages

In the case of sequentially thrusting rocket stages, the equation applies for each stage, where for each stage the final mass in the equation is the total mass of the rocket after discarding the previous stage, and the initial mass in the equation is the total mass of the rocket just before discarding the stage concerned. For each stage the specific impulse may be different.

For example, if 80% of the mass of a rocket is the fuel of the first stage, and 10% is the dry mass of the first stage, and 10% is the remaining rocket, then

\begin{align} \Delta v \ & = v_\text{e} \ln { 100 \over 100 - 80 }\\ & = v_\text{e} \ln 5 \\ & = 1.61 v_\text{e}. \\ \end{align}

With three similar, subsequently smaller stages with the same for each stage, we have

and the payload is 10%*10%*10% = 0.1% of the initial mass.

A comparable SSTO rocket, also with a 0.1% payload, could have a mass of 11% for fuel tanks and engines, and 88.9% for fuel. This would give

If the motor of a new stage is ignited before the previous stage has been discarded and the simultaneously working motors have a different specific impulse (as is often the case with solid rocket boosters and a liquid-fuel stage), the situation is more complicated.

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