Rocket Motor Efficiency

If we look closer at the rocket equation

\displaystyle \Delta v = v_\text{e} \hspace{0.1em} \ln \left( \frac{m_\text{i}}{m_\text{f}} \right),

we see that \Delta v is proportional to v_\text{e}. This means that if the motor is throwing out gas molecules faster, the force per unit mass of propellant is higher. Thus, v_\text{e} is a measure of how efficient the rocket motor is. The exhaust velocity parameter is very important when designing a rocket motor, and we can actually calculate it since we know that it is the same as dividing the total impulse by the propellant mass m_\text{p}. Since it is specific to propellant mass we often call it specific impulse, and it is thus given as

\displaystyle I_\text{SP} = v_\text{e} = \frac{I_\text{TOT}}{m_\text{p}} = \frac{\int_0^{t} F \, \text{d}t'}{m_\text{p}} = \frac{\bar{F} \, t}{m_\text{p}}.

Since I_\text{SP} is the same as the exhaust velocity, the unit is, in fact, \text{m}/\text{s}. We have here described I_\text{SP} as being the same as the exhaust velocity. In some cases the specific impulse is defined differently, as

\displaystyle I_{\text{SP}}^* = \frac{I_\text{TOT}}{m_\text{p} g_0} \neq v_\text{e},

where g_{0}=9.81 \; \text{m}/\text{s}^2. This definition is often used in the US, and the unit is seconds.

Common values of the specific impulse on expensive rocket motors is 2500–4500 m/s. The most efficient motor to date (2016) is the RL10 motor, an upper stage on the American Delta IV rocket. It has a specific impulse in vacuum of 4 567 m/s.

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This article is part of a pre-course program used by Andøya Space Education in Fly a Rocket! and similar programs.