# Combustion Jet

The equation of motion for rockets is $$m_\textrm{r}\,\dot{v}_\textrm{r}= u_\textrm{e}\, \dot{m}_\textrm{r},$$ where $$u_\textrm{e}$$ is the speed of the exhaust shooting out the back. The final velocity of the rocket is a few multiples of the exhaust speed, so it's quite important, but in physics textbooks, $$u_\textrm{e}$$ is usually treated like a freely adjustable parameter whose origin and precise value is a detail left for engineers to worry about. With a little bit of approximation, we can do better than that.

Estimate $$u_\textrm{e}$$ for the combustion of methane $$(\ce{CH4}),$$ the propellant used by SpaceX for its Raptor rocket engine program.

Details and Assumptions:

• The specific enthalpy of combustion for methane is $$\Delta_c H \approx \SI[per-mode=symbol]{890}{\kilo\joule\per\mole}.$$
• Assume that all the energy released in combustion goes into the kinetic energy of the gas.
• Beyond its performance, methane is nice because it could potentially be produced on other planets via the Sabatier reaction $$\ce{CO2 + 4H2 -> CH4 + 2H2O}.$$
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