Combustion Jet

Classical Mechanics Level 1

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 $\ce{CO2 + 4H2 -> CH4 + 2H2O}.$

\approx\SI[per-mode=symbol]{50}{\meter\per\second}

\approx\SI[per-mode=symbol]{500}{\meter\per\second}

\approx\SI[per-mode=symbol]{5000}{\meter\per\second}

\approx\SI[per-mode=symbol]{50000}{\meter\per\second}

\approx\SI[per-mode=symbol]{500000}{\meter\per\second}