User:Tweesdad/sandbox

Limit of finite mass "pellet" expulsion

The rocket equation can also be derived as the limiting case of the speed change for a rocket that expels its fuel in the form of $N$ pellets consecutively, as $N\rightarrow \infty$, with an effective exhaust speed $v_{\rm {eff}}$ such that the mechanical energy gained per unit fuel mass is given by ${\tfrac {1}{2}}v_{\rm {eff}}^{2}$.

Let $\phi$ be the initial fuel mass fraction on board and $m_{0}$ the initial fueled-up mass of the rocket. Divide the total mass of fuel $\phi m_{0}$ into $N$ discrete pellets each of mass $\phi m_{0}/N$. From momentum conservation when ejecting the $j$'th pellet, the overall speed change can be shown to be the sum

$\Delta v=v_{\rm {eff}}\sum _{j=1}^{j=N}{\frac {\phi /N}{\sqrt {(1-j\phi /N)(1-j\phi /N+\phi /N)}}}$

Notice that for a large number of pellets, $\phi /N\ll 1$ to give

$\Delta v\approx v_{\rm {eff}}\sum _{j=1}^{j=N}{\frac {\phi /N}{1-j\phi /N}}=v_{\rm {eff}}\sum _{j=1}^{j=N}{\frac {\Delta x}{1-x_{j}}}$ where $\Delta x={\frac {\phi }{N}}$ and $x_{j}={\frac {j\phi }{N}}$.

As $N\rightarrow \infty$ this Riemann sum becomes the definite integral

$\lim _{N\to \infty }\Delta v=v_{\rm {eff}}\int _{0}^{\phi }{\frac {dx}{1-x}}=v_{\rm {eff}}\ln {\frac {1}{1-\phi }}=v_{\rm {eff}}\ln {\frac {m_{0}}{m_{f}}}$ since the remaining mass of the rocket is $m_{f}=m_{0}(1-\phi )$.