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


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

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


 * $$ \Delta v \approx v_{\rm eff} \sum^{j=N}_{j=1}\frac{\phi/N}{1-j\phi/N} = v_{\rm eff}\sum ^{j=N}_{j=1}\frac{\Delta x}{1-x_j} $$ 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 $$ m_f = m_0(1-\phi)$$.