= Two-stage rocket with proportional fuel and body masses
{title2=$v_2=u\log\frac{M+m_0}{M+(1-k)m_0}+u\log\frac{M+m_0}{M}$}
For zero initial <velocity>, body <masses> $kM,(1-k)M$ and fuel <masses> $km_0,(1-k)m_0$, the <rocket equation> gives the displayed final second-stage <speed> after burning the first-stage fuel, discarding its body without impulse, and burning the second-stage fuel. It assumes $M>0$ and $0\le k<1$. The speed exceeds the single-stage value when $0<k<1$ and fuel is positive. Its limit as $k\to1^-$ concerns a surviving second stage of vanishing <mass>; at $k=1$ no such stage exists.
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