A two stage rocket is fired as follows in deep space ( no gravity ).
Stage 1:
Mass of the frame+ engine = $M_1$
Mass of the fuel + oxidizer =$m_1$
gas velocity relative to the rocket = $v_r$
Stage 2:
After burning $M_1$ the second stage is ignited with
Mass of the frame+ engine = $M_2$
Mass of the fuel + oxidizer =$m_2$
gas velocity relative to the rocket = $v_r$
Show that the velocity achieved is $$v_f= - v_r \ln \frac{(M_1+M_2+m_1+m_2)(M_2+m_2)}{(M_1+m_2+m_2)M_2} $$
Source::HSM 70
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4727:Diamond Point,
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