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\begin{align}

MV_0 &= MV_0' + mv'\\

M(V_0 - V_0') &= mv'\qquad (*)\\

MV_0^2 &= MV_0^{'2} + mv^{'2}\\

M(V_0 - V_0')(V_0 + V_0') &= mv^{'2}\qquad (**)

\end{align}

So let's take \(\frac{(**)}{(*)}\Rightarrow V_0 + V_0' = v'\)

How do I write \(v'\) and \(V_0'\) in terms of their masses and \(V_0\)?