How do I solve for $v_2$ where $mv_1^2 + MU_1^2 = mv_2^2 + M U_2^2$ and $MU_1 - Mv_1 = MU_2 - mv_2$ by eliminating $U_2$?

Question

I was trying to solve the head on collision slingshot problem where the rocket moving with speed $v_1$ approaches a planet which is moving with speed $U_1$. I wanted the final speed of the rocket ($v_2$). $U_2$ is the final speed of the planet. Mass of planet is $M$. Mass of rocket is $m$. So I made two equations-

$$ M(U_1)^2 + m(v_1)^2 = M(U_2)^2 + m(v_2)^2 $$ $$ M(U_1) - m(v_1) = M(U_2) - m(v_2) $$

However, I am unable to eliminate $U_2$ to get ($v_2 = 2U_1 + v_1$) as the answer by also taking $\frac{m}{M} = 0$

Note :- This is a head on u-turn slingshot.

Answer 1

I am getting $ U2=-U1-v1-v2 $ (Remember, these are added according to vector rules)

writing the two equations as, \begin{equation} M(U1)^{2}-M(U2)^{2}=m(v2^{2})-m(v1^{2}) \end{equation} \begin{equation} \implies M(U1-U2)(U1+U2)=m(v2-v1)(v2+v1) \end{equation} \begin{equation} M(U1-U2)=m(v1-v2) \end{equation} Divide last two equations to get the relation.