Q 11-07-136JEE MainJEE Main 2021 (31 Aug, Shift 1)Medium
The masses and radii of the earth and moon are $(M_1, R_1)$ and $(M_2, R_2)$ respectively. Their centres are at a distance $r$ apart. Find the minimum escape velocity for a particle of mass $m$ to be projected from the middle of these two masses:
Answer: (A) $v_e = \sqrt{\dfrac{4G(M_1 + M_2)}{r}}$
Potential energy at the midpoint (distance $\dfrac r2$ from each):
$$U = -\frac{GM_1m}{r/2} - \frac{GM_2m}{r/2} = -\frac{2G(M_1 + M_2)m}{r}$$
To escape: $\dfrac12mv_e^2 = \dfrac{2G(M_1 + M_2)m}{r} \Rightarrow v_e = \sqrt{\dfrac{4G(M_1 + M_2)}{r}}$.
Solution by Sreeraj P, M.Sc Physics