The masses and radii of the earth and moon are M 1 ,   R 1 and M 2 ,   R 2 respectively. Their…

1. $v_e=\sqrt{\frac{4G\left(M_1+M_2\right)}{r}}$
2. $v_e=\frac{1}{2}\sqrt{\frac{2G\left(M_1+M_2\right)}{r}}$
3. $v_e=\frac{1}{2}\sqrt{\frac{4G\left(M_1+M_2\right)}{r}}$
4. $v_e=\frac{\sqrt{2G}\left(M_1+M_2\right)}{r}$

Solution

On conserving mechanical energy. The initial total energy is equal to final total energy,  $\Rightarrow -\left(\frac{G{M}_1}{{\displaystyle \frac{r}{2}}}+\frac{G{M}_2}{\frac{r}{2}}\right)m+\frac{1}{2}m{v}_e^2=0$
$\Rightarrow v_e=\sqrt{\frac{4G\left(M_1+M_2\right)}{r}}$. 

Asked in: JEE Main 2021 (31 Aug Shift 1)

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