The masses and radii of the earth and moon are $\left(M_1, R_1\right)$ and $\left(M_2, R_2\right)$ 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 :

• Correct answer
  A

  $v_e=\sqrt{\frac{4G\left(M_1+M_2\right)}{r}}$
• B

  $v_e=\frac{1}{2}\sqrt{\frac{2G\left(M_1+M_2\right)}{r}}$
• C

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

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

Solution

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

Hence, the correct answer is option: 'A'.