Question 15 of 223

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 :

A${v}_{e}=\sqrt{\dfrac{4G\left({M}_{1}+{M}_{2}\right)}{r}}$B${v}_{e}=\dfrac{1}{2}\sqrt{\dfrac{2G\left({M}_{1}+{M}_{2}\right)}{r}}$C${v}_{e}=\dfrac{1}{2}\sqrt{\dfrac{4G\left({M}_{1}+{M}_{2}\right)}{r}}$D${v}_{e}=\dfrac{\sqrt{2G}\left({M}_{1}+{M}_{2}\right)}{r}$

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Answer: (A) ${v}_{e}=\sqrt{\dfrac{4G\left({M}_{1}+{M}_{2}\right)}{r}}$

On conserving mechanical energy. The initial total energy is equal to final total energy, $\Rightarrow -\left(\dfrac{G{M}_{1}}{\dfrac{r}{2}}+\dfrac{G{M}_{2}}{\dfrac{r}{2}}\right)m+\dfrac{1}{2}m{v}_{e}^{2}=0$
$\Rightarrow {v}_{e}=\sqrt{\dfrac{4G\left({M}_{1}+{M}_{2}\right)}{r}}$.