Masses and radii of earth and moon are $M_{1}, M_{2}$ and $R_{1}, R_{2}$ respectively. The distance between theircentre is 'd'. The minimum velocity given to mass 'M' from the mid point of line joining their centre so that it will escape.

\sqrt{\frac{4 G (M_{1} + M_{2})}{d}}

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\sqrt{\frac{4 G}{d}} \frac{M_{1} M_{2}}{(M_{1} + M_{2})}

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\sqrt{\frac{2 G}{d}} (\frac{M_{1} + M_{2}}{M_{1} M_{2}})

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\sqrt{\frac{2 G}{d} (M_{1} + M_{2})}

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Solution

The correct option is B $\sqrt{\frac{4 G (M_{1} + M_{2})}{d}}$
From conservation of energy, since the velocity will be 0 at infinity,

$\frac{1}{2} m v^{2} - \frac{G m M_{1}}{d / 2} - \frac{G m M_{2}}{d / 2} = 0$

$v = 2 \sqrt{\frac{G (M_{1} + M_{2})}{d}}$

Answer is option A.

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