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.

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$V = \frac{1}{2} \sqrt{\frac{2 G (M_{1} + M_{2})}{r}}$

$V = \frac{\sqrt{2 G} (M_{1} + M_{2})}{r}$

$V = \sqrt{\frac{4 G (M_{1} + M_{2})}{r}}$

$V = \frac{1}{2} \sqrt{\frac{4 G (M_{1} + M_{2})}{r}}$

answer is B.

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Detailed Solution

To get escape total energy at midpoint should be zero.

$K.E + P.E of M_{1} + m + P.E of M_{2} + m = 0$

$\frac{1}{2} m V^{2} - \frac{G M_{1} m}{r / 2} - \frac{G M_{2} m}{r / 2} = 0$

$\frac{1}{2} m V^{2} = \frac{2 G m}{r} (M_{1} + M_{2})$

$V = \sqrt{\frac{4 G (M_{1} + M_{2})}{r}}$

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