The masses and radii of earth and moon are $M_{1}, R_{1}$ and $M_{2}, R_{2}$ respectively. Their centres are at a distance $d$ apart. The minimum speed with which a particle of mass $m$ should be projected from a point midway between the two centres so as to escape to infinity is:

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

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

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

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

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Solution

The correct option is C $\sqrt{\frac{4 G (M_{1} + M_{2})}{d}}$

On conserving mechanical energy

Initial total energy $=$ final total energy

$\frac{- {G M}_{2} m}{d / 2} - \frac{{G M}_{1} m}{d / 2} = \frac{1}{2} {m V}^{2}$

$\Rightarrow V^{2} = \frac{4 G}{d} (M_{1} + M_{2})$

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

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The masses and radii of the earth and moon are $M_{1}$ , $R_{1}$ and $M_{2}$ , $R_{2}$ respectively. Their centres are a distance d apart. The minimum speed with which a particle of mass m should be projected from a point midway between the two centres so as to escape to infinity is given by