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:

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Correct Answer

(B) \({V}=\sqrt{\frac{4 {G}\left({M}_{1}+{M}_{2}\right)}{{r}}}\)

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Detailed explanation

\(\frac{1}{2} {mV}^{2}-\frac{{GM}_{1} {m}}{{r} / 2}-\frac{{GM}_{2} {m}}{{r} / 2}=0\)

\(\frac{1}{2} {mV}^{2}=\frac{2 {Gm}}{{r}}\left({M}_{1}+{M}_{2}\right)\)

\({V}=\sqrt{\frac{4 {G}\left({M}_{1}+{M}_{2}\right)}{{r}}}\)

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