Two bodies, each of mass M, are kept fixed with a separation 2L. A particle of mass m is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is G. The correct statement(s) is (are)

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The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is $4 \sqrt{\frac{G M}{L}}$

The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is $2 \sqrt{\frac{G M}{L}}$

The energy of the mass m remains constant

The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is $\sqrt{\frac{2 G M}{L}}$

answer is B, D.

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Detailed Solution

From conservation of mechanical energy,

$\frac{- G M m}{L} - \frac{G M m}{L} + \frac{1}{2} m v^{2} = 0 + 0$

$o r, \frac{1}{2} m v^{2} = \frac{2 G M m}{L} ∴ V = \sqrt{\frac{4 G M}{L}} = 2 \sqrt{\frac{G M}{L}}$

Total energy of mass ‘m’ is conserved as there is no external force involved

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