Two bodies, each of mass M , are kept fixed with a separation of 2L...

Two bodies, each of mass

M

, are kept fixed with a separation of

2L

. A particle of mass

m

is projected from the midpoint of the line joining their centers, perpendicular to the line. The gravitational constant is

G

. The correct statement (s) is (are) : Two bodies, each of mass M , are kept fixed with a separation of 2L . A particle of mass m is projected from the

Options

AThe minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $4\sqrt{\dfrac{GM}{L}}$ .
BThe minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $2\sqrt{\dfrac{GM}{L}}$ .
CThe minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $\sqrt{\dfrac{2GM}{L}}$ .
DThe energy of the mass $m$ remains constant.

Correct answer

B. The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $2\sqrt{\dfrac{GM}{L}}$ .

Step-by-step solution

Applying mechanical energy conservation,

${U}_{i} + K{E}_{i} = {U}_{f} + K{E}_{f}$
$- \dfrac{GMm}{L} \times 2 + \dfrac{1}{2}m{v}^{2} = 0 + 0$
$⇒ v\text{=}\sqrt{\dfrac{4GM}{L}}$

$⇒ v\text{=}2\sqrt{\dfrac{GM}{L}}$

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