Question

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 centers, perpendicular to the line. The gravitational constant is $G$. The correct statement(s) is (are)

• A. the minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $4\sqrt{\frac{GM}{L}}$
• B. the minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $2\sqrt{\frac{GM}{L}}$
• C. the minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $\sqrt{\frac{2GM}{L}}$
• D. the energy of the mass $m$ remains constant

Answer 1

Answer: (B), (D)

the energy of the mass $m$ remains constant

Gravitational potential energy of $m$ due to two bodies of mass $M$,
$U_m=\frac{-2GMm}{L}$
As bodies of mass $M$ are fixed and final velocity of mass $m$ is zero, therefore by conservation of mechanical energy,
$\frac{2GMm}{L}=\frac{m{v}^2}{2}$
$v=2\sqrt{\frac{GM}{L}}$
Total energy of the mass $m$ is also conserved as there are no non-conservative forces involved.