Question

Two bodies, each of mass $M$, are kept fixed with a separation $2L$. A particle of mass m is projected from the mid-point of the line joining their centres, 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)

Let v is the minimum velocity. From energy conservation,
$U_c + K_c = U_{\infty} + K_{\infty} $
$\therefore \, \, \, m V_c + \frac{1}{2} m v^2 =0+0 $
$\therefore \, \, \, \, \, \, \, \, \, v= \sqrt{-2V_c } = \sqrt{(-2) \bigg ( \frac{-2GM}{L} \bigg) } = 2 \sqrt{\frac{ GM}{L }}$