Escape Velocity of a Particle between Two Fixed Masses – Rankers Physics
Topic: Gravitation
Subtopic: Gravitational Field

Escape Velocity of a Particle between Two Fixed Masses

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
 
a, b
b, d
a, c
a, d

Solution:

At the midpoint, potential energy is \(U = -\frac{2GMm}{L}\). For escaping to infinity, total mechanical energy must be at least 0:

\(\frac{1}{2}mv^2 - \frac{2GMm}{L} = 0 ⇒ v = 2\sqrt{\frac{GM}{L}}\). Mechanical energy remains conserved as only gravity acts.

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