The masses and radii of the Earth and Moon are M₁, R₁ and M₂, R₂, respectively. Their centres are a distance d apart. Find the minimum speed with which a particle of mass m must be projected from the midpoint of the line joining their centres so that it escapes to infinity.
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Question

The masses and radii of the Earth and Moon are M₁, R₁ and M₂, R₂, respectively. Their centres are a distance d apart. Find the minimum speed with which a particle of mass m must be projected from the midpoint of the line joining their centres so that it escapes to infinity.

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G1 Physics · Accepted Answer

At the midpoint, the particle is at distance d/2 from both bodies. Its gravitational potential energy is U = -GM₁m/(d/2) - GM₂m/(d/2) = -2Gm(M₁+M₂)/d. For minimum escape speed, final energy at infinity is zero: (1/2)mv² + U = 0. Therefore v = 2√[G(M₁+M₂)/d]. Correct answer: A.