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Earth and Moon have masses and radii (M1, R1) and (M2, R2), with their centres separated by distance r. A particle of mass m is projected from the midpoint between them. Find the minimum escape speed for the particle.

1. V = 1/2 * sqrt(4G(M1 + M2)/r)
2. V = sqrt(4G(M1 + M2)/r)
3. V = 1/2 * sqrt(2G(M1 + M2)/r)
4. V = sqrt(2G(M1 + M2)/r)

Correct answer: V = sqrt(4G(M1 + M2)/r)

Solution

At the midpoint the distance to each body is r/2, so the potential energy is U = -2Gm(M1+M2)/r. To escape (reach zero total energy from this point) (1/2)mV² = 2Gm(M1+M2)/r ⇒ V² = 4G(M1+M2)/r ⇒ V = sqrt(4G(M1+M2)/r).

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