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A tunnel is bored straight through the centre of the Earth (radius R, mass M) along a diameter. A particle of mass m is placed exactly at the centre of the tunnel. What minimum speed must be given to the particle so that it just reaches the surface of the Earth?

1. sqrt(GM/R)
2. sqrt(GM/(2R))
3. sqrt(2GM/R)
4. it will reach the surface even with negligible initial speed

Correct answer: sqrt(GM/R)

Solution

Inside a uniform solid sphere, the gravitational potential at the centre is V_centre = -3GM/(2R) and at the surface V_surface = -GM/R. To just reach the surface, give the particle kinetic energy equal to the rise in potential energy: (1/2) m v² = m(V_surface - V_centre) = m[-GM/R - (-3GM/(2R))] = m[GM/(2R)]. So (1/2)v² = GM/(2R) -> v² = GM/R -> v = sqrt(GM/R).

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