The escape velocity for a planet is V. A tunnel is dug along its diame

1.	The escape velocity for a planet is V. A tunnel is dug along its diameter and a particle is dropped into the tunnel. At the centre of the planet, the speed of the particle will be (a) V (b) V/2 (c) V/√2 (d) V/2√2
Answer» Correct Answer is: (c) V/√2 Let M, R be the mass and radius of the planet, and g be the acceleration due to gravity on its surface. Then, V = √(2Rg) and GM = R²g. Gravitational potential at the surface is - GM/R and at the centre is - 3GM/2R. In going from the surface to the centre, loss in gravitational PE = m [ - GM/R - ( - 3/2 GM/R)] = 1/2 GMm/R = 1/2 mv² or v² = GM/R = Rg =V²/2 or V/√2.

The escape velocity for a planet is V. A tunnel is dug along its diameter and a particle is dropped into the tunnel. At the centre of the planet, the speed of the particle will be (a) V (b) V/2 (c) V/√2 (d) V/2√2

Answer»

Correct Answer is: (c) V/√2

Let M, R be the mass and radius of the planet, and g be the acceleration due to gravity on its surface.

Then, V = √(2Rg) and GM = R²g.

Gravitational potential at the surface is - GM/R and at the centre is - 3GM/2R. In going from the surface to the centre, loss in gravitational PE

= m [ - GM/R - ( - 3/2 GM/R)] = 1/2 GMm/R = 1/2 mv²

or v² = GM/R = Rg =V²/2 or V/√2.