The escape velocity for a planet from the surface is v e . A particle starts from rest at a large distance from the planet, reaches the planet only under gravitational attraction, and passes through a smooth tunnel through its centre. Its speed at the centre of the planet will be

Step-by-step solution

Taking the potential at a large distance from the planet as zero, the potential at the centre of the planet = - 3 GM 2 R . 1 2 mv 2 = m 0 - - 3 GM 2 R or v 2 = 3 GM 2 R = 3 Rg ⇒   v 2 = 3 2 2 RG = 3 2 v e 2       ∵   v e = 2 R G ⇒   v = 1 . 5 v e