A large solid sphere with uniformly distributed positive charge has a smooth narrow tunnel through its centre. A small particle with negative charge, initially at rest far from the sphere, approaches it along the line of the tunnel, reaches its surface with a speed v, and passes through the tunnel. Its speed at the centre of the sphere will be?

Potential at infinity = V∞ = 0
Potential at the surface of the sphere, Vs = kQ/R
Potential at the centre of the sphere, Vc = (3/2)kQ/R
Let m and -q be the mass and the charge of the particle respectively.
Let v0 = speed of the particle at the centre of the sphere.
(1/2)mv² = -q[V∞ - Vs] = qkQ/R (i)
(1/2)mv0² = -q[V∞ - Vc] = q(3/2)kQ/R (ii)
Dividing eqn. (ii) by eqn. (i),
v0²/v² = 3/2 = 1.5
or v0 = √1.5v.