A ball is kept at a height h above the surface of a heavy transparent sphere made of a material of refractive index μ. The radius of the sphere is R. At t = 0, the ball is dropped to fall normally on the sphere. Find the speed of the image formed as a function of time for t<. Consider only the image by a single refraction.

refractive index of the transparent sphere =μ

Radius of the sphere = R

Height at which ball is kept= h

At t= 0, the ball drops normally on the sphere

Let the time taken by the ball from A to B = t

Distance travelled by the ball AB is given by,

Now, the distance BC is given by

Assuming that this is the distance of object from the lens at any time t. Hence, here u is given by

We know that,

Refractive index of air, =1

Refractive index of sphere,

Thus, using the lens maker’s formula

the velocity of the image is given by