Question

prove that the ratio of displacement to amplitude when kinetic energy of body is rheice its potential energy

Answer 1

Well strictly speaking it is not free if it is accelerating, rather it is being acted on by some force or other.

You do not say what type of force is causing this acceleration. If you mean that it is falling under gravity but is otherwise free, then you have still not specified the problem enough to be solved.

If you further specify the problem to say that it is falling freely under a constant gravitational field { for example as we have to a good degree of approximation if the full distance of the fall is all close to the earth's surface} - then then you might still think it will could depend on the size of that field i.e. is it falling near the surface of the earth or near the surface of the moon?

However as Vinod Gohel's treatment - which assumes such a constant gravitational field, g, - shows, the result is independent of the size of g and amounts to asking for the solution of the simple expression 3x3 = 2T-1 so that T = 5s.