With an example derive the equation for kinetic energy of a moving body.

Let us consider a trolley which is initially at rest on a horizontal, smooth surface having negligibly small friction such as that of ice. If the trolley is pushed with a constant horizontal force, it begins to move. But if the force is withdrawn, the trolley still continues to move with a uniform velocity v. The trolley is set into motion by the initial work done on it to move from rest. When we push the trolley with a constant force, it gets accelerated. Suppose it moves a distance s in the time interval t. Thus
W = Fs ----(1)
By, Newton's second law of motion
F = ma ----(2)
m being the mass of the trolley. The distance s moved by the trolley through the acceleration, a, during the time t is given by
s = ut + at² ----(3)
Further, the velocity v it attains after time t is given by
v = u + at ----(4)
Since the trolley has started from rest, u = 0 in equations (3) and (4). The velocity v is thus the velocity, which the trolley attains when the force F is withdrawn.
Substituting equation (2) for F and equation (3) for s in equation (1), we get
W = ma × at²
= m (at)²
= mv² [from equation (4)]
The work done is stored in the form of kinetic energy of the body.