Question

derive a mathematical expression for kinetic energy

Answer 1

K. E = 1/2 mv2
If it is wrong so sorry for that

Answer 2

Derivation for the equation of Kinetic Energy:
The relation connecting the initial velocity (u) and final velocity (v) of an object moving with a uniform acceleration a, and the displacement, S is 
                             v2 - u2 = 2aS
This gives  
                             S = v 2 - u 2 2a 

We know F = ma. Thus using above equations, we can write the workdone by the force, F as

                            W = ma × v 2 - u 2 2a 
                                  or
                             W = 1 2 m( v 2 - u 2 )


If object is starting from its stationary position, that is, u = 0, then
                            W = 1 2 m v 2 
 
It is clear that the work done is equal to the change in the kinetic energy of an object.

If u = 0, the work done will be  W = 1 2 m v 2 .

Thus, the kinetic energy possessed by an object of mass, m and moving with a uniform velocity, v is Ek=  ½  mv2