define kinetic energy. derive an expression …

The kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. 

Consider a body of mass "m" starts moving from rest. After a time interval "t" its velocity becomes V.
If initial velocity of the body is Vi = 0 ,final velocity Vf = V and the displacement of body is "d". Then

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 = ½a(v² - u²) 

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

                            W = ma × ½a(v² - u²) 
                                  or
                             W =m( v 2 - u 2 ) ½


If object is starting from its stationary position, that is, u = 0, then
                            W = ½mv²
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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 = ½mv²

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