Question

derive the equation of kinetic energy of a body of mass moving with velocity

Answer 1

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 Ek=  ½mv2

Answer 2

let a body with mass,m is moving with a velocity,u along a distance,s. a force,F acting on it changes its velocity,v.
we know that W=Fs--------------(1)

F=ma--------------(2)

2as=v^2-u^2
s=v^2-u^2/2a---------------(3)

from equations 1,2,3

W=ma×v^2-u^2/2a
W=1/2mv^2-u^2

if initial velocity is 0
then

W=1/2mv^2

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