The kinetic energy of a body of mass m and moving with a velocity v is given by $\frac{1}{2}m{v}^2$. 

Suppose a body of mass m is moving with a uniform velocity u. Let an external force be applied on it so that it gets displaced by distance s and its velocity becomes v. In this scenario, the kinetic energy of the moving body is equal to the work that was required to change its velocity from u to v.

Thus, we have the velocity−position relation as:

v² = u² + 2as

OR

$s=\frac{v^2-u^2}{2a} .....\left(i\right)$

Where, a is the acceleration of the body during the change in its velocity

Now, the work done on the body by the external force is given by:

W = F × s

F = ma …(ii)

From equations (i) and (ii), we obtain:

$W=ma\times \left(\frac{v^2-u^2}{2a}\right)=\frac{1}{2}m\left(v^2-u^2\right)$

If the body was initially at rest (i.e., u = 0), then:

$W=\frac{1}{2}m{v}^2$

Since kinetic energy is equal to the work done on the body to change its velocity from 0 to v.

Hence, the kinetic energy of a body of mass m and moving with a velocity v is given by $\frac{1}{2}m{v}^2$.