**Define K.E. and derive its relation.**

**OR**

**Prove that K.E. = K.E. = $\frac{1}{2} m v^{2}$**

Difficulty: Medium

**Kinetic Energy:**

** **The energy possessed by a body due to its motion is called its kinetic energy.

**Derivation of K.E:**

Consider a body of mass *m* moving with velocity *v*. The body stops after moving through some distance *S* due to some opposing force such as the force of friction acting on it. The body possesses kinetic energy and is capable to do work against opposing force *F* until all of its kinetic energy is used up.

K.E. of the body = Work done by it due to motion

K.E = FS.......... (i)

V_{i }= v

V_{f } = 0

As F = ma

A = $-\frac{F}{m}$

Since motion is opposed, hence, *a* is negative. Using 3rd equation of motion:

$2aS=V_f^2- V_i^2$

$2 (- F/m) S = (0)2 – (v)2$

F S =$\frac{1}{2} m v^{2}$........(ii)

From Eq. (i) and (ii), we get

K.E. = $\frac{1}{2} mv^{2}$.......(iii)

Equation (iii) gives the *K.E.* possessed by a body of mass m moving with velocity v.