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)

Vi         =        v

Vf        =        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.