Define K.E. and derive its relation.


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.