Question

derive an expression for the kinetic energy of a moving object

Answer 1

as we know that,
work done = f×s
( where f=force & s=displacement)
w=f×s
w=ma×s
(f=m×a) w=ma×v²-u²/2a
( s=v²-u²/2a)
w=½mv²
( if u=0)
work done is equal to change in kinetic energy of an object.
K.E=½mv²

Answer 2

Consider of 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 force of friction acting on it. The body posses kinetic energy and is capable to do work against opposing force F until all its kinetic energy is used up

Derivation:

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

                       K.E.= FS

                         Vi = V

                         Vf = 0

As                       F = ma

∴                         a = -F/m (retardation)

           Since motion is opposed, hence, a is negative.

Using 3rd equation of motion:

                    2 a S = Vf² - Vi²

              2 (-F/m) S= (0)² - (Vi)²

                         F S= 1/2 m v²

As K.E, is equal to FS we get

                         K.E.=1/2 m v²