Derive the formula for kinetic energy: K E = 1 2 m v 2

Let the body of mass m be moving along a straight line with a constant velocity v . For this, the kinetic energy of the body must be equal to the work done in imparting it a velocity v . Suppose initially the body is at rest at A (That is, initial velocity u = 0 ) as shown in the figure. Let a force F be applied on the body till it attains a velocity v . Let the acceleration produced in the body be a and the distance moved by the body as it attains this velocity v be s . From the equation of motion, v 2 - u 2 = 2 a s v 2 - ( 0 ) 2 = 2 a s ( ∵ u = 0 ) v 2 = 2 a s s = v 2 2 a . . . ( 1 ) According to Newton's second law of motion, the force applied on the body is, F = m a … ( 2 ) The work done by this force is, Work = Force × Displacement W = F × s . . . ( 3 ) Substituting the values of F and s in equation ( 3 ) , we get ⇒ W = m a × v 2 2 a ⇒ W = 1 2 m v 2 . Energy spent in doing this work by a moving body to attain a velocity v from rest ( u = 0 ) is stored in the body as its kinetic energy. That is, work done = Kinetic energy gained by the body. Thus, K E = 1 2 m v 2