Start with the formula for work: work done = mf × s , where m = mass of the object, f = constant acceleration, and s = linear distance over which the force is applied.
Relate velocity, acceleration, and distance by Applying the equation v² = u² + 2 fs where v is the terminal velocity of the object and u is the initial velocity . Assuming the object starts from rest, u = 0.
Solve for acceleration to get a = v² / 2x.
substitute into the equation for work done = m × v² / 2s × s = kinetic energy or kinetic energy ( k.e.) = 1/2 m v².
Since the work required to accelerate an object exactly equals the kinetic energy imparted to it, this expression, 1/2 m v ², gives the energy in joules or ergs according to the system of units used above.