Work Energy Theorem : Classical Mechanics

If you push a stationary car, the car will start moving. That means, the car has gained some velocity. We say that the car has gained some kinetic energy. Kinetic energy is the energy associated with velocity of an object. You have to do some work to push the car and work is a kind of energy. But energy cannot be destroyed. So this work is converted into another type of energy.

According to the work energy theorem, this work is converted into the kinetic energy of the car.

Let the mass of the car is m and its initial velocity v1 = 0 m/s. You pushed the car with a force of F. According the newton’s second law of motion, the car will be accelerate and the acceleration a = F/m. So, its velocity will be increased. Its final velocity is v2, after crossing a distance of x. According to the definition of work, the work done by you is :

According to Newtons laws of motion:

Now, we replace a*x in equation (1)

But k = (1/2)*m*v² is the kinetic energy of an object with velocity v and mass m. So,

This is the work energy theorem. The theorem says,

Work done by the net force on an object equals the change in the object’s kinetic energy.

If there are multiple forces working on the object, then W is the sum of the works done by each force. Let us see the following example :

Let there is a constant friction force f on the road that works opposite to the direction of the car’s velocity.

Now there are two force working on the car, F and the friction force f. We need to calculate work for each force and sum them up.

So, the total work done by both forces is the sum of the two works.

So, according to the work energy theorem,

In general the work energy theorem is written as,