When a force is conservative, the work done by the force on an object equals the change in the potential energy of the object. If the work done is positive, there is a decrease in the potential energy of the object. Similarly, if the work done is negative, there is an increase in the potential energy of the object. Hence,
Energy is always conserved. Essentially the sum of the kinetic energy and the potential energy of a system is constant.
- where is a constant
This also implies that if no external energy is added or removed from the system, the initial energy of a system is equal to the final energy of a system, and hence
This is easy to prove in the case of a conservative force like gravity. Let’s say an object is dropped from rest in free fall, now the work done, by gravity on the object is, as I’ve already established,
and since the force of gravity accelerates the object from rest to a speed , the work done by the force of gravity on the object increases the Kinetic energy of the object. In other words,
If the initial Kinetic energy and potential energy of the system were and and
Then the final kinetic and potential energies, and of the system can be written as the sum of the initial energies and the change in energies. Hence,
We know that , hence
Quod erat demonstrandum.