Classical Mechanics

Consider a man applying a force to push an object from position 1 to position 2, causing a small displacement.

The total work done by this force is the sum of all the infinitesimal amounts of work performed throughout the displacement.

Applying Newton’s second law to the equation and integrating it relates this work done to the kinetic energy change. This relation is called the work-energy theorem.

This total work can also be expressed in terms of a change in potential energy.

When an object is thrown up into the air, its potential energy increases while its kinetic energy decreases. On the other hand, as the object falls back to Earth, it gains kinetic energy while simultaneously losing an equivalent amount of potential energy.

By setting the change in kinetic energy equal to the change in potential energy, it follows that the sum of kinetic and potential energies — collectively known as mechanical energy — remains constant, provided only conservative forces act on the object.