Consider a mass which is falling vertically under the influence of gravity. We already know how to analyze the motion of such a mass. Let us employ this knowledge to search for an expression for the conserved energy during this process. (N.B., This is clearly an example of a closed system, involving only the mass and the gravitational field.).
. Suppose that the mass falls from height to , its initial velocity is , and its final velocity is . It follows that the net vertical displacement of the mass is . Moreover, and . Hence, the previous expression can be rearranged to give
The above equation clearly represents a conservation law, of some description, since the left-hand side only contains quantities evaluated at the initial height, whereas the right-hand side only contains quantities evaluated at the final height. Now, let us define the kinetic energy of the mass,
and the gravitational potential energy of the mass,
Note that kinetic energy represents energy the mass possesses by virtue of its motion. Likewise, potential energy represents energy the mass possesses by virtue of its position.
Here, is the total energy of the mass: i.e., the sum of its kinetic and potential energies. It is clear that is a conserved quantity: i.e., although the kinetic and potential energies of the mass vary as it falls, its total energy remains the same.