The Conservation of Energy and Potential Energy

One of the most fundamental tenets of physics is the set of conservation laws. We will study four of the six conservation laws (energy, charge, linear momentum, angular momentum, and also color charge and probability). The most basic conservation law is the law of the conservation of energy, which states that in a closed system, the sum of all energy present in all forms is constant.

For example, when the Road Runner drops an anvil from the air, aiming for Wile E. Coyote, the anvil falls and, in doing so, gains speed. Therefore its kinetic energy must increase. Where does this energy come from? The anvil must have had the energy before it was dropped, since there’s no way it could have gained the energy in flight—the anvil-earth system is closed.

The answer is that the anvil had stored the energy by virtue of its configuration. It had gravitational potential energy, . For objects in a constant gravitational field (to which the earth’s reduces when you don’t go very high above it), the relevant equation is . It really only makes sense to speak of a change in , but neither the AP test nor the AP teachers are picky about the distinction, so we may as well say . In this equation, represents the acceleration due to gravity, and as we have previously discussed, is approximately close to earth’s surface.

As a demonstration, let’s work through the derivation of the equation for . We know that and . The only energy we need to consider presently is , so . Now , so . We have assumed that and are constant, so we pull them out of the integral: . Although we haven’t proven that the gravitational force is a conservative one, trust me that it is; then, we can choose any path that is convenient. In this case, a straight line will be easiest. Therefore we evaluate , so . Since our displacement is a height for the case of gravity, we write this as .

Potential energy in general can come from many things. In Physics C, we will deal principally with three types: , gravitational potential energy; , spring or elastic potential energy, and , electrical potential energy. Each of these, as we will examine in that order, is an energy derived from the arrangement of materials relative to one another. Any shift in the arrangement of the materials changes the potential energy of the system.

Going back to our example of the Road Runner dropping an anvil, as the anvil falls, potential energy is converted into kinetic energy as a result of the work done by gravity. Suppose the anvil has mass and is dropped from . To calculate the speed at any given height, we could use the constant accelerations we previously discussed. However, it is generally much easier to use energy equations. We simply need to remember that the initial gravitational potential energy is equal to the sum of and at any time: . First, we’ll need to find a value for : . Now we plug in the equations for and : . Now given a height we can easily find the speed when the anvil is at that height.