Potential energy

The introduction of work has simplified the description of motion considerably: we can use the work–kinetic energy theorem even if the acceleration is not constant, without the need to use complicated calculus. In certain systems, we can establish an even more convenient framework if we incorporate the potential work external forces may do on the system into the system itself as a form of energy. We call this type of energy potential energy, and we can conceive of it as energy due to position (whereas kinetic energy is energy due to motion).

Consider a small car with a spring attached to the front of it. Initially, it has some kinetic energy, then it stops for a moment when hitting a wall, losing all its kinetic energy, then it rebounds, regaining almost all of its kinetic energy. What happened to the kinetic energy in between?

There are two ways of looking at this process. The one we have already seen is to use the work–kinetic energy theorem: the  original kinetic energy is lost in the first step, because being compressed, the spring does negative work on the car (displacement is to the left, the force is to the right), whilst in the second step, the positive work done by the spring (both the displacement and the force are to the right) restores the kinetic energy. A new approach can be to say that the spring is a part of the system, it is able to store energy in the form of potential energy, which is energy due to its compressed or expanded state, and if we incorporate this energy value in the total energy of the car–spring system, this total energy stays constant in the process, it just gets converted to and fro between the kinetic and the potential form.