As mentioned previously, kinetic energy is the energy of objects in motion. Remember that we don’t really know what energy is, we just know that objects in motion have it. There are some peculiarities about this framework that are worth considering.

Suppose you are riding a moving train and holding a ball in your hand. From your point of view, the ball has no kinetic energy because it appears to be stationary relative to you. However, if a person is standing on the ground as the train goes by, they will measure the ball moving at the speed that the train is moving, and they will determine that the ball has some kinetic energy. Who is correct?

They are both correct, at least from their own points of view. This only adds to the mystery of energy, because it suggests that kinetic energy isn’t something that the object actually has. It’s a number that we ascribe to the object, and it depends on how we perceive its motion.

The formula for kinetic energy is \(KE = \frac{1}{2} m v^2\text{,}\) where \(m\) is the mass of the object and \(v\) is its velocity. Using SI units, kinetic energy is measured in \(\text{kg} \cdot \frac{\text{m}^2}{\text{s}^2} = \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2}\text{.}\) This is a derived unit known as Joules (J).

Gravitational potential energy is the energy stored in an object by moving it to a different height while under the influence of gravity. It makes sense that objects that are higher up have more potential energy that objects that are lower, and this is reflected in the formula for potential energy: \(PE_\text{g} = mgh\text{,}\) where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity, and \(h\) is the height. Notice that the units here are \(\text{kg} \cdot \frac{\text{m}}{\text{s}^2} \cdot \text{m} = \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2}\text{,}\) which are the exact same units as kinetic energy. This makes sense since energy is energy, but it’s not immediately clear from the formula that it works out that way.

Similarly to kinetic energy, this value will depend on how you measure it. If you are on the top floor of a building, the height at which you are holding an object will seem to you to be relative to the floor. But to someone on the ground floor will say that the object is much higher up. Even more confusingly, a person on the top floor of a taller building will say that the height of the object is negative, which means that the object can be seen as having a negative energy.

The reason that the physics still works out correctly is that we basically never talk about the absolute amount of gravitational potential energy an object has. In any practical setting, we only care about the change in that potential energy. So if an object falls a distance of 3 meters, it doesn’t matter whose perspective you’re taking because both people will measure the same change in height.

An even more confusing aspect of this idea is that it allows for negative potential energies. A person in a taller building might measure the height of the ball to be negative. If you plug in a negative value of \(h\) into the formula, the energy is negative. And this fact still doesn’t change how the physics works out because as the ball falls, its energy simply gets more negative.

Elastic potential energy is the potential energy stored in a spring that has either been compressed or stretched from its natural length (which is the length of the spring when it’s just left alone). The formula for elastic potential energy for basic springs is \(U = \frac{kx^2}{2}\text{,}\) where \(k\) is known as the spring constant and \(x\) is the change in length from the natural length. The spring constant is a value that basically measures the stiffness of the spring. When the spring constant is large, it requires a lot of effort to stretch or compress it.

Springs can be an entire chapter on their own due to the fact that we can use springs as a model for a range of behaviors in physics. We are going to look at just a small piece of the picture. Imagine that you have a spring with one end attached to the ceiling and the other end hanging downward. Now imagine putting a small weight on the end of the spring, pulling it downward a bit, and then letting it go. What do you think is going to happen?

You should be anticipating that the weight will bounce up and down. But if we ignore friction and look more closely, what we will see is energy conservation in action. As the spring reaches its shortest and longest lengths, the mass is coming to a rest, leading to there being no kinetic energy. At those moments, all of the energy is stored up in the spring as potential energy. As the spring starts to move back towards its equilibrium length, the mass picks up speed and the potential energy is converted to kinetic energy. And this back-and-forth behavior is known as simple harmonic osciallation, and it will continue forever because we are ignoring friction.