50 Introduction to Energy
You have doubtless heard the statement that “energy is conserved.” This statement is, it turns out, a consequence of Newton’s laws. Before we do so, however, it will be useful to have a rough sense of what it is we are working towards first.
So- energy. The are different kinds of energy. The first is called kinetic energy, and it is the energy associated with motion. It is defined via 
, so big objects moving fast have lots of energy. The second kind of energy is called potential energy, which is denoted by the letter 
. The simplest example is gravitational potential energy, 
where 
is the height of the object. As an object falls, goes down while 
goes up, meaning the potential energy goes down while the kinetic energy goes up.
We can think of this “falling down” as a transfer of energy. We have two cups at our disposal, one for potential energy and one for kinetic energy. An object that starts high up starts with all its energy in the “potential energy” cup. As it falls down, we the energy in the potential energy cup gets poured into the kinetic energy cup. Consequently, the total energy 
is conserved:
.
Like momentum, this is a conservation law that we can use to solve before-and-after problems quickly. Let’s try it now.
Exercise 50.1: Using Conservation of Energy
Suppose you throw a rock straight up with an initial speed 
. Use conservation of energy to figure out how high the rock will go.
We’ll do this together.
Step 0- what’s the story? Make sure you identify the beginning and end of the story.
Step 1- Draw a picture that shows the beginning of the story, and the end of the story. Make sure your picture contains all the information you know about both the beginning and end.
Step 2- Find relations. Calculate the total energy at the beginning of the problem, and at the end of the problem, and then set them equal to each other. Finally (step 3), solve for the height of the ball at the end of the story.
As it happens, we can also add or subtract energy from a system by pouring or removing energy into the kinetic energy cup. The amount of energy that gets added or subtracted is called work, which we denote with the letter 
. Thus, the full conservation of energy equation reads:
.
To change the kinetic energy of an object, I need to change its velocity, which means I have to push/pull on the object. Clearly, “forces” and “work” must have something to do with each other. Let’s take a closer look.
Exercise 50.2: Work
The point is this: forces parallel to the velocity can add or remove energy, while forces perpendicular to the velocity of an object do not. Thus, to understand energy, we will need to learn about a new mathematical operation that “picks out” the component of a force that is parallel to the velocity. This new mathematical operation is called the dot product.
To end, I would like to leave you with this video from Mark Rober, which beautifully illustrates how powerful this idea of energy can be for understanding the world around us. You don’t have to watch it if you don’t want to, but it’s quite fun!