Relationships Between Variables

When you're working a physics lab for a class, you'll often find yourself making graphs of a couple variables. You change one of the variables yourself, and track the corresponding change in the other. For instance, you might be tasked with placing a sealed container filled with gas in a pot of water, and measuring the change in pressure of the gas as the water is heated. Here, the temperature of the water is the variable you are changing, and the gas pressure is the second variable you are tracking. When you make a graph of pressure vs. temperature you are exploring the relationship between these two variables.

In an introductory physics course, there are four different common relationships between variables you are bound to run into: they are linear, direct, quadratic, and inverse relationships. Here, we'll go over both quadratic and inverse relationships, and a couple examples of places they pop up in a physics course.

Quadratic Relationships

A quadratic relationship is a mathematical relation between two variables that follows the form of a quadratic equation. To put it simply, the equation that holds our two variables looks like the following:

Here, y and x are our variables, and a, b, and c are constants. If you didn't have this equation, and only had some data points for a graph, you'd be able to tell it's a quadratic relation if the graph's curve forms a parabola, which on a graph looks like a dip or a valley. Even if we have an equation like this where b and c both equal zero, it's still considered quadratic. If that happens, we get the simplest form of a quadratic relationship:

This works because it turns out that it's the x squared component that's absolutely necessary for a relationship to be quadratic. So, unlike b and c, a must never equal zero because that will remove the x squared from the formula since zero multiplied by anything is zero.

One of the first places you'll encounter a quadratic relation in physics is with projectile motion. This makes sense if you think about how a projectile travels through the air over time. Imagine you're tossing a baseball straight up in the air. Let's try visualizing this with a height vs. time graph. Over time the ball goes up to a maximum height, and then back down to the starting height again when you catch it.

A parabola is a good indication of a quadratic relationship

We can see our graph creates an upside-down parabola, which is the sort of thing you might expect from a quadratic relation. To make absolutely sure the relation between height and time is quadratic, we'll also look at the vertical equation for projectile motion that deals with position and time:

Does it look familiar? Let's try rearranging the equation a bit:

You might not know this yet, but in this equation the only two variables are height (y) and time (t). Everything else is constant over the course of a single throw. So, the equation we're seeing here is really in the exact form of a quadratic equation:

Inverse Relationships

You might remember that in a direct relationship, as one variable increases, the other increases, or as one decreases, the other decreases. In an inverse relationship, instead of the two variables moving in the same direction they move in opposite directions, meaning as one variable increases, the other decreases.

Often in a physics course, the type of inverse relationship you'll run across is an inversely proportional relationship. For inversely proportional relationships, we specify that as one variable increases the other decreases at the same rate.