Let's imagine that you are in a car with no windows. I know that's crazy, but just hold on. Although there are no windows, you can see the speedometer. So, here is the question. Is it possible figure out how far you have traveled just by looking at the speedometer? This is a classic physics problem—and we are going to do it in real life. It's going to be fun.
I'm going to start with some idealized situations so that we can figure out how to work this problem. Then we can get to try it on real data—a video of my car's speedometer. It will be a real-life physics problem.
Let's start with a simple case to make sure we know what's going on. Suppose I have a car moving at a constant speed of 10 meters/second for 5 seconds. Since the car is moving with a constant velocity (in one dimension), I can write the following as the definition of velocity.
In this expression, Δx is the change in position (the displacement) and Δt is the amount of time (the time interval). If I algebraically solve this for Δx, I get:
With a velocity of 10 m/s and a time of 5 seconds, that gives a displacement of 50 meters. See, that was simple. You probably could have done that in your head. But wait, there's another way to look at this problem. What if I create a graph of the velocity as a function of time? Yes, this would be a boring graph—but let's do it anyway. Here's what it would look like.