How to use your speedometer to determine where your car is


Imagine that you are in a car with no windows. I know this is crazy, but wait. Although there is no window, you can see the speedometer. So here is the question. Is it possible to determine the distance you’ve traveled just by looking at the speedometer? It’s a classic physics problem – and we’re going to do it in real life. It’s gonna be fun.

I’ll start with a few idealized situations so that we can figure out how to fix this. Then we can try on real data – a video of my car’s speedometer. It will be a real physics problem.

Let’s start with a simple case to make sure we know what’s going on. Suppose a car is moving at a constant speed of 10 meters / second for 5 seconds. Since the car is moving with constant speed (in one dimension), I can write the following as a definition of speed.

Illustration: Rhett Allain

In this expression, Δx is the change in position (displacement) and Δt is duration (time interval). If I algebraically solve this for Δx, I get:

Illustration: Rhett Allain

With a speed of 10 m / s and a time of 5 seconds, this gives a displacement of 50 meters. See, it was simple. You could probably have done this in your head. But wait, there is another way to look at this problem. What if I graph the speed against time? Yes, that would be a boring graph, but let’s do it anyway. Here is what it would look like.

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