2.6: Velocity and Position by Integration
We haven’t gone over integrals in the “Calculus!” section yet. We will eventually, and I don’t want to muddle anything by attempting a 2-3 sentence explanation. So for now I’m going to treat it like you already understand the integral.
Recall that the derivative of position is velocity and the derivative of velocity is acceleration. I like to think of it this way: Taking the derivative allows you to go “up” in how complicated your concept is. Position is just a location. Velocity is a change in location per second. Acceleration is a change per second in the change in location per second.
By taking the integral, you can go “down” in how complicated the concept is. When you take the integral of acceleration, you’re essentially multiplying time back in. So, you go from distance per time per time, to just distance per time (aka velocity). By multiplying in time again, you can get to position.
This is the way you should think about integrals, in my opinion. You’ll often be told “the integral is finding the area under a curve.” This is technically true, but not terribly useful. As Kalid from BetterExplained notes, saying that be like saying multiplication is finding the area of a rectangle. It’s a way to think about it, but it isn’t what’s actually going on.
Remember, calculus is the study of changing quantities. The integral, for all the complexity it can present, is essentially the multiplying of changing quantities. When you take the integral of acceleration with respect to time, you’re taking a function of acceleration and multiplying by time at each infinitesimal step, resulting in velocity.
Got it? If not, I heartily recommend Kalid’s article. It’s one of my favorite insights of all time.
Overall, this is fairly straightforward stuff if you get integrals. The important thing to realize is that by using calculus, you can go from position to velocity to acceleration to jerk and then back down again fairly simply.