Mathematical Jerk

I just learned an interesting factoid today that I never heard before.

One of my biggest complaints when they learned me some math way back when was that, occasionally, they explained you the formula without necessarily telling you the real-life practical meaning behind it.

For a very basic example, I didn’t know that integration was a method to determine the area under a curve until I got to college math. But as I write this I realize a goodly portion of the blame for things like this lie at my feet. After all, in high school I had a lot more on the mind than calculus.

Anyway, imagine a polynomial equation with variable t representing time and x representing position. Let’s say it’s the result of computer analysis of a typical kick-off run back. It zigs and zigs, speeds up, curves this way and that, and ends. The equation describes the player’s position at any given t.

Now get the first derivative of the polynomial. What does this describe?

The first derivative describes the speed (velocity) of the player at any given t. I knew this from my physics classes.

What does the second derivative of the polynomial represent?

Again, from physics, I learned that this describes the player’s acceleration.

Now, how about the the third derivative? What does this tell us?

Answer: the jerk. This is what I learned the other day.

Jerk is the highly technical term for acceleration acting on acceleration.

In my example, Santonio Holmes fields the ball at t = 0. Then he speeds up the field in a zig-zagging pattern, accelerating here and there. Let’s assume his velocity suddenly increases (accelerates) between, say, t = 4 and t = 5 as he’s trying to squeeze through a closing hole between tacklers. Suddenly he’s hit from behind by an opponent, as sometimes happens when all the players converge on the ball carrier. This acceleration on his acceleration would be the “jerk”, and we can see it when Santonio’s crumpled form is slammed to the ground.

Under constant velocity, acceleration is zero. When velocity increases or decreases, that’s acceleration.

Under constant acceleration, jerk is zero. When acceleration increases or decreases, that’s jerk.

At least, I think.