Jerk

What is jerk? Jerk is to acceleration as acceleration is to velocity. And, acceleration is to velocity as velocity is to position. So let’s start there. The instantaneous velocity at which you travel tells you how fast your position is changing over time, i.e., it is the rate of change of position. Mathematically, the rate of change is given by the derivative. Hence, the instantaneous velocity \(v\) of an object with a position \(s\) that varies over time \(t\) is given by

\[v=\frac{ds}{dt}.\]

Acceleration is comparable, but with respect to velocity instead of position. Your instantaneous acceleration tells you how fast your velocity is changing. Acceleration \(a\) is given by

\[a=\frac{dv}{dt}.\]

Although acceleration and velocity are well know, this is less true for jerk. If you’re not an engineer, you might not have heard of it. An easy practical demonstration of jerk is when you brake steadily with your car, without slowly raising your foot just before it stops. If you do that, you experience a small jolt, which is actually jerk. Jolt, incidentally, is indeed another name for jerk.

The reason for this jolt is that the deceleration changes abruptly from whatever value it had to zero. Jerk \(j\) is the rate of change of acceleration, given by

\[j=\frac{da}{dt}.\]

This implies that jerk is the third derivative of position, because velocity is the first and acceleration is the second, so that

\[j=\frac{d^{3}s}{dt^3}.\]

That said, if people call you a jerk, they are probably just rude, and not talking about the third derivative of position…