What are the purposes of higher order time-derivatives of a position?

Question

I’ve been thinking a bit about rate of change and I read that there are higher order derivatives of a position (like jerk, snap, crackle and pop) which had me confused. The concepts of velocity and acceleration are easier to understand because they are common. I realize though that acceleration etc also ought to be able to have a rate of change but I don’t understand the purpose or usage.

It also had me thinking, is there a limit to what order or how “far” a position can be recursively differentiated? It seems to me that the higher order derivatives perhaps would explain the origins of motion in more detail?

In: Physics

Answer 1

For a real-world example, imagine two cars at a red light, ready to go.
When the light goes green, driver A mashes the pedal to the floor. Driver B does it like he was taught in driver ed — pushing the pedal down to the floor smoothly, ‘like you have an egg under your foot’.

B will trail A, but not by much, in reaching peak acceleration, and thus in speed and distance travelled.
But his passengers will have had a much more comfortable ride without that jerk at the beginning.

Answer 2

Each derivative is how fast the prior one is changing with time.

1st derivative = how fast position is changing with time = velocity

2nd = how fast velocity is changing with time = acceleration

3rd = how fast acceleration is changing with time = jerk

Past that it gets unintuitive really fast because our bodies only sense acceleration and time, so we have a tough time directly sensing anything past the 3rd. We usually express the 3rd derivative in terms of “smoothness”, the opposite of jerkiness.

For engineering purposes, the 3rd derivative can be really important because it determines how fast your actuators need to respond in a control system.

Past that I’m not sure of the physical significance but when you’re designing control systems you need to model the system and lots of real-world systems have really high orders (helicopters get up into the 6s or more, I recall) so the math of higher order derivatives still matters.