- #1

- 19

- 0

How many levels of derivatives (rates of change) typically occur for objects in nature?

For instance, a car has a position, velocity (1st derivative), and acceleration (2nd derivative), but it can also be said to have a rate of change of acceleration (3rd derivative), 4th derivative, and so on. For example if you push down on your accelerator faster and faster then you are creating a 3rd derivative of the cars motion.

Is there a threshold at which further derivatives stop being common in the real world? I would imagine so considering that it would likely require more energy with each level of differentiation applicable.

As far as practice is concerned is there a point at which higher derivatives either stop being useful or stop occurring?

In the meantime, this reminds me of a related thought: If an object has a constant non-zero 3rd derivative will an person or device inside or attached to that object always be able to feel the force of increasing acceleration? Am I right in saying that the person/device would perceive a gravitation-like effect of increasing magnitude?

My current thoughts are perhaps: A person can "get used to" (stop "feeling") the force of acceleration (2nd deriv.) because a person's body is familiar with constant acceleration (e.g. because gravity is always there). However in the case of the 3rd derivative a person would always perceive a "tugging" force on their body and eventually be squashed if a (positive) 3rd derivative remains in effect long enough. Is this the right way to think about it?

Anyway, that was a tangent from the topic. The main question remains how many levels of differentiability typically occur in natural forces, and also perhaps what are the practical consequences.