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## Main Question or Discussion Point

Here's a question I've thought about on several occasions:

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.

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.