A mental model I employ is: from a frequency response perspective, humans are 30Hz high pass filters. That means we end up conducting more high-frequency force (either by actively fighting it or by having it act against our inertia by transmitting thru our bodies), and this is work. In the lower frequencies, we can generally more actively participate and e.g. spread out the energy transfer over longer time periods to decrease instantaneous forces in joints/etc. Picture jumping off a 3 ft ledge, you can "eat" most of the impact with your legs bending, but some of the energy is going to affect you. The mental model is that it's the high-frequency content that human bodies don't handle well.

> Is there something specific about jerk that makes it important to optimize for, or are all position derivatives of order 3+ the same?

I think of it in terms of neck muscles. If your car is accelerating at a constant rate, you feel that as a force pushing your head back. Your neck muscles activate to compensate and keep your head still.

If the acceleration changes suddenly and aggressively (i.e. high jerk), so does the force on your neck. So either your neck muscles react quickly to counteract the new force, or your head bounces around.

Higher order derivatives also matter, but mostly inasfar as they act on the acceleration (and hence force) that you feel.

I believe a way to view it is as such

1) coffee is always under constant acceleration (g).

2) constant acceleration (say in the X direction instead of Y) would just mean a constant "tilt" within the cup. compare this to an airplane that is not "accelerating" and just at a constant X velocity, coffee would look "still/flat" in your cup.

3) its only the jerk that changes the "tilt" within the cup (and hence causes the spill)

no, and its simple to visualize even

If you move in a circle around a corner, you have constant acceleration, otherwise you would go straight (centripetal force needs acceleration to exist). yet it is possible to move a coffee cup in a corner, as long as you tilt it a little bit. So acceleration is not the issue

However if you suddenly change the direction of the coffee cup, you introduce jerk, because you accelerate the acceleration (you change the size of the circle means you change the acceleration, therefore you introduce jerk = coffee spilled on the floor)

Constant acceleration will cause the surface of your coffee to tilt, but not spill it unless the cup is almost full. Jerky motion (non-constant acceleration) causes it to slosh around and can create resonances that disturb the fluid level far more than a constant acceleration of the same magnitude would.