Why does jerk have units of mm/s rather than mm/s³? In 3D printing firmware and slicers, jerk settings are expressed in units if mm/s. This is contrary the physical definition of jerk, which is in units of mm/s³, being the second derivative of speed with respect to time (or the third derivative of position). What is the reason for this discrepancy and how does one interpret jerk in this contect?

The jerk setting in 3D printing G-code and firmware represents a concept similar to, but distinct from, the physical definition of jerk. Rather, it's a [limit on] instantaneous change of speed.

Mathematically, one way to make sense of this is to think that, rather than being the second derivative of speed with respect to time, this "jerk" is the entire remainder of the first-order expansion of speed with respect to time - it corresponds to the second-order term _and all higher order terms_. Such terms cannot be combined just as coefficients, since they all have different units corresponding to different powers of time; rather, they can be combined only _with their corresponding powers of time_ , in which case the resulting unit is mm/s.