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 gcode 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.
The units for jerk should be meters per second cubed or m/s3.
Meters are the basic unit for distance. The first derivative is speed, or velocity, m/s. The second derivative is acceleration, m/s2. The third derivative is jerk, m/s3.
It is rate of change in acceleration.
While seldom used, I've only heard it once concerning the Hubble Space telescope, there is a fourth derivative call jounce, m/s4.