# 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?

• I'm not sure what other tags would be appropriate for this question; please suggest or edit them in if you have good ideas. – R.. GitHub STOP HELPING ICE Jun 24 '19 at 19:07

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.

• Confused me for a sec - I always look at the Nth derivative of position, not speed. Just noting this in case anyone else is as poor at reading as I was. – Carl Witthoft Jun 25 '19 at 14:34
• @CarlWitthoft: Of position. I specifically said of speed. – R.. GitHub STOP HELPING ICE Jun 25 '19 at 14:36
• @CarlWitthoft Speed (v) is the derivate of positon (s) over time. because of that the physical jerk = v'' = s''' – Trish Jun 27 '19 at 11:00

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.

• While this expands on the physics definitions in the original question, it does not address what jerk means in 3D printing or why the units are different. – Davo Jun 25 '19 at 21:40
• The units are wrong, we are talking about derivatives of time, hence the units are in m/s^x (where x is the derivative), not m*s^x (unless x is negative). Please update your answer and make the units consistent. – 0scar Jun 26 '19 at 9:11
• Also velocity is m/s not m/s^2 – Greenonline Jun 26 '19 at 12:06