**Velocity**is change in position: $\vec v=\vec s^{\prime}$.**Acceleration**is change in velocity: $\vec a =\vec s^{\prime \prime}$.

Acceleration can of course change as well; and its change can change. Which can also change… Let’s invent some more names:^{[1,2]}

**Jerk**^{1}is change in acceleration: $\vec s^{\prime \prime \prime}$.**Snap**^{2}is change in jerk: $\vec s^{\prime \prime \prime \prime}$.**Crackle**is change in snap: $\vec s^{\prime \prime \prime \prime \prime}$.**Pop**is change in crackle: $\vec s^{\prime \prime \prime \prime \prime \prime}$.*…*

And we could continue if we wanted.^{3} Let’s not invent symbols for these rather rare properties.

Note that the units will have added one more ‘per-second’ for each deeper derivative. Acceleration adds extra ‘metres-per-second’ every second, so ‘metres-per-second-per-second’, $\mathrm{m/s^2}$. Jerk adds extra ‘metres-per-second-per-second’ every second, $\mathrm{m/s^3}$. Snap will have units of $\mathrm{m/s^4}$, crackle of $\mathrm{m/s^5}$, pop of $\mathrm{m/s^6}$ etc.

What happens if we look the opposite way? What is position a derivative of? And the double and triple derivative of? Let’s invent names for some of those as well:^{[1]}

- Position is the derivative of
**absition**.^{4} - Position is the double derivative of
**absity**. - Position is the triple derivative of
**abseleration**. - …

These are even rarer to see in use.

Although rare, we should not completely disregard them: To overtake on the motorway, you press down the gas pedal to some new **position**, which causes a constant **acceleration** of the car. Car acceleration corresponds to pedal position; what then does car **position **correspond to? It corresponds to pedal **absity**! Absity is how much you have pressed the pedal down *combined with *how long time you have done so – knowing this lets you find out where the car is right now. Also, *while* the pedal was moved down, the acceleration was growing; how fast it grew is **jerk**.^{5}

Not entirely useless properties. But rare.

References:

- ‘
**Beyond velocity and acceleration: jerk, snap and higher derivatives**’ (article),*David Eager, Ann-marie Pendrill and others*, European Journal of Physics, IOP Publishing, vol. 37, issue 6, 2016, www.iopscience.iop.org/article/10.1088/0143-0807/37/6/065008, DOI 10.1088/0143-0807/37/6/065008 - ‘
**LOG#053. Derivatives of position.**’ (web page),*Amarashiki*, The Spectrum Of Riemannium, 2012, www.thespectrumofriemannium.com/2012/11/10/log053-derivatives-of-position (accessed Aug. 7th, 2019) - ‘
**What is the difference between jolt and jerk?**’ (web page), WikiDiff, 2019, wikidiff.com/jolt/jerk (accessed Sep. 24th, 2019)