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Skill 11 of 14
In Progress

Changes of changes

  • 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]

  • Jerk1 is change in acceleration: $\vec s^{\prime \prime \prime}$.
  • Snap2 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:

  1. 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
  2. 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)
  3. What is the difference between jolt and jerk?’ (web page), WikiDiff, 2019, wikidiff.com/jolt/jerk (accessed Sep. 24th, 2019)