Snap, Crackle, Pop!

It's (snapping) turtles all the way down.

Image credit to KaiThePhaux

Sometimes, while helping classmates with Mechanics questions, I find that they think of Displacement, Velocity, and Acceleration as disconnected quantities. If you press them for a definition, they recite "Velocity is the rate of change of displacement against time." However, it’s easy to lose track of what that means, particularly if the structure of SUVAT (constant acceleration equations) gets taken away.

When I think it’s worthwhile, I always take the top few (displacement, velocity, and acceleration) and tie them in with a longer chain of derivatives of position. It’s not particularly useful, but it is rather fun, and quite memorable too. Derivatives, while sometimes tricky to understand in themselves, serve as an excellent tool for understanding other things, and should be used more in teaching Physics.¹

For anyone reading who isn't familiar with differentiation (or would like a refresher), it's a calculus method for looking at the rate of change (commonly the slope of a graph). The neat thing is that it’s a tool you can roll out for anything that’s changing relative to something else.

Displacement

Displacement (or position) is a measure of where something is. Some formulae use ‘s’ as a symbol but for my purposes, I’ll write ‘r.’ It means the same thing in this context. Imagine your house on a map - it has coordinates, which identify where it is located. Displacement can be a grid reference (2d), a point along a line (1d), or a position in space (3d). Mathematically, you can keep stacking dimensions, but that's for another article.

Displacement can change. You could park your car half a mile down the road from your house, and measure the displacement between the two. Park somewhere else, and the displacement between the house and the car changes - even if it's still half a mile, the distance in a different direction, so the displacement is different.

The final thing to think about is time - when you move around, time is ticking, and your displacement is changing. Now we can make use of differentiation, and look at how quickly our displacement is changing relative to time, as seen in the formula below.

\(\frac{dr}{dt}=\dot{r} (=v)\)

Velocity (First Derivative)

You are moving and time is ticking. Except, now you are moving with a speed and in a direction. Change one or the other and your velocity changes. The formula above is what is actually going on beneath the hand-wavy descriptions that talk about “rise over run.”

Now, we can also differentiate this, and examine what happens while you change your velocity:

\(\frac{dv}{dt}=\frac{d^2r}{dt^2}=\ddot{r} (=a)\)

(The dots are a shorthand count of the differentiations)

Acceleration (Second Derivative)

This doesn't need much explanation - the rate of change of velocity against time is summed up by the gas and brake pedals in a motor vehicle - press either one and your velocity changes at a given rate, dependent on how hard you're pressing the pedal.

School exam boards quit here. After all, the further derivatives aren't overly useful to anyone - but they’re still worth looking at.

Beware, those still at school - beyond this point is the nonsensical humour of physicists who don't need to follow serious naming conventions. Its primary use case is making friends laugh. And so we dive down another layer, to what happens as you change acceleration…

\(\frac{da}{dt}=\frac{d^2v}{dt^2}=\frac{d^3r}{dt^3}=\dddot{r}\space(=j)\)

Jerk (Jolt if you're British) (Third)

This name makes sense - slam on the brakes and the acceleration needs to change faster. You feel a jolt. There's a niche use-case for Jerk in making smoothly accelerating vehicles, and rollercoasters that can either jolt or run smoothly with complex changes of acceleration.

But, Jerk is also a variable measured against time. It can change too, can't it? Yes, yes it can. Guess what we call the feeling of a quickly changing rate of change of acceleration?

\(\frac{d^2a}{dt^2}=\frac{d^3v}{dt^3}=\frac{d^4r}{dt^4}=\ddddot{r}\)

These dots are getting out of hand….

Snap (Fourth)

Yup - this just about describes the feeling of Jolt changing quickly. Snap your fingers, and they decelerate quickly as they hit your palm (high Jolt). That Jolt needs to increase from 0 quickly, which means there's also high Snap. Some aspects of train braking systems, and other systems designed to run smoothly for a long time, use this.

Nearly there! We only have a couple of layers to go.

The rate of change of snap is:

\(\frac{d^5r}{dt^5}=\overset{..... }{r}\)

Crackle (Fifth)

The name doesn't relate to any aspect of changes in motion we can feel. Any practical intuition you had for the first few derivatives stops here.

For context, this is the

rate of change of
          rate of change of
                    rate of change of 
                              rate of change of
                                        *breathes*
                                                    rate of change of displacement against time.

Now, you might see where Snap and Crackle are going, given how bad physicists' senses of humour is. Since I gave the game away right at the start, I'll write out the algebra again, to emphasise how far from displacement we've gone.

\(\frac{d^6r}{dt^6}=\overset{......}{r}\)

Which, you guessed it, is called…

Pop (Sixth)

I don't think there's anything left to say, beyond this - supposedly - being completely serious physics.²

The joke is complete.

Since differentiation is what it is, you can keep playing this game. The ‘dot’ shorthand for derivatives has become very impractical, and is now harder to both read and write than the ordinary form.

\(\begin{align} \text{full:}\space\space\frac{d^7r}{dt^7}, \frac{d^8r}{dt^8}, \frac{d^9r}{dt^9}, \frac{d^{10}r}{dt^{10}}\\ \\\\ \text{shorthand:}\space\space\overset{.......}{r}, \overset{........}{r}, \overset{.........}{r}, \overset{..........}{r} \end{align}\)

There are even more names: Lock (7th), Drop (8th), Shot (9th), and Put (10th).

That’s about enough for anyone’s appetite. If not, just keep naming! I already came up with one for the 11th derivative of displacement, and given the sheer number of dots above it, I’ll justify a longer name. Here’s the algebraic representation.

\(\frac{d^{11}r}{dt^{11}}=\overset{...........}{r}\)

Which I'm going to call:

Subscribe!

Not even corny at this point - the competition for worst name has been won.

Thank you for making it this far down an utterly useless rabbit-hole. I hope it was somewhat enjoyable. At the very least, you now have something to joke about.

1. (A personal gripe of mine is A-Level physics doing its utmost to avoid calculus - which I think does more harm than good. No ranting for now, though.)

2. Wikipedia: Fourth, Fifth, and Sixth derivatives of Displacement

Although it’s about as serious as some scientific spider names