Snap, Crackle & Pop! Cereal or Calculus?

Calculus is all about rates of change. The rate of change of a function is found by taking its derivative. One of the first concepts we encounter in classical mechanics is applying differential calculus to help us understand motion. 

You might be wondering where Rice Krispies fit into all of this – bear with me, we’ll get there. But first let’s go back to basics.

We denote our position vector as x(t) – position, x, as a function of time, t. Taking its derivative with respect to time, we get velocity, dx(t)/dt, the rate of change of position. Taking the second derivative, we get acceleration, the rate of change of velocity. So far, so familiar.

But here’s a thought: why stop there?

What comes next? That would be jerk – the rate of change of acceleration. Just think of the worst driver you know, you probably experience jerk when you’re in their car. If acceleration isn’t constant, you feel jerk. Since we know constant force relies on constant acceleration (thanks to Newton’s F=ma), then a change in acceleration implies a change in force, and vice versa, provided mass remains constant. So that sudden jolt or jerk forward you feel on a bumpy car or rollercoaster ride? Jerk. 

What about the fourth derivative of x with respect to time, or equivalently, the third derivative or velocity, the second derivative of acceleration, the first derivative of jerk? This is snap, the rate of change of jerk. In practical terms, the minimisation of snap is important in civil and aerospace engineering. The design of railway tracks and roads, especially around bends with changing curvature, involves minimising snap. When snap is constant, jerk changes linearly, allowing for a smooth increase in radial acceleration, and even better, when snap is eliminated completely, the change in radial acceleration is linear.

Now comes the fun part: velocity doesn’t just magically switch on, it has to grow from zero. So, there must be some acceleration involved. In a similar manner, acceleration itself has to grow from zero, meaning there’s jerk. But the same goes for jerk, so in turn, there’s snap, and so on…

And that’s how we get to our Rice Krispies mascots. The fifth derivative of position is crackle, and the sixth is – you’ve guessed it – pop! 

Beyond snap, there aren’t many known useful physical applications of the higher derivatives as of yet, but they’re still present mathematically nonetheless.

However, I think the main lesson we’ve all learnt here is that cereal is the language of the universe – classically speaking anyway.

Sources of inspiration:

https://en.wikipedia.org/wiki/Fourth,_fifth,_and_sixth_derivatives_of_position

https://iopscience.iop.org/article/10.1088/0143-0807/37/6/065008/pdf

Image reference:

https://onegoodthingteach.wordpress.com/2018/09/20/snap-crackle-pop-3/