# Jerk, Jounce, Snap, Crackle and Pop

Physicists are obsessed by rates: how quantities change over time. The rate of change of the number of nuclei in a radioactive sample tells us how radioactive something is; the rate at which the chemicals in a reaction change tells us how reactive something is; and so on.

If we start by looking at the displacement of an object (i.e. the distance from where it started to where it currently is) then when we look at the first derivative (by time) of displacement, (i.e. dividing the displacement of an object for how long it took to be displaced) we have calculated the object’s velocity.

$v = \frac{dx}{dt}$

If we look at the rate of change of velocity, the second derivative (by time) of the object’s displacement (i.e. the rate of change of the rate of change of its displacement), then we have calculated the object’s acceleration.

$a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$

If we now look at the rate of change of acceleration, the third derivative of the object’s displacement (i.e. the rate of change of the rate of change of the rate of change of its displacement) then we have calculated the object’s jerk.

$j = \frac{da}{dt} = \frac{d^2v}{dt^2} = \frac{d^3x}{dt^3}$

The first two derivatives of displacement, velocity and acceleration, are well known and reasonably well-understood by most people. But jerk is a little bit more difficult to understand. If we apply a force to an object it will accelerate, and we usually assume that this force is applied instantaneously. But this is not correct – it takes time to apply a force. As a result, the rate of acceleration will not be constant, and thus we have the jerk.

It may be easier to understand the concept of a third derivative by looking at an example from economics: inflation. US President Richard Nixon once famously said “the rate of increase of inflation is decreasing”, using a third derivative in the process.

The rate of inflation is the rate at which prices increase over time, and this is therefore the first derivative of price. The rate of the increase in inflation is a second derivative, and if this itself is decreasing then that is a third derivative. That is, in Nixon’s case, prices were increasing (i.e. inflation was positive), and this rate of inflation was itself also increasing, but the rate at which it increasing was decreasing.

The fourth derivative of an object’s displacement (the rate of change of jerk) is known as snap (also known as jounce), the fifth derivative (the rate of change of snap) is crackle, and – you’ve guessed it – the sixth derivative of displacement is pop. As far as I can tell, none of these are commonly used.

## 22 thoughts on “Jerk, Jounce, Snap, Crackle and Pop”

1. luc says:

Beautiful. Especially the part about Nixon. If I have seen him say it, it would certainly take me some time to understand the way he was trying to hide what he was really saying :D

2. Yep, that’s what the post says.

3. “then we have calculated the object’s jerk (also known as jounce).”
Jounce is not the same as jerk.

4. Oh. Right. I’ll fix that.

5. morteza says:

I’ve started to search about a reference book about snap or jounce but I could not find a reference book or paper about it and its definition. I ‘d be thankful if any one help me.

6. Jerk is easy to explain because it’s literally the jerk you feel due to changes in acceleration.
When a car jerks, you’re experiencing sudden shifts in acceleration as the driver shifts from gas (jerk+) to brake (jerk-).
When they press hard on the gas, acceleration grows linearly (velocity increases quadratically) and jerk is pressing you back into the seat against your inertia.
After peeling away from the light, velocity grows linearly, acceleration becomes flat, and jerk becomes 0 when you no longer feel pressed against your seat.
Upon merging on to the highway, you let up on the gas to maintain the speed of traffic and your body tilts forward slightly as jerk goes negative while your acceleration drops toward 0.

7. Soumya Dasgupta says:

Is it possible to represent Jerk into a linear mathematical notation?
Can jerk give us the acceleration if we divide it by time?

8. I think saw thse terms explained long ago in an open university TV broadcast years ago. Apparently these terms are vectors with very practical application in roller coaster design and.construction.

9. Anonymous says:

Simple acceleration is sufficient for one to feel pressed against the seat of a car. Jerk can be 0 and still the seat can be felt.

10. Amarnath Chowdhury says:

Beautifully simplified! But how do you explain Snap?