# Non-physical Jounce Examples in Nature

What are some good examples of jounce, the fourth derivative of position, in the non-physics arena?

The reason I ask is that A) it's already difficult for a lay to visualize it in the physical arena and B) you never hear of too many examples past the second derivative outside of said physical arena.

Jerk is relatively easy to perceive when one slams on the breaks and then lets them go, and the best way to describe jounce is an amusement park ride since you're always being jerked around.

Outside of physics, it's a little hard to come by, so I'd like to see if there are other ways of perceiving higher order derivatives in other disciplines.

Many thanks in advance!

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What is the question? Money is not the problem. –  Ross Millikan Nov 9 '12 at 4:50
Aargh. The question could use a lot of improvement, sure, but it's not all that hard to figure out what Joe wants (or at least to make a reasonable guess). I was in the middle of writing an answer when it got closed. Come on, Eric, easy on the moderator superpowers. –  Gerry Myerson Nov 9 '12 at 4:55
In determining stability of nonhyperbolic fixed points of one-dimensional discrete dynamical systems, it may be necessary to evaluate the so-called Schwarzian derivative. The formula for the Schwarzian derivative involves the third derivative of the function in question. –  Gerry Myerson Nov 9 '12 at 5:10
@GerryMyerson: It looks like Austin Mohr and you have understood OP better than me. Congrats and sorry for voting to close. –  Ross Millikan Nov 9 '12 at 5:21
A related question. –  Ｊ. Ｍ. Nov 9 '12 at 12:00

## 3 Answers

Constant acceleration (m/s^2) occurs when riding a rocket sled, by lighting a single solid rocket motor. Light a series of the same rockets, one after another each second, whilst they're all burning, gives you the experience of jerk (m/s^3) as you'd feel a steady increase in g or acceleration. If you light the rockets quicker each time, instead of a steady one second interval, you'll get a rate of change in jerk called jounce or snap (m/s^4), feeling your head pushed back harder each time and with more force than the previous rocket. If you then repeated the jounce experiment but with a bigger rocket each time, lighting each one quicker than the previous, you'll experience crackle (m/s^5). Now if those progressively bigger rockets use solid rocket fuel that gets steadily more powerful as they burn (an accelerating burn rate), running the crackle experiment again, you'll experience pop (m/s^6). If you run the pop experiment again but use solid rocket fuel that accelerates in power (has a jerk burn rate), you'll experience crispy (my name this time) or (m/s^7). Using rocket fuel with snap burn rate, where the fuel is burning with a rate of change in acceleration of the burn front, and using progressively more volatile fuel as it burns through, you'll experience toast (my name again) or (m/s^8). You can see this is a chain reaction thought experiment, the more rates of change you add to rockets, solid fuel, and fuel pellets etc, you can define more orders of acceleration. In practice I doubt there are many real instances of pop and beyond outside detonations, which would not leave much of the rocket sled or passenger left :-)

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A way I think of jounce is to imagine you are at the gym and you start to lift a weight you start off with a decreasing accelration (jerk), we'll say $3m/s^3$ (or slowing acceleration) until all of a sudden you get quite tired and your jerk changes. That moment when you hit the wall and your decreasing acceleration (or the change in jerk) goes to, we'll say $1m/s^3$ is jounce.

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I think there should be $3m/s^2$ in second line –  iostream007 Jun 19 '13 at 19:55

You write of "examples past the second derivative," so I'm going to assume the third derivative is of interest to you. As I noted in a comment, "In determining stability of nonhyperbolic fixed points of one-dimensional discrete dynamical systems, it may be necessary to evaluate the so-called Schwarzian derivative. The formula for the Schwarzian derivative involves the third derivative of the function in question."

To expand on this a bit:

Given differentiable $f:I\to I$ where $I$ is an interval, and $x$ such that $f(x)=x$, it can be shown that $x$ is an asymptotically stable fixed point of $f$ (roughly: if you start near $x$, and iterate $f$, you converge to $x$) if $|f'(x)|\lt1$, and unstable if $|f'(x)|\gt1$. Things are more complicated if $|f'(x)|=1$ and, in the case $f'(x)=-1$, the determining factor is the value of the Schwarzian derivative evaluated at $x$. You can look up the Schwarzian derivative --- you'll find it involves $f'''(x)$.

EDIT: Maybe you'd prefer Wilson, Matthews, Greasham, Will, and Copeland, Application of fourth derivative absorption spectroscopy to protein quantitation during purification, Anal Biochem. 1989 Oct;182(1):141-5.

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You asked for examples "in the non-physics arena," "outside of said physical arena," "Outside of physics," and I have presented one. If what you really want is examples "where math isn't the perceived prime concern," then you should consider editing the body of your question to say so. –  Gerry Myerson Nov 11 '12 at 23:16