# the nth derivative of velocity

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What is the difference between acceleration and jerk? Jerk and the fourth derivative? What happens as n approaches infinity?

How does the energy of something change as it experiences jerk or 4th derivative motion in a certain direction? It seems like higher derivatives of motion could describe vibration or other frequencies.

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Vibration is well-explained using the first two derivatives of position wrt time. When you get past the third derivative, you basically stop getting physical meaning — I don't think anybody has bothered to name higher derivatives, since they don't tend to show up in any solutions to motion.

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IIRC the fourth fifth and sixth derivatives are called snap crackle and pop

I am having trouble understanding why these derivatives don't show up in solutions to special types of motion. Surely they have some physical significance?

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Imagine a weight tied on a string moving with constant speed in a circle.

The acceleration is a vector pointing from the weight to the middle of the circle.

The first derivative of that is a vector moving in a circle in the same direction as the weight. The second derivative is a vector pointing to the centre of the circle and the thrid derivative is a vector moving in a circle in the same direction as the weight.

As n tends to infinity this sequence gets infinitely boring.

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As n tends to infinity this sequence gets infinitely boring.

Does it always? IF you have simple harmonic motion, then yeah, all you get is a long series of sines and cosines. But what if you have some other function? Presumably you'd reach a point where you had something like

$\frac{d^n v(t)}{dt^n} = const$.

You're saying there's no physical significance in this?

Edit: I thought tex tags worked here?

Edited by swansont
fixed math tags
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The particular series I gave gets very dull .

Other sequences may be more interesting but I suspect that for most practical ones you end up with the nth derivative being zero (for high enough n).

If the motion of the object can be described by an nth order polynomial then the nth derivative is pretty dull.

A whole ot of functions can be aproximated reasonably well by polynomials so I think it's fair to assume that these things generally get dull for large n.

If anyone can do the maths for an object sliding down a cateneray curve it might be more interesting.

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Edit: I thought tex tags worked here?

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If we do it backwards and integrate a constant, we get velocity. If we integrate that we get acceleration. If we integrate that we get the acceleration of an acceleration. This occurs in nature and would be due to a force that is accelerating in magnitude, such as a forming star where the mass is building up the gravitational force with time. The result is a distant falling object accelerating it's acceleration. One can do that with magnetics by boosting the electricity in an acceleration of current.

If we integrate that we get an acceleration of an acceleration of a force. This may also occur in nature. Here is one possible scenario. Say you had an exploding star, not an impulse explosion but building as gravity resistance is decreasing. This give us the acceleration of force. This is going into the expansion acceleration of the universe.

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Well, the integral of distance is a sort of concept space-time, able to be expressed in meter-seconds. For instance, an object at rest will still have a different "time-coordinate" after five seconds, but say during that five seconds the object moved two meters. After this, the object is 10 m-s from the origin.

Edited by Kyrisch
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...

Back on topic..

(sorry man, you aren't helping)

Imagine a weight tied on a string moving with constant speed in a circle.

The acceleration is a vector pointing from the weight to the middle of the circle.

The first derivative of that is a vector moving in a circle in the same direction as the weight. The second derivative is a vector pointing to the centre of the circle and the thrid derivative is a vector moving in a circle in the same direction as the weight.

As n tends to infinity this sequence gets infinitely boring.

Given the large mass and small number of objects in the solar system I digress back to my initial presumption that nth derivatives of motion can be used to describe existing physical phenomena. How is gravitational force related to motion?

I am somewhat familiar with Kepler's laws and the two and three body problem as well as the big question mark that comes in when describing the exact mechanisms of "planetary accretion." Perhaps when the masses involved in a system are in ratio similar to sun:planet the force is so large as to cause high derivative motion, accounting for the shape of the solar system?

Kyrisch: Would the space-time description work when describing something that is at rest in all reference frames?

Thanks!

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• 4 years later...

Wouldn't a car pedal be a simple example ? if I keep it constant I'm applying a constant force and have a constant acceleration, if I push it at constant speed we have yank and jerk, if I accelerate the push the car experiences the 4th derivative and so on.. I bet you can even feel those physical quantities in a good sports car...

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