Questions on Jerk: What Occurs in Nature?

[Ja4Coltrane]

Questions on the "jerk"

I have two questions relating to the third derivative of displacement with respect to time. My main question is this. Does a jerk ever occur in the natural universe? I know that there are many situations in which acceleration changes over time, but all that I can think of do not directly relate change in force to time, the all are with respect to position. For example, an object falling towards Earth from an extremely long distance or a hooke's law spring.

My second question is more complicated. It is pretty obvious that you can not suddenly change velocity. You must first accelerate to reach some final velocity. Does this mean that you can not suddenly reach some acceleration? Must you first undergo some jerk? (I can't believe this easily because this would mean that an object freely falling would have to spend some period of time to reach 9.8 m/(ss)). But if an object does need to have some jerk for some acceleration to be reached, does this mean that in order to have some jerk, you must undergo some fourth derivative of displacement, and some fifth for that, and so on for infinate derivatives?

I'm only a high school senior, so try to answer in a way that is not too complicated.

 

[phlegmy]

that an interesting take! I've never thought about it before.
you can relate an acceleration directly to a force
so i guess the instant a force is applied, the full acceleration due this force will instantly exist.
the reason you can't instantly have a velocity is because of inertia, eg a bodies mass. the inertia resists the acceleration of the mass. but i don't think there is a corresponding "inertia" resisting the acceleration of the acceleration.

what I'm trying to say is that no, the INSTANT you apply gravity or any other force, to something, (and no other force) it will have an INSTANT acceleration proportional to the force, and its velocity will gradually increase.

 

[actionintegral]

I think you are asking the same question twice. Does acceleration change in time? Yes. A "jerk" is called a discontinuity and discontinuities don't really exist. If you zoom in enough you will find a gradual increase.

And your observation that force is often dependent on position is true, but not pertinent to this discussion.

 

[leright]

Of course jerks occur in the natural universe. I meet them everyday. :p

Jokes aside, yes, they can occur. If you apply a constant net force on an object then it will have constant acceleration. The jerk (rate of change of acceleration) is zero.

Now, if you simply have a time varying force on an object then it will have time varying acceleration, right? The jerk (rate of change of acceleration) is non-zero!

Technically, all applications of net forces result in jerks, since force cannot change instananeously. There must be a gradual transition from zero force to some finite force when a force is applied. This results in a gradually changing acceleration, or jerk.

But, yes, IF all forces changed instantaneouly then the acceleration would also change instantaneously resulting in discontinuities in the acceleration curve which are interpreted as infinite acceleration changes, or INFINITE jerk between intervals of constant acceleration. But forces NEVER change instantaneously!

 

[leright]

that an interesting take! I've never thought about it before.
you can relate an acceleration directly to a force
so i guess the instant a force is applied, the full acceleration due this force will instantly exist.
the reason you can't instantly have a velocity is because of inertia, eg a bodies mass. the inertia resists the acceleration of the mass. but i don't think there is a corresponding "inertia" resisting the acceleration of the acceleration.

what I'm trying to say is that no, the INSTANT you apply gravity or any other force, to something, (and no other force) it will have an INSTANT acceleration proportional to the force, and its velocity will gradually increase.

But you can never have an instantaneously applied force! The force will smoothly transition from 0 to some value resulting in transitioning acceleration, or jerk.

I can certainly cause time varying accelerations without discontinuities.

 

[WMGoBuffs]

I suggest you look into Taylor Series and compare it to your normal displacement formula in terms of time. There are plenty of terms beyond the jerk!

 

[Ja4Coltrane]

Wow, thanks for all the fast replies!
So I have one more question that is more mathematical. Take gravity, a force that varies with displacement rather than time. If an object is falling towards a distant planet, assuming that it is heading straight for the planet, and that no forces other than that planet's gravity act on the object, how would you give a mathematical model for acceleration as a function of time? Since I have only taken a quarter of calculus, I don't know how to do this, but I do actually have some ideas how how it would be done.
Thanks again for all the help.

 

[Cyrus]

How do you go from displacement to velocity, velocity to acceleration, acceleration to jerk... Extrapolate...

 

[actionintegral]

how would you give a mathematical model for acceleration as a function of time?

most beginning calculus problems dealing with gravity's acceleration treat gravity as a constant acceleration. the equation you set up for gravity's 1/r^2 form is a nonlinear second order equation and not easily solved d2x/dx2 = 1/x^2

 

[Cyrus]

Take gravity, a force that varies with displacement rather than time.

Are you sure about this? Think a little bit harder what that is really a function of.

 

[marlon]

Take gravity, a force that varies with displacement rather than time.

I don't get that. The force itself is not directly depending on time but that does not imply the fact that the corresponding movement of objects is NOT depending on time.

It is quite obvious that is NOT the case.

Besides, when trying to solve Newton's second law for this case one has the benefit of the associated conservation laws (total energy and angular momentum)

That's TWO conservation laws for a second order partial differential equation. Mathematically, that means "all differential problems are over". My point is that when studying planetary movements (like calculating the trajectory of a sattelite with given initial speed etsc etc, from Earth to let's say Venus) one can caracterise the entire trajectory using ONLY these two laws.

This also answers your second question, btw.

regards

marlon

 

[desA]

Fluid flows are interesting examples of changes in the acceleration field. The momentum equations are essentially Newton's 2nd Law, & thus describe force balances. Plots of this information as an acceleration vector field will show regions of changing acceleration - both in space & time. As I understand it, 'jerk' is the temporal change in acceleration. Would the spatial change in acceleration perhaps be similar to a 'shock'?

 

[Karlisbad]

from a "more" theoretical framework..using Euler-Lagrange equations you can deduce the equation of motion by taking:

$$\delta S =0$$ the question is..How the hell can you quantize a theory with a lagrangian involving "acceleration"?..in fact if you wished to construct the quantum version of a fluid motion that depends on acceleration how you could do this?...in wikipedia and other pages spoke about using some kind of "Dirac constraints" but i couldn't work out how this was used.