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Actually in Mechanics rate of change of acceleration with time is not considered.Why is it so?Is the rate of change of acceleration with time not important? Or there exist any cases, where we can consider the rate of change of acceleration?

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The important thing in classical mechanics is that Newton’s Second Law says force is proportional to the second time derivative of position, not the first, or third, or fourth, or fifth, or any other.

So the time derivative of acceleration is not particularly important most of the time. However, the Abraham-Lorentz force in electrodynamics is a rare example of where it is.

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Newtonian Mechanics can be easily dealt with via considering forces and there effects only. As you may get from $\mathbf F =m \mathbf a$ effects of the forces are instantaneous (which can vary from one time to another) and hence for considering the dynamics of a situation you need the instantaneous information ( i.e., force as some function of time or position) about the forces, and if you have so then you are done for it.

But if you consider the quantity "jerk" it may seem to be at advantage at first sight but isn't so, as in a real life situation (consider the case of gravity here) it may not be constant and may itself vary with time and/or position. Then you may consider the higher derivative of position ( you may notice where this is going) and then you would find that the position function would itself not be some polynomial function, then you would have no choice but to leave this hunt (for higher derivatives) whereas a simple knowledge about the function of the force acting would be most helpful in understanding the dynamics of the situation (though this itself would be hard to do but would be easier than former).

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As others have mentioned in their answers, I'd like to say that while studying mechanics we are interested in the positions of the object, how fast they move and what causes them to move faster or slower. In many cases, we can get what we need without invoking "Jerk".

The concept of "jerk" isn't that useful here. I mean, we can apply a constant force and still cause a change in the state of motion of an object while having a jerk of $0$. Moreover, you can take the example of gravity where an object suffers acceleration from a constant force (if the change in height is sufficiently small during the fall) under it's influence. There you could see the object's velocity change with a jerk of $0$ because

$$\text{jerk=}\frac{d}{dt} a(t)=\frac{d}{dt} \left (\frac{F(t)}{m}\right)=\frac 1 m \frac{d}{dt} F(t)$$

But for significant distance changes you can have a non zero jerk and that data may be useful to you in some way.

In a similar manner you can define $n$ th derivative of position w.r.t time (other people have linked to the wiki article) but not all of them are useful to us every time we want to do (know) something. We can use them when necessary.

Like this, there are many other things and formulae which are defined and could be used for specific purposes. If they're used simply depends on if that's useful for the current computation or not.

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To answer this question, one could think what would be the effect of modified equations of motion where jerk would appear (in place or in addition to the usual acceleration). Well, theory of differential equations tells us that in that case, a well-posed problem, i.e. the possibility of predicting the future motion from the present state, would require position, velocity, and acceleration at a given time as independent state variables.

Since in most of the mechanical problems one is facing at macroscopic level(*) position and velocity seem enough to provide a unique evolution, we have no experimental basis for thinking that jerk would play a role in the equations of motion.

Moreover, at the extent Newton's equations of motion can be derived from Eherenfst's theorem in the case of strongly localized quantum states, on could say that a modification of Newton'a laws would imply that something should be changed also at the quantum level (not very likely, taking into account the precision of spectroscopic measurements).

(*) There exist mechanical problems where the equations of motions go beyond the usual newtonian form, requiring a different determination of the mechanical state. Usually this happens as an effect of eliminating some degrees of freedom which results in some memory effect on dynamics.

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