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When I got more into physics, I started asking myself if just like acceleration represents the growth of speed, something else could also represent the growth of acceleration itself. And it came that is exists and is called jerk. Before I thought about this, I thought that acceleration was the highest derivative that made sense on physics.

My question is, why can we perform accurate calculations without account for jerk? I never needed it in a single physics class! I've also read that it is seldom used.

Also, while position, speed and acceleration seem familiar to me due to everyday experience, why doesn't jerk seem familiar to me? I never tell someone when driving a car "you're jerking too much", whereas I could say "you are accelerating too much".

Is it that jerk has little influence because it is almost constant? Even if it is not, what about higher order derivatives? Why don't derivatives higher than acceleration have much influence on calculations? We perform a lot of reasoning in everyday life without resorting to it!

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marked as duplicate by ACuriousMind, Brandon Enright, Kyle Kanos, tpg2114, John Rennie Jan 22 '15 at 7:16

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I think the main reason you don't hear about $\dddot{x}$ very often is that it rarely appears in equations that model physical processes, whereas $x$, $\dot{x}$ and $\ddot{x}$ do.

For instance, the following is sufficient to fully specify the evolution of a particle of mass $M$ in a static density field $\rho(x)$ (Newtonian gravity), provided with an initial position $x_0$ and velocity $\dot{x}_0$.

$$\nabla^2\Phi(x) = -4\pi G\rho(x)$$ $$M\ddot{x}=-\nabla\Phi(x)$$

Since the evolution of the system is fully specified, there's no reason to get $\dddot{x}$ involved. The same is true of most of the usual suspects (electrodynamics, basic fluid dynamics, quantum mechanics, etc.). Once in a while a specialized case where jerk plays a meaningful role comes up, but it's not the norm.

One thing jerk does do is provide a decent measure of exactly what it sounds like - if your friend driving their car causes many rapid changes in acceleration, you would probably describe the motion as "jerky".

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There are examples where jerk is very familiar from everyday experience. Large jerk is what might cause you to stumble or fall when standing in a bus or a tram. If the acceleration only changes slowly (i.e. when the jerk is small) you are easily able to balance the acceleration and the consequent force by shifting your center of mass, i.e. by leaning slightly in the direction of the acceleration. If the acceleration however changes quickly, you are not able to balance quickly enough and you fall.

The reason, why third derivatives show up so rarely in physics, is that they do not show up in the fundamental laws of physics. One can in fact show that theories that lead to equations of motion with higher derivatives suffer from instabilities (cf. Ostrogradski instability). This concerns fundamental theories only, which means that higher derivatives can in fact feature in effective theories.

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The physical laws are tested against experiments, so two first derivatives suffice according to experiments. It is the main reason.

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    $\begingroup$ Non-constant forces can be very important in describing collisions of objects, especially if strength of materials or human body reactions are involved. Car companies spend a lot of money analyzing the jerk in their vehicles. $\endgroup$ – Bill N Jan 21 '15 at 22:15
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    $\begingroup$ +1. Vladimir is not writing about constant forces. Forces can vary with time. He is just saying that those variations don't need to be explicitly taken into account in the equations of motion. $\endgroup$ – David Hammen Jan 21 '15 at 22:19

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