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I'm trying to understand how high-order derivatives (Jerk, Snap, Crack, Pop and onward) impact a system and their implications to analysis. I have used Jerk in the past when considering how "smooth" a system "feels" but that is a super jazz-handsy interpretation of it.

For example in a simple spring-mass system:

Force = mass * acceleration + damping coefficient * velocity + spring constant * position

Where acceleration is the second derivative of position with time, and velocity is the first derivative of position with time.

My question is if you were to add in Jerk (the third derivative of position with time), what would you need to measure or know about your physical system to account for jerk

So the resulting equation would be something like this:

Force = (Jerk coefficient??) * Jerk + mass * acceleration + damping coefficient * velocity + spring constant * position

Mass is a function of the density and geometry, the spring constant is a function of geometry, and Young's modulus, what would the "Jerk coefficient" be a function of?

My third follow-up question would be how to account for Snap, Crack, and Pop in a similar manner.

Qmechanic
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Diesel
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  • Your equation “Force = ...” is incorrect. Newton’s Second Law is $F=ma$, and that’s it. There are no additional terms, unless you use a non-inertial reference frame, – G. Smith Apr 18 '21 at 17:17
  • Possible duplicates: https://physics.stackexchange.com/q/4471/2451 , https://physics.stackexchange.com/q/136880/2451 and links therein. – Qmechanic Apr 18 '21 at 17:18
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    Your equation should have been “mass * acceleration = force = -damping coefficient * velocity + spring constant * position”. There are two different forces acting. In a damped oscillator, there are no forces depending on jerk, snap, etc. – G. Smith Apr 18 '21 at 17:26

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In a simple mass-on-spring system, the motion has time derivatives of all orders. You can compute all of them from $x(t)=A\cos{\omega t}+B\sin{\omega t}$. (Let’s consider the undamped case; the damped one is not conceptually different.) Since $\omega$ is expressible in terms of just the spring constant $k$ and the mass $m$, and $A$ and $B$ are determined by the initial position and velocity, all the time derivatives are fully determined. You need to know nothing else about the system.

The equations of classical mechanics do not involve jerk, snap, etc. so you do not generally need to ever think about them. General motion involves nonzero time derivatives of all orders, but that doesn’t change how you find the motion of an object. Newton’s Second Law $F=ma$ is only a second-order differential equation, since $F$ is generally a function of only position and velocity. If you know the force and the mass, you know the acceleration. From only the initial position, initial velocity, and acceleration at each instant, you can find the motion.

If you are doing engineering, you might have constraints on these higher derivatives, but they have essentially nothing to do with physics, until you get to electrodynamics, where the Abraham-Lorentz force depends on the jerk.

G. Smith
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