Beyond velocity and acceleration: jerk, snap and higher derivatives (2016)

Do you have proof for that, or is this like audiophiles asking for gold connectors because "they make the sound better"?

Not proof, but jerk is a factor when bringing a car to a smooth stop. You have to learn how to brake smoothly in order to avoid the "drivers ed stop" where the car and its passengers lurch forward and then bounce back. But the controls for automated vehicles like airport trams have to be designed to avoid this. The underlying reason is that some components such as the tires and suspension are elastic.

This is in fact an issue for the designers of controls for mechanical systems. I learned about it in Process Control class, albeit 40 years ago.

Anecdotal evidence:

Ever experienced that a bus is braking (near-constant deacceleration), and people seem fine; but then the bus comes to a halt and thus stops deaccelerating, and people suddenly fall on the floor?

I think at least the derivative or acceleration is important for how well people can compensate. Not sure about higher derivatives though.

It's broadly recognized that minimizing jerk and snap is important to comfort in roller-coasters, so there is evidence for that proposition:

e.g. https://iopscience.iop.org/article/10.1088/1361-6552/aba732

Acceleration equals force, so yeah, if you abruptly change acceleration then this equals abruptly changing the force on people in the bus. Acceleration should thus be continuous (not necessarily differentiable). I don't know how you would justify constraints on higher derivatives. Perhaps they mess with our own internal control mechanism?

is it physically possible to have non-continuous acceleration?

Cars 20+ years ago vs more recent cars - I've definitely noticed them auto-doing what I was taught to do with older cars: ease off the brakes right at the end.

I used to work at a self-driving car company, and all the vehicle's motion was planned around how much jerk to apply.

Your muscles are pretty good at applying a constant force (or responding to a constant acceleration). Hold your arm out straight: it's no effort to keep your arm still and counteract the force of gravity. Now imagine gravity varies quickly and randomly between 0.5g and 2g. I guarantee your arm won't stay still.

The same prinicple applies on a bus or in a car, except this time the forces are smaller, and it's your neck keeping your head still!

In railroad design it is important for the track to not be a curved segment of a circle (starting from a straight line), as the acceleration forces start suddenly (aka a high jerk). So this concept exists and is well known in some circles (heh).

It is an active research topic in train engineering.

Imagine a multistage rocket and the changes in acceleration.

Figure 4-3 in https://www.ibiblio.org/apollo/Documents/lvfea-AS506-Apollo1... shows this for Apollo 11.

>Do you have proof for that, or is this like audiophiles asking for gold connectors because "they make the sound better"?

The proof is that roughly 100% of cars have components designed to limit this.

I imagine if you zoom in far enough on those points you have the acceleration continuously changing as pressure slowly builds in the engines over several microseconds.

I was thinking more of the instant you shut off engines and disconnect 130,000 kg of mass of stage one.

There is an interesting Δa/Δt while fuel is consumed and mass changes.

There are discontinuities to the graph when engines are shut down and stages decoupled.

If we zoom in on a single electron absorbing the momentum of a single photon, it will accelerate “instantly”. The same goes for e.g. an unstable atomic nucleus that ”splits”.

At macroscopic scales, I’m not aware of exactly instantaneous acceleration, since you would need some time to “sync” the movement of each atom in the object. But some processes will of course look instantaneous at any given time scale.

Ok, minimizing jerk makes sense, but how about snap and crackle? Because GP said:

> (...) jerk, snap and crackle. Turns out if you minimize those, you get a far more comfortable ride.

Voltages can change abruptly. Therefore, forces can change abruptly, and hence acceleration as well.

Snap and crackle I couldn't tell you about. But jerk is definitely important.

I wonder if this is a change in braking material, specifically a reduction in difference between dynamic coefficient of friction and static coefficient of friction between the pad and rotor (or equivalently, the shoe and drum).

If older cars had a higher differential, you’d need to let up more as the brake finally locks up.

That’s the essence of a legitimate question: over small enough time periods (as the bolts explode over a non-zero period of time), is it continuous or discontinuous?

Over a macro scale, it’s discontinuous, of course.

Don't forget vehicles got heavier, rims got bigger/rubber has thinner sides, suspensions geometry evolved and got stiffer (and possibly non-linear, at least on the high end) and so on and so forth, reducing the amount of elastic energy.

There's mechanical braking assistance (not just ABS) which means pressing the same pedal distance may produce different breaking strength depending on the speed at which the pedal is pressed; e.g pressing hard triggers force assistance from, say, a vacuum reservoir that reuses engine pump loss, which means conversely pressing lightly for a normal stop does not need to exert as much pressure, hence an eased in stop.

Also with more stable vehicles with better chassis, suspension, and overall balance, I feel like rear braking has been tuned upwards over time, making for a more stable stop: notice how lightly pulling the handbrake has a straight-rolling car "sitting" instead of "diving". More consistent use of disc brakes instead of drums on the rear end certainly helps, as well as the ability for the vehicle to remain stable even when braking while in a turn.

Regarding brake friction itself, I can think of at least one major change: the ban of materials such as copper or asbestos in brake pads.

I think bus is braking with a constant breaking force.

But, the bus has a non-constant kinetic energy (going up with the velocity*velocity, down as velocity goes down.)

So, you're actually producing a non-linear acceleration. This is jerk, but you can also think of it as just a non-linear acceleration and people are reacting to the fact it's not at all near constant deacceleration, and this is most noticable as velocity hits zero.

So, yes, it's jerk, but no, I think it can be intuitively better understood with pure acceleration terms and no jerk needed

It's nature, it's continuous at small enough scales.

But, checkout Zeno's paradox for more on your philosophical questions

I was thinking of both the changing of material composition of existing organic or semi-metallic pads, but also the general drift towards ceramic pads for low-dust.

Some of the German marque factory pads have exceptional initial bite, coupled with exceptional high levels of dust.

Aren’t you describing infinite acceleration, or discontinuous velocity?