If I remember my uni engineering/calculus maths class correctly, the third derivative of position is used in planning these sort of curves.
The first derivative of postion (with respect to time) is velocity. The second derivative is acceleration (ie rate of change of velocity). And the third derivative is jerk (rate of change of acceleration).
And 'jerk' has to be kept below a certain threshold for humans to find movement comfortable.
A very similar thing is done in the creation of reflective surfaces on car bodies (typically in CAD software).
They call these constraints by G and a number.
G1 would be a positional constraint: the two surfaces meet each other at the same point
G2 tangential: same as G1, but the surfaces are tangential
G3: same as G2, but the curvature (radius^-1) of the surfaces is the same at the point where the two meet. This essentially means the curvature combs of the surfaces shall meet at the same position (G1)
G4: same as G3, only now the meeting curvature combs have to be tangential as well
G5: same as G4, only now the curvature combs of the curvature combs have to meet at the same position
And so on. The goal is to create smooth transitions between two separate mathematical surfaces that cannot be seen in the reflections in the sheet metal. E.g. if you think about the connection of straight sheet of metal (curvature: 0) and a cylindrical surface (curvature: 1/radius) the curvature will go from zero to some different value immidiately on the transation you will definitly see this as a hard corner on the reflection or when light falls onto the surface.
Also seen in the planning of curves in roads (where jerk corresponds to the rate at which a steering wheel must be turned) and railways.
And this is also why the passengers jerk of a vehicle jerk backwards after it comes to a complete stop. Their muscles statically counter the relative forwards acceleration of their torsos during braking and require time to react to the acceleration suddenly going away. This effect can be prevented by gradually letting off the brake before reapplying it fully upon stopping, but few drivers and rapid transit systems seem to be aware.
> but few drivers or rapid transit systems seem to be aware
I find that amazing. What the heck are drivers ed instructors doing? It's not just hard on the passengers, it's hard on the machinery.
It's the same with the clutch. I've driven with enough people who fancy themselves as great shifters, but they jerk the hell out of the clutch every time, never attempting to match the shaft speed with the engine speed. If I comment on it, they always deny doing that :-/
If I'm on my game, I can shift smoother than an automatic. The bonus is the clutch will last a very long time.
This effect can be prevented by gradually letting off the brake before reapplying it fully upon stopping, but few drivers and rapid transit systems are aware.
This is surprising to read. Everyone whose car I've ridden in knows to do that, and it's only in extremely urgent and unexpected stops where it's neglected. Also, when fully stopped, only minimal pressure should be necessary to keep the car still.
There's definitely confirmation bias causing me to not notice stops where there isn't a jerk. And perhaps by the Baader-Meinhof effect you'll begin to notice them more frequently, too. But stops without any perceptible jerk are rare enough that when they happen I get an odd feeling of floating forwards like when the train beside your own at a station departs—perhaps it's because my subconscious, still anticipating the jerk, believes that I'm still in motion.
In some cars there's hysteresis in the brake pedal (perhaps caused by the booster or self-energizing system) that makes it hard to smoothly release the brake even when the driver tries to. But metros seem to increase their braking force—visible in standing passengers leaning progressively more—as speed decreases. Is there some physical cause to that?
> only minimal pressure should be necessary
I gathered that it was best practice to fully brake when stopped in case someone hits you, especially at a light where you might be rear-ended and roll into the intersection.
If rear-ended with your brakes engaged more of the impact energy will go into crumbling your car than if it's allowed to transfer into forward speed.
If you see it coming, let go of the brake, and then step on it after impact. That's the advise I got.
It depends on if you're trying to minimize damage to your car, or the passengers. For a light impact where you aren't going to sustain any injuries it might be optimal to let off the brake. But if the accelerations are going to cause injury, then you would want to apply the brakes to minimize the acceleration of the car.
Strategy sounds legit but my concern would be that the (previously distracted) incoming driver attempts to dodge at the last second. Their car will hit your car with torque that could send it into oncoming traffic. But that angled impact also means less push in your car's forward direction, so maybe this concern is overblown.
> Everyone whose car I've ridden in knows to do that
I also do that, and don't know anyone who doesn't. I vaguely remember that I learned it in driving school, most likely because my driving instructor didn't want to be jerked around on the passenger seat for an hour every week.
Train/tram drivers here also usually do that in stations, except when they try to make up for delays, or when they have wrongly estimated the breaking distance.
Re train/tram drivers: Especially with more modern(ish) rolling stock that can also depend on how well the manufacturer has set up the braking system.
For one instance multiple units (especially electrically powered ones) commonly have computerised braking controls, often transition from dynamic to friction braking shortly before coming to a standstill, and might possibly have some sort of automatically applied parking brake.
If the manufacturer didn't properly adjust this whole system, the friction brake as it takes over for the last few kph might be applied with too much of a "bite" and therefore cause a jerky stop which even a skilled driver might not be able to fully prevent.
More pressure won't do any harm, it's just not necessary if you only want to keep the car still (on a flat road, in neutral, with nobody trying to tow you, etc.).
It's a little more complicated than that in a passenger car. The deceleration compresses the front springs. When the car comes to a stop, the springs decompress and the front of the car pops up and the body of the car moves slightly backwards even though the wheels are now stationary.
1) lots of comments are attributing the jerk felt by occupants to the vehicle suspension. But that isn't the case! The occupants visibly move backwards relative to the car body, but the backwards motion of the car itself should rather have the opposite effect.
2) a commenter contradicts what several commenters here have noticed:
> I wager this has to do with the driver; most drivers I've noticed don't ease up off the breaks when slowing down, and so the 'slowing force' feels like it ramps up along the deceleration profile, up until the point when the car comes to a complete start and there's a 'jerk'.
> The occupants visibly move backwards relative to the car body
The only video linked to in that thread has been removed so I wasn't able to see this, but I will point out that the timing is critical to interpreting this observation. During deceleration, the occupants will move forward relative to the car body so that the car can apply a decelerating force to their bodies (via seat belts or friction against the seats and floor of the car). At some point after the car has stopped, they will necessarily move back to their neutral position. This will have nothing to do with the transient motion of the car when acceleration drops to zero.
I will also note that it is not that difficult to come to a complete stop with no perceivable transient if you ease off the brakes as the car comes to a halt. It's actually a useful skill to cultivate IMHO.
This was particularly noticeable to me riding San Francisco light rail to work. When trains run underground they start and stop under computer control. Nice, smooth acceleration and deceleration.
On the surface (or when the computer control system was borked) the starts and stops were a lot less pleasant.
This is fun to try in cars. The drivetrain can have a bit of twist under deceleration, and you can feel it spring back after the wheels stop. For best comfort you need to gradually reduce the braking force not just for human comfort but also to relieve that twist.
OT: that reminds me of an interesting physics problem.
If you have a ball sitting on the floor in the aisle of a stationary bus and the bus starts accelerating forward the ball rolls toward the back of the bus. If you have a bus moving at a constant velocity and it start decelerating the ball rolls toward the front of the bus.
Suppose you also have a helium balloon floating in the bus. Does it also move toward the back of the bus when the bus accelerates and toward the front when the bus decelerates? Or does it stay where it is? Or does it move toward the front when the bus accelerates and toward the back when the bus decelerates?
The balloon would probably move toward the front when accelerating and toward the back when the bus slows, if it moved at all.
It helps (me) to imagine an air bubble in a sealed, nearly-full fish tank on that same accelerating bus. The heavier water gets "flung" harder away from the direction of acceleration, and the bubble gets pushed out of the way in the opposite direction. Same principle.
Given that train drivers have cameras pointing at passengers, I'd argue that they are indeed aware of the effect, but don't always roll the stop due to delays on the line and/or personal reasons.
> This effect can be prevented by gradually letting off the brake before reapplying it fully upon stopping, but few drivers and rapid transit systems seem to be aware.
Is this why it seems to be mostly Americans that are into the idea of self-driving cars, because the standard of driving is so low?
I saw a shirt the other day that said there’s no place like G28 0 0 0. I think for now that wins my nerdshirt championship (I still like my “velociraptor = distraptor / timeraptor” short though, even if it fails dimensional analysis. =)
And 'jerk' has to be kept below a certain threshold for humans to find movement comfortable.
It's not strictly a matter of threshold -- people might tolerate a higher jerk if it's for a much shorter duration, for example. In practice it doesn't much matter which metric you minimize; you'll end up with similar results. The simplest option is to minimize the mean absolute jerk, which has the side benefit of utterly confusing any non-physics-literate people listening in. (You want to do what to whom?)
Note that 'mean absolute jerk' can also be described as 'total variation of acceleration over time'. That implies it does not depend on how fast the acceleration changes, only about the difference between minimum and maximum (as long as the acceleration increases/decreases monotonically to/from the maximum). This may or may not be what you want.
It feels wrong to me that pop comes after crackle. Crackle seems like the ultimate high-frequency effect. In fact, "pop" seems like it should come before "snap". But I guess it is somewhat arbitrary.
Note that the narrow-loop shape in modern roller-coasters means that the tightest curve is at the top, where the G-forces are partially countered by gravity. That's an 1-G that they can subtract, and it means the radius can be significantly larger on entry to the loop, resulting in smaller G-forces.
This free 1g of turning force is apparently known among pilots as “God’s G”. It needn’t be produced as lift while you’re inverted at the top of a loop, so you get a good turn rate despite losing speed and hence lift as you get up there.
Not just gravity, but the train also loses speed at the top, so the loop needs to tighten if you want to maintain the same force (you might not want this).
An inverted loop, where the loop is lower than the track, and with a very long train, where the center of mass continues to decrease in height due to the part of the train not presently looping.
Certain modern(-ish) roller coasters do have more circular loops than others. Specifically Schwarzkopf[1] coasters are famous for having more circular loops (and the more intense positive Gs that come with it). Anyone in the Bay Area might remember Zonga[2] at Six Flags Discovery Kingdom which featured the more circular loops.
Also maybe of interest is Blue Flash [3], a backyard roller coaster that has a loop that reminds me of old school circular loops.
> More people paid to watch others ride these early coasters rather than ride themselves
I had no idea roller coasters have been around this long. The photos are laugh-out-loud terrifying. I was shocked that anyone would pay to ride them until I read the quote above. Now it makes sense. I’d pay to watch that too!
Note how large the cars relative to the tiny people standing in them. This thing was unimaginably massive. It's easy to think of 1893 as being before the modern technological era, but we were more modern than most people like to think.
Having first read your comment, the photo wasn't as bad as I expected. The supports don't look too unreasonable for a single wooden car with two passengers. Modern trains are an order of magnitude heavier--~20 people and a gross weight of ~20 tonnes.
If you're trading a horse for the car, a seatbelt might not be the first thing you'd think of. I don't think our views on danger have changed, it's just that any new technology requires a period of adjustment to it.
Idk how old you are, but I have ridden on public roads in the back of a pickup certainly as late as the 2005 as a child. I think it might be more common than you think in some parts of the country
We have a very substantial part of the population becoming chronically ill after an infection of a disease labeled “mostly mild”, yet we decided low-key interventions such as masking and air filtration are not worth it.
When I was a kid, we used to rid maybe 6 kids in the back of a pickup truck flat bed.
where i live, because of paucity of transportation means, it is not uncommon to have the back door of a car straight open and 3 kids sit there, dangling their feet. that is considered very normal in these parts of the world.
We also used to knock on wood to prevent bad things happening. I think an argument could be made that we shifted from interventions that dont work to interventions that do.
We have more science, more education. 10% of kids under age 1 would die at the turn of the century (1900), which means, everyone knew a family with a child that had died, and that's traumatic.
Our views of safety and risk have evolved.
My grandfather, he worked in a lumber camp, the stories ... my god.
We also react to it differently now. Even in our language i.e. 'I feel unsafe' or people talk about 'harm' or 'violence' with just language.
Thus is similar to corners in roads not working well when they're circular arcs. Going from straight to circular arc means your steering wheel needs to jump instantly from one fixed position to another. Using something like a linear change in curvature (i.e. a clothoid curve) is much smoother. This kind of thinking applies to other areas too: https://en.wikipedia.org/wiki/Euler_spiral
Smooth often means that higher derivatives are continuous as well. A straight line changing into an arc has continuous first derivative, but the second is discontinuous.
A "G force spikes" is called a "jerk" in physics; as acceleration is the rate of change of speed, jerk is the rate of change of acceleration. The jerk is huge in transitioning from a straight track to a circular one immediately, since the acceleration goes from zero to nonzero instantly.
If I recall correctly the list of derivatives and derivatives of derivatives goes like this: position, velocity, acceleration, jerk, snap, crackle, pop.
A similar principle works in UI design and industrial design. Rounded rectangles often just slap circle arcs on the corners—but that looks unnatural because you don't get such a shape if you bend a straight rod. In this approach the turning radius changes from infinity to a constant at one point, but the proper way is to change it gradually. Bezier curves accomplish that, I think.
Once you see this, you begin to notice it everywhere, just like with kerning. Apple used both ways on iPhones for rounding the phone's corners, iirc. Also I've been told that the principle applies to road turns: you don't want people to have to suddenly turn into the curve. (In related news, I wish a month of bad hiccups on Herman Tilke.)
I once implemented rounded rectangles by drawing circles with regards to arbitrary p-norms [1,2] – I did not think about it at the time, but the derivatives are probably quite nice. (It did have the advantage of using the same code for rounded rectangles and for circles.)
Pretty sure that's how squircles are supposed to be mathematically implemented—at least in the option based on superellipses: https://en.wikipedia.org/wiki/Squircle
IIRC from when I read about this concept for the first time: Apple managed to revert to bad rounding on the phone shape at some point. And probably reverted back later, but I can't confirm that. Basically, the reader is advised to check the rounding before buying their next phone.
I really want to like Vox's video format but just can't. Which is a shame because they have lots of topics that are quite interesting.
I can't pin down which part that I hate most, the "stylish" presentation with random SFX and virtual pen circling around, or the fact they always insert some super low quality video conference footage instead of just letting the narrator paraphrasing (I get it they're the ___domain experts, but still..).
I'm not high now and I read the article they cited for that fact. Nowhere in the article does it say the first loops were a circle. In fact, it says the opposite:
>New York City's Coney Island, home to several amusement parks, followed with its own looping coaster in 1901. Using an ellipse rather than a circle for the loop,
I remember the Corkscrew at Knotts as a kid, no longer exists. The Revolution at Magic Mountain arrived in the late 70s with the new parabolic shape and still running. (Southern California)
I looked up this term to be sure and I'm convinced it's as meaningless as I thought it was and is a strange way of saying “force” or in this case a centrifugal force.
G-force is just measuring acceleration in units of G. It's a commonly used measurement because you've got an intuitive feeling for what one G feels like.
14 G's! That would knock you out. There's no way it was that high. I looked it up and it says they tested it on chimpanzees. I'm sure the monkeys were absolutely thrilled.
The first derivative of postion (with respect to time) is velocity. The second derivative is acceleration (ie rate of change of velocity). And the third derivative is jerk (rate of change of acceleration).
And 'jerk' has to be kept below a certain threshold for humans to find movement comfortable.