It is a simplified version of the formula...and not simplified like Navier Stokes to Euler but more like something you'd learn in high school physics. It is very good at illustrating the top level concept and getting the first order answer, and specifically does that by hand-waving away of all the perturbations (which is exactly where things like the nuances of the braking system come into play).
Since you posted it though we can still use it, because it actually does in fact support the notion of bigger brakes being better anyway. See that "f" in the equation? That's the coefficient of friction. That's not a constant. In fact, that's where many of those pertubations roll up. Going back again to first year high school physics, friction comes in two forms: static, which is higher, and dynamic, which is lower. You see this all the time in life--its a lot harder to get something moving (Static) than it is to keep it moving (dynamic) . A rolling tire has no relative motion to the ground (more or less) and thus falls under the "static" realm. Conversely, a locked up tire is moving relative to the ground and thus is dynamic and has a lower coefficient of friction. Apply that to a car, and it explains why it takes longer to stop a car when you lock up the brakes than it does when the tires are just at the limit of 'still rolling'--the higher friction in the rolling tires enables more braking force. (There's even more nuance when you add in things like tire slip, but since that just supports my position even more we can skip past all that.)
In fact all that is why ABS was invented in the first place--to better keep the tire near that 'still rolling' limit and not beyond into the locked up condition.
So far so good... here's where you go off the rails...
As previously discussed we know that bigger brakes enable the ABS to more finely control the wheel speed;
Except no, we don't.
We didn't "discuss" that.
You
claimed that- and then when asked for sources to support that claim (which would be contrary to all the sources I provided) you failed to provide any- or even a physical explanation of how or why that would be true- instead you simply repeated the claim and pretended anyone "discussed" it.
that of course will increase the amount of time the tire can spend near its maximum braking force, and that enables the car to slow down faster.
It would if somehow bigger brakes actually did that- they don't of course (and physically can't)
I again suggest you re-read pulp friction.
It explains what bigger brakes can and can NOT actually do.
An adjacent analogy to this point is taking two identical modern cars and magically replacing one car's ABS computer and control unit/modulator with units from, say, the 80's. Same tires, rotors, calipers, suspension, master cylinder, vehicle mass, etc. Both versions of that car will lock up the brakes and engage ABS in an emergency stop. One version of that car is going to stop shorter than the other in an otherwise identical emergency stop, and that's because (in this example) technology has enabled much finer control over wheel speed.
How does that help your point?
That's due to better ABS sensors taking more readings more often and the computer controlling the system being improved- in your example the
actual physical brakes are the same.
So it's an incredibly weird way to try and suggest upgrading the
physical brakes would do that since that's not, at all, what happened in your example.
Indeed, you're absolutely right that more force doesn't help once you've applied enough brake force to maximize stopping force. Nothing I've said suggests otherwise.
Except the part where you claim larger brakes help. Several times. Never once offering any evidence of that actually being correct- and in the fact of source after source telling you it's not.
Again, the value in the bigger brake is the additional control they provide.
In court this is called "assuming facts not in evidence"
When ABS 'engages', what its really doing is relieving system pressure by a small amount over and over again very quickly. It basically lets off the brakes for a split second when it senses the tire has transitioned from static friction mode (rolling) to dynamic friction mode (locked up), which allows the tire to start rolling again and transition back to the higher friction mode. Importantly, that cycle is a function of the system's ability to control based on the inputs.
Here again you provide a fairly accurate explanation then go off the rails with-
Adding more control--like, by adding bigger brakes--reduces the cycle time and magnitude
This is utter nonsense you've once again provided no evidence for.
How does "bigger brakes" let you come OFF the brakes faster (the bit you just described in ABS)
We know it doesn't help fully engaging the brakes- since even you admitted more force once you have enough to lock the wheels does not help.
So coming off is the only bit left in your story.
How, specifically, do "bigger brakes" help you REMOVE braking force quicker?
Make no mistake, I'm fully aware that this definitely does not move the needle on the first order answer. We're probably taking a few inches...maybe a foot at best.
Or it's zero. Because that's what even the people who
sell big brakes say it is. Also the people who design braking systems for OEMs for a living. Also Road and Track, Car and Driver, the folks who tested the $10,000 Porsche PCCB upgrade, and basically everyone else who understands brakes and physics.
To that end I wholeheartedly stand by a statement like "bigger brakes won't really make a difference in an emergency situation on a modern car".
Then it's
really weird you've spent multiple posts trying to insist otherwise...including most of this one I'm replying to now.
It is the resolute insistence in turning that kind of first-order, hand-wavey statement into a binary truth that I find damaging in this kind of fourm. As it were, I'm a sucker for calling out misinformation.
Then again- please provide a source for your claim "bigger brakes" make even the small difference you claim they do- in contradiction of expert after expert saying otherwise.
Because if not you appear to be posting exactly the sort of claim you just said you like to call out.
Anyway, back to sourced facts
Which are that in some cases a big brake kit will make ABS performance WORSE
Here's stoptech explaining why-
http://www.stoptech.com/technical-support/technical-white-papers/abs-and-big-brake-kits
It explains (a bit simplified, though in more detail than your explaination) how ABS works- and why aftermarket brake kits that change the factory pressure-torque (P-T) relationship of the caliper/pad/rotor assembly and the pressure-volume (P-V) characteristic of the system, on which the ABS computer is depending can cause you problems.
It also explains that the better big brake kits specifically are engineered to offer the
as similar as possible P-T and P-V setups as the factory brakes... so that you'll be able to get the
same ABS performance as factory brakes, not worse.
At no point do they suggest the bigger brakes which, again, they are
trying to sell you can
in any way actually
improve ABS performance over the factory ones.
They can only match them, or for cheaper kits, make them worse.
If they could
improve stopping distance, at all, they'd be
highly motivated to tell you.
They don't because they're an honest brake manufacture. (As is Brembo if you go back and read the bit I posted from them about why they don't provide stopping distance test numbers- because they admit bigger brakes
can't stop the car any shorter than factory ones.)
