Seventy versus One Hundred
solipsys.co.uk
solipsys.co.uk
It illustrates an important principle, the quadratic dependance of KE. It encourages the argument to lower speed limits.
All good things.
But the vehicle dynamics are wrong.
Vehicle stopping force depends on tire traction not the brakes [1]. You'd think it's a wash but tire traction depends (non linearly) on the normal force which is vehicle weight plus aerodynamics. At 100 kph, aero matters.
Aero can go either way. A ferrari or a rice racer with a massive wing will get decent down force and have improved braking performance lowering final impact speed. A van will probably have lift and therefore worse traction.
Finally there are drag effects, not at all negligible at 100kph.
[1] obviously; brakes have hydraulic calipers. As long as they're at operating temperature, they're fine. Brakes are sized for far more abuse than a single emergency slam. If your brakes can skid the tires, the brakes are not the limiting factor.
A memory of a groundbreaking car in the history of racing there...
A video on the the Chaparral 2J - which has some interesting things with its downforce. https://youtu.be/WDaX3gFop7I
From Wikipedia - https://en.wikipedia.org/wiki/Chaparral_2J
> The skirting produced a zone within which the fans could create a vacuum producing downforce on the order of 1.25 to 1.50 g when the car was fully loaded (fuel, oil, coolant). Tremendous gripping power and greater maneuverability at all speeds were produced. Since it created the same levels of low pressure under the car at all speeds, downforce did not decrease at lower speeds. With other aerodynamic devices, downforce decreased as the car slowed or reached too much of a slip angle.
I was thinking more of a road going Ferrari. But, wing, fan or whatever, the principle is the same: as soon as the normal force on the tires is anything other than the weight of car, the quadratic model brakes down.
We also know that the average velocity for each car equals the distance travelled over time elapsed, where the distance is equal for both cars, but the time isn't. So, 70/2 = d/t1 and (100+v)/2 = d/t2.
Can then solve these equations for v ~= 71.4
The gotcha is that it's much easier to solve if you take both distance and time as constant for both cars, but that would be a mistake.
Of course you could get even more meta with it. Since the faster car is going to reach the obstacle first and smash into it, it probably won't be in the same position when the slower car reaches it either!
But again, it doesn't matter, because road cars are traction limited in braking.
Work is by definition (work) = (force) x (distance), or the same thing expressed as an integral.
Distance is the same, that’s a parameter of the problem. If we say that the brakes apply constant deceleration, then that implies that the force is the same, between both cars. Then the amount of energy shed by both cars is the same before they reach the barrier. (We don’t need to use the integral definition for work here, because force is a constant.)
This explains why the people who assumed that both cars dump the same amount of energy arrived at the same result, even though some of the people in this thread appear to be using faulty reasoning.
For example, from here:
https://www.automotive-fleet.com/driver-care/239402/driver-c...
At 50 mph braking distance is 125 ft.
At 60 mph it's 245 ft
At 80 mph it's 439 ft
Now, when you consider the emergency circumstances where you might be flooring the brakes, this is not good ...
e.g.
A deer or child runs out in front of you, so close that emergency braking is called for .. how far away are they going to be from you ?
Or, say the car in front of you on highway slams on their brakes, and they were travelling at 50mph, with you approaching at 60mph. Unless you are at least 120 ft behind them (vs the typical highway following distance of a few car lengths), you are going to be in a bad head-on collision.
> At 60 mph it's 245 ft
> At 80 mph it's 439 ft
It looks like you're jumping between columns and sometimes transposing rows here. 125' @ 50 mph is from the raw braking distance column, 245' is from the same column but for 70 mph rather than 60 mph, and 439' @ 80 mph is from the overall stopping distance column. I'm not pointing this out just to nit, rather because:
> say the car in front of you on highway slams on their brakes, and they were travelling at 50mph, with you approaching at 60mph. Unless you are at least 120 ft behind them (vs the typical highway following distance of a few car lengths), you are going to be in a bad head-on collision.
You've concluded nearly twice the stopping footage as the table actually says and, as is more directly said in the last column, it's really roughly a 4 car braking length difference between the two speeds (including reaction time).
I think the point still stands though - by definition in an emergency situation there is not going to be much distance between you and that deer/whatever, and an extra 10mph can make all the difference!
From the follow-up post, the author gives a justification for that.
> the force applied by the brake pads on the brake disc is going to be roughly constant, and so the work done, which in this case is the energy shed, is the force times the distance over which the force is applied.
There will of course be confounding factors like temperature, etc. But it's probably pretty close.
The first car loses kinetic energy equal to 1/2 * m * 70^2 while the second car starts with 1/2 * m * 100^2.
Assuming the second car loses the same energy as the first car while breaking, the formula is: 1/2 * m * 100^2 - 1/2 * m * 70^2 = 1/2 * m * v^2 or v = sqrt(100^2 - 70^2) = 71.4 mph
Interesting (and I think very reasonable) first-order assumption. Maybe to second-order, the calculation could assume the braking force is proportional to velocity, which I think is roughly true of friction generally, but is also harder to model.
> Interesting (and I think very reasonable) first-order assumption.
Really? That triggered alarm bells in my head immediately. I mean, it might be true, but I'd need to do the math to figure it out one way or another.
> the calculation could assume the braking force is proportional to velocity, which I think is roughly true of friction generally
Again, not my intuition at all. I'd have gone with the braking force being constant at non-zero velocity. (And force is variable up to the limit of static friction when at zero velocity.)
* It may be reasonable to assume the brakes produce the same force on both cars. With both cars weighing the same, this produces the same acceleration. Then the speeds decrease at the same rate, which means the faster car is going 100 - 70 = 30 kph when the slower car stops.
* The faster car must dissipate more energy to do this. We may assume that braking is limited by heat dissipation and that the brakes shed energy at the same rate. Then the answer is sqrt(100^2 - 70^2) ~ 71 kph.
Adding the obstacle:
* As noted in the text, brakes don't shed energy at the "same rate per unit distance". So the faster car hits the obstacle first, at greater speeds than above in each case. We aren't given the information required to calculate the stopping time or distance of the slower car, so the problem is underspecified.
There's too many unknowns in this, to me. Do brakex primarily shed speed, kinetic energy, or momentum? Is there a secondary effect and if so, how large? Is any of this linear? There's friction here and that can be notoriously non-linear. Would this change for disc brakes versus the other kind? I know this varies for older versus newer vehicles. What about the tendency of the car body to lurch forward and tilt? Does that affect things?
...
"(was that far enough?) "
oh man, i was hoping this was going to be one of those tricks of no matter what your window size was, this would always be just below the fold with CSS. i just got lucky for it to line up. after trying it again with a smaller window, it was just in the scroll. =(
then again, looking at the site's layout, it does not really suggest heavy use of CSS. at. all.
(But probably read it because it's really short)
Stopping distances are part of the theory test in the UK, albeit with nonsense values. The frustrating part for me is how they completely ignore differences due to vehicles and gloss over road conditions.
- For most drivers, more tread is better, and more channels (and a variety of sizes thereof) are better. CA roads don't shed water very well for example, so in heavy rains you need the tire to get the water out from between you and the road.
- Assume tire manufacturers are pareto optimal. Grippier, safer tires have lower MPG and expected lifetimes. The latter is usually somewhat advertised. Buy the 30k tires instead of the 80k mile options.
- If you're not driving in cold (<15F or so) weather, don't get all-season tires. If you are driving in cold weather, get snow tires (better, Nokian studded options don't tear up the road and offer excellent grip if they're legal in your locale). If you are driving in cold weather and not springing for snow tires, do get the all-season option.
- Properly inflated tires matter a lot more than you might think, making cheap tires outperform good options. Checking (or asking a mechanic to check for free) after any major temperature swings or every 3k miles or so is a good idea.
- Several tire sizes fit most rims. The wider options perform a bit better (total tire grip scales sub-linearly with road pressure, so distributing the weight over more area slightly improves grip). You won't pick up more than 3-5% with this parameter, but all-else equal (and if you don't mind worse fuel economy) it's good to know.
The best tires vary from year to year, and there's an inherent tradeoff between the different driving conditions you'll subject them to and how many sets of tires you want to keep around (most consumers want 1 set, most northern consumers at least recognize that 2 sets isn't a bad tradeoff if you have the space for as much of a grip improvement as proper snow tires offer). If you have a competent mechanic you trust, just tell them how you drive and ask what they would do for their car. They test drive a dozen cars a day and have a pretty good feel for what works well enough and gives you a good bang for your buck.
Actually answering your question now, AutoBild, Consumer Reports, Tire Rack, MotorTrend, and a number of other organizations test at least a few models (some of those test lots of models) of tires each year. Actually ranking the best from second-best is hard and maybe prone to bribes. Generally though, they provide objective measurements, and those measurements correlate pretty well with how the tire will handle when placed on your car. If a few choices are close, choosing based on what you can easily find locally or price or shipping or other factors is probably a good idea.
I always assumed friction and thus deceleration scaled linearly with contact surface area
1) Main slowing effect is braking system converting kinetic energy to heat dissipated by the brakes, and maybe back to chemical energy in the battery in the case of an EV with regenerative braking.
2) As the vehicle slows due to braking effect there's likely to be a mismatch between cars forward speed due to momentum and the rotational speed of the slowing wheels, and the role of tires is to try to maintain a grip on the road and prevent this from turning into a skid. Presumably there's energy being dissipated in the tires (and road) in the form of heat too, but only so much the tires can do if braking is too aggressive.
3) Either the driver, or anti-lock braking system, can pump the brakes to prevent the wheels from locking up since that's not the quickest way to stop.
So, tire contact area with road, and tread pattern, and tire material/construction will all play a role, but overall deceleration of the vehicle is based on a combination of all three of the above factors.
The usual metric describing that transition is the "coefficient of friction." It describes the ratio of normal forces (your car pressing down) to frictional forces (your tire slowing down your car) at which you transition from static (good) to kinetic (bad) friction. Tires have a CoF which depends on the pressure (roughly car weight divided by tire surface area, complicated by the fact that wider tires don't necessarily have more surface area on their contact patches). As pressure increases in rubber tires, CoF decreases, so the same normal force (the gravitational force of your car pushing downward) results in less peak frictional force (the ability of your tires to stop you moving forward). Increasing surface area decreases that pressure, increasing CoF, increasing braking ability.