Braking retardation..MathsGraph Help needed

[anonymous-user]

panholio said:

Most striking thing about this thread (aside from the fact that I love an excel graph) for me is the realisation that a 2s gap at 70mph is about 65m.

Indeed, i would say about 12 or so car lengths. In reality, the vast majority of people are within probably 5 car lengths at 60mph, and some even less!


The crucial fact for me however is to link your distance with observational distance. As your field of view, both in terms of distance and time shrinks, your following distance should increase commensurately!

[anonymous-user]

Toltec said:

This then opens up the interesting idea that as the performance* of modern cars improves the behaviour of the driver becomes even more critical in maintaining primary safety.

Indeed, as technology improves and the max performance envelope of a passenger car grows, unfortunately the gap between the best and worst performing drivers grows too. When no one could stop for toffee back in the 1960's on cross plies and drum brakes, everyone trundled around, left big gaps etc. Now that we "could" stop on a dime, people assume they can too, and hence feel "safe" tailgating etc. Yet in reality, the average driver is just that, average.

A few years ago when my sister bought her first car with stability control, we took it to a quiet empty trading estate on a sunday afternoon and i showed her how it worked. First time around, she simply wasn't even engaging the system at all, and i could get the car stopped, when braking halfway around a wet roundabout at less than 20mph, in about half the distance she could! After just a few minutes practice, she could just about match my performance. That's typical for the average driver imo.

The lack of training and understanding of how a modern car actually works and performs is shocking, especially for new / learner drivers!

[walm]

Max_Torque said:

If the reaction time is less than the "following time" then you always avoid a crash.

Is the right answer.
(Source: it's bloody obvious.)

Say it takes these cars 100m to stop.

So the distance the lead car travels from the point he applies the brakes is 100m.
(Source: that's the definition of stopping distance.)

The distance the following car travels is that same 100m plus however far he goes before hitting the brakes.

So as long as the following car applies the brakes before the point in the road where the lead car started braking he is fine.

[AJB]

PaulD86 said:

AJB said:

Not true. In our theoretical ideal model, both cars are doing the same as each other, whether that's linear or not. They'll track each others non-linearities, the second car just doing the same thing slightly later. It shouldn't affect impact/non-impact.

Are you really sure on this? I really am not. Indeed after some googling I think it really does matter. Thinking about it, we have established that the gap does close I believe and if when the gap is closing the vehicle in front gains an increase in deceleration velocity then this will cause the gap to shrink/potentially become a collision.

Yes I'm pretty sure. Think of it another way. If there was a non-linearity which could cause a collision, then that would mean that if, at the time of the collision, the second car moved across and they went side-by-side, then at that point the second car would be level with the first car but going faster than it. If the cars are identical, then this would mean that the second car would have to stop further down the road than the first car.

Given that the second car started braking at or before the point on the road where the first car started braking, this would mean that the seconds car's overall stopping distance would be longer. But they're identical cars braking from the same speed, so the stopping distances will be the same, and therefore they can't overlap.

In other words, if they collide at any point during their braking, then the second car would stop further down the road if it could overtake instead of colliding. If the second car stops short of where the first car stops then they can't collide at any point.

[AJB]

PaulD86 said:

Also from a bit more research it does seem that a car will start to slow quicker and quicker as it decelerates. This makes sense to me on a logical level as the cars energy is half its mass times its velocity squared so its energy decreases logarithmically as it slows.

I'm not sure why this would be? Without downforce, assuming the driver is braking to the limit of grip, that limit is a result of the tyre/road friction and the weight of the car. Neither of those will change, and so the deceleration force shouldn't change, and so the deceleration itself shouldn't change.

The real world is a lot more complicated than simple theory though, so I could be wrong!

Edited by AJB on Friday 29th November 13:58

[AJB]

panholio said:

Most striking thing about this thread (aside from the fact that I love an excel graph) for me is the realisation that a 2s gap at 70mph is about 65m.

A lot of people on motorways are exceeding 70mph and probably travelling closer than that when it is busy. Maybe we should stop calling it the 2s rule as it is a lot more than it sounds.

Yes - the distance is surprisingly big!

The advantage with "2s" rule rather than "65m rule" is that it automatically compensates for speed - it kind of works at 40mph and 80mph, as it's allowing for a relatively fixed reaction time. Also I'm not great at estimating distances, but I'm better at counting seconds!

[PaulD86]

AJB said:

PaulD86 said:

Also from a bit more research it does seem that a car will start to slow quicker and quicker as it decelerates. This makes sense to me on a logical level as the cars energy is half its mass times its velocity squared so its energy decreases logarithmically as it slows.

I'm not sure why this would be? Without downforce, assuming the driver is braking to the limit of grip, that limit is a result of the tyre/road friction and the weight of the car. Neither of those will change, and so the decceleration force shouldn't change, and so the decceleration itself shouldn't change.

The real world is a lot more complicated than simple theory though, so I could be wrong!

I don't have time at the mo to respond to everything, but quickly for now, what you say about limits of adhesion of the tyre etc are correct, but, say the car stops from 60, when it starts breaking the car has a forward kinetic energy of X, by the time it has slowed to 30 it now has less energy to overcome to stop. The complexity is that the amount of energy it has isn't 1/2 X as it is half the speed but rather it will be more like 1/3rd or less X (I could do an example with numbers later) so the cars tyres have a much smaller amount of energy to stop, hence why the car slows every more rapidly as it decelerates. Sorry, I'm not articulating this brilliantly, but I will try and write up an example later and will see if I can solve mathematically whether the increase in deceleration forces increasing as the car slows would increase the chances of a collision. I could be wrong on this, but the more I think about/read about it, the more I think my theory stands. Maybe someone with a better physics background can add to this.

[Kozy]

AJB said:

I'm not sure why this would be? Without downforce, assuming the driver is braking to the limit of grip, that limit is a result of the tyre/road friction and the weight of the car. Neither of those will change, and so the decceleration force shouldn't change, and so the decceleration itself shouldn't change.

The real world is a lot more complicated than simple theory though, so I could be wrong!

Correct. Assuming the braking is maximum from the outset, it does not increase. The energy dissipated decreases, but that's something different.

[AJB]

PaulD86 said:

I don't have time at the mo to respond to everything, but quickly for now, what you say about limits of adhesion of the tyre etc are correct, but, say the car stops from 60, when it starts breaking the car has a forward kinetic energy of X, by the time it has slowed to 30 it now has less energy to overcome to stop. The complexity is that the amount of energy it has isn't 1/2 X as it is half the speed but rather it will be more like 1/3rd or less X (I could do an example with numbers later) so the cars tyres have a much smaller amount of energy to stop, hence why the car slows every more rapidly as it decelerates. Sorry, I'm not articulating this brilliantly, but I will try and write up an example later and will see if I can solve mathematically whether the increase in deceleration forces increasing as the car slows would increase the chances of a collision. I could be wrong on this, but the more I think about/read about it, the more I think my theory stands. Maybe someone with a better physics background can add to this.

You're completely right about the energy being speed squared, and so the initial braking will have to dissapate a lot more energy than the lower-speed braking at the end.

This works, though, because it just means the brake pads/discs get a lot hotter at the start. They're applying the same torque to the wheel (the limit of its grip), but moving rotating twice as fast at 60 as at 30. That means that twice as much power is going into them as heat (power is force x speed).

So, I think the braking force is constant, the braking power goes down over time at that constant force, and this allows for the fact that the amount of kinetic energy to get rid of also goes down over time.

[Kozy]

AJB said:

Thanks again! I think there must be something slightly wrong though - maybe not enough steps in its calculation or a rounding error. If you put in the same reaction time as the gap, meaning that both cars start braking from the same speed at the same point on the road, they should both stop at the same point. IE the gap should be 0m once they've both stopped. Yours is giving a small, but non-zero gap.

I think this might be the difference between the 8m apart I came up with for the initial example and the 12m or so which your graph showed.

If you're doing it in a spreadsheet, then each interval assumes a fixed speed. Actually the speed is dropping throughout that interval. The more intervals you have the more accurate it should get (I think). Or maybe it would work to have the average speed during that interval instead of the inital speed when calculating the distances.

All interesting though, and good job with the interactive tool!!

Yea you might be right, something doesn't seem quite right with it. Will check it out later.

[anonymous-user]

A conventional passenger car, which has both lift (caused by it's aerodynamics), a large moment of inertia, and a brake system that is a cost compromise, will always brake "harder" in the later stages of braking.

As, respectively:

high speed lifts reduces total normal force applied to contact patch

It takes a significant period of time for the longitudinal decel to load the front tyres full, and the brake system is naturally front biased for stability reasons

The system almost certainly cannot apply enough pad clamping force to reach the limit of tyre adhesion at high differential speeds (pads to disc)

[AJB]

Max_Torque said:

high speed lifts reduces total normal force applied to contact patch

True. I might be wrong as I've never looked at the figures, but I'd have thought that lift was fairly insignificant compared to the car's weight at the 60/70mph type speeds we were talking about though.

Max_Torque said:

It takes a significant period of time for the longitudinal decel to load the front tyres full, and the brake system is naturally front biased for stability reasons

Interesting. I'd never considered that - it does make sense though. I'd have thought that the load transfer would have settled in within maybe the first 0.5s or 1s (complete guess), but I can see that it will make a difference to the initial braking force available until that's happened.

Max_Torque said:

The system almost certainly cannot apply enough pad clamping force to reach the limit of tyre adhesion at high differential speeds (pads to disc)

That seems unlikely to me. I'm sure that if I stamped on the brake pedal of a car without ABS I could lock the wheels at 70mph.

[mikeveal]

Max_Torque said:

Modern cars can stop from 100mph in well under 5sec (even crappy ones, let alone something like a GT3 etc) so even 100ms worth of delay means a big differential impact speed.

Oh no they can't.
I was on a speed awareness course last month and the two highly qualified ex-emergency response drivers patronising us lecturing were adamant that modern cars can brake no better than a four up Ford Anglia.
"Yes, brakes have got better, but cars have got heavier, so the net result is about the same."

[Kozy]

Lift might reduce the reactive effort at high speed but I'd guess aero drag would cancel that out.

[anonymous-user]

mikeveal said:

"Yes, brakes have got better, but cars have got heavier, so the net result is about the same."

Maths fail for those tutors! ;-)

[Froomee]

walm said:

Max_Torque said:

If the reaction time is less than the "following time" then you always avoid a crash.

Is the right answer.
(Source: it's bloody obvious.)

Say it takes these cars 100m to stop.

So the distance the lead car travels from the point he applies the brakes is 100m.
(Source: that's the definition of stopping distance.)

The distance the following car travels is that same 100m plus however far he goes before hitting the brakes.

So as long as the following car applies the brakes before the point in the road where the lead car started braking he is fine.

Assuming the car in front doesn't stop quicker than the car behind..... The reality is depending on conditions, tyres, car type, etc the results will vary.

I was in a friends Corsa (94/95 model) and a friend in a MK6 Fiesta Zetec S stopped fairly swiftly in front for a zebra crossing. My cousin the driver of the Corsa only stopped with inches to spare and he literally slammed the anchors on immediately but the front wheel locked up and it slid just shy of my friends (new at the time) car.

Staying 1s off the bumper of someone is just asking for trouble to be honest and it's only a matter of time before they crash especially if you do it all the time especially at 60-80mph

[Cockey]

So all this assumes both cars brake at the same rate.

How far back would you have to be to avoid going up the back of a fully braking GT3, if you were driving an average car?

[silverfoxcc]

To all who have contributed.
first of all a very heartfelt thanks for the time you have all taken on this, looks like there is going to be a long discussion this Sunday!!

Second, and not meant in any way derogatory, it was refreshing to read each and every post, that was written in a well mannered way, no smart arse replies or observations, just well thought out questions re the problem.
Kozy, a great graph, a vitual bottle of malt, should you partake, for your effort.

makes me feel proud to be part of the PH collective

Each and everyone of you have a safe and enjoyable weekend

Once again Many many thanks

Ron

Edited by silverfoxcc on Friday 29th November 20:04

[anonymous-user]

Cockey said:

So all this assumes both cars brake at the same rate.

How far back would you have to be to avoid going up the back of a fully braking GT3, if you were driving an average car?

You perhaps be surprised at how little difference there is in that respect at least when we are talking about a single controlled stop in a straight line.

(quick googling suggests 60-0mph for a 997 GT3 of 94 feet, vs 110 feet for something like a GT-86)

Basically, all modern cars use their brakes and tyres well!

Obviously, once we get up to mega speed above 120mph, then huge carbon brakes and aero downforce are going to start making enormous differences in braking power

[TA14]

Max_Torque said:

Cockey said:

So all this assumes both cars brake at the same rate.

How far back would you have to be to avoid going up the back of a fully braking GT3, if you were driving an average car?

You perhaps be surprised at how little difference there is in that respect at least when we are talking about a single controlled stop in a straight line.

(quick googling suggests 60-0mph for a 997 GT3 of 94 feet, vs 110 feet for something like a GT-86)

Basically, all modern cars use their brakes and tyres well!

Obviously, once we get up to mega speed above 120mph, then huge carbon brakes and aero downforce are going to start making enormous differences in braking power

That's the first time I've heard a GT-86 called an average car.

Some stopping distances here: http://www.monteverdiclub.com/rahmen/stoptbl.htm average seems to be about 140 feet.