Braking retardation..MathsGraph Help needed

[AJB]

Kozy said:

AJB said:

Don't think this is true. 1s gap = 27m. If 0.7s thinking time, 2nd car starts braking 0.3s further up the road. 0.3s = 8m. Both cars do exactly the same thing, but with the second one 8m further back and 0.3s later. I think the second car will stop 8m behind the first one, 0.3s after it does.

There is no thinking time for the front car if the following car is simply reacting the brake lights.

Every 0.7s, both cars drop 7m/s off their speed. With the following car starting earlier, the gap will close.

Yes they'll close. When the second car starts braking they're 8m + however far the first car went in 0.7 seconds apart. By the time they stop they're only 8m apart. So closing, but I don't think they'll hit.

[Kozy]

Did it properly on a spreadsheet.

[AJB]

Kozy said:

Did it properly on a spreadsheet.

Good job!!! That's what I should have done!

From a quick glance, should green gap line be offset by how far apart they are at the start? The gap isn't 0 at time 0 (because the 2 distances start from different places on the road). I may have missed something though - like I said I've only looked quickly.

Thanks for putting the effort in!

[Kozy]

Good point.



It looks like 1 second would be enough, but there's not much margin for error.

I'll knock up an interactive version of this for my website I think.

Edited by Kozy on Friday 29th November 11:10

[anonymous-user]

What you'll find is that as long as the "reaction time" is less than the "gap time" then the second car will always miss the first car, what ever the speed or brake decel used

This is because both cars take the same distance to stop once the brakes are applied, but the second car continues to travel at the initial speed for the reaction time. Hence, the speed cancels and if the gap is less than the reaction time, the cars overlap at some point during the braking!

[anonymous-user]

Also, a much more 'real' simulation is to base the brake force application for the second car on the closing speed to the first car. For example, car 1 starts braking at full effort, car 2 waits for reaction time (say 0.5sec) but only applies 0.3g brake force, then, only as the differential speed between then increases does the second car gradually apply full brake force!

That one^^ is a bit of an eye opening as to the differential speed of the impact when gap = zero!

(this is because braking hard is more important at high speed, where you travel further for every second than you do at low speed. Hence it is proportionally more difficult to "make back some distance" as the average speed of the two cars falls

[Kozy]

Here we go then, tweakable version of the above chart:

http://blackartdynamics.com/StoppingDistances/Arti...

[Monty Python]

What's "brake acceleration"?

[AJB]

Max_Torque said:

Also, a much more 'real' simulation is to base the brake force application for the second car on the closing speed to the first car. For example, car 1 starts braking at full effort, car 2 waits for reaction time (say 0.5sec) but only applies 0.3g brake force, then, only as the differential speed between then increases does the second car gradually apply full brake force!

That one^^ is a bit of an eye opening as to the differential speed of the impact when gap = zero!

Completely agree with both of Max_Torque's posts. The above is also one reason why you need a bigger gap if you're following something which you can't see through (van, high 4x4, car with blacked out windows etc). If your only point of reference is your approach speed to the vehicle directly in front, then you don't brake hard enough early enough and the gap disappears scarily fast. If you can see what it's slowing for, then you get a lot more of a heads up and can base brake pressure on your approach speed to the car 2 or 3 in front and things end a lot better.

It's one thing I don't like about dark tints on the rear windows of cars these days - I hate following them on busy roads.

[AJB]

Kozy said:

Here we go then, tweakable version of the above chart:

http://blackartdynamics.com/StoppingDistances/Arti...

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!!

[PaulD86]

Just to add, an assumption that is being made is that a car will decelerate in a linear fashion losing speed at a constant rate. It won't. This effects the calculations on whether the cars will impact. [edit - I think the rate of deceleration actually increases as you slow making the car behind hitting the one in front even more likely if I have got my sums right!]

My work make us do an annual driver assessment as driving has been identified as the most dangerous thing we do at work (well that's what we are told the stats say at any rate). The guy who owns the company that does the assessing (as well as offering skid car training, advanced driver training etc) and I were chatting at length the other day and safe following distances were brought up. Our company policy is that a 4 second gap must be left. To me this seems a fair bit (at 60 it's about 100m). The guy I was speaking to who is ex traffic police and does advanced driver training said that for the most part he agreed that 4 was more than needed and indeed that a 4 second gap on a motorway, for example would probably be quickly filled by another vehicle. He said that the 2 second rule is a very good one and that as a minimum is fine for most instances. But he also said one of his colleagues is a fan of the 4 second rule. Both highly qualified and advanced drivers and both with slightly differing opinions. But the consensus is clear that a minimum of 2 is sensible.

Edited by PaulD86 on Friday 29th November 12:49

[anonymous-user]

PaulD86 said:

Just to add, an assumption that is being made is that a car will decelerate in a linear fashion losing speed at a constant rate. It won't. This effects the calculations on whether the cars will impact.

The basic assumption is that once braking is commenced, both cars have exactly the same deceleration profile vs time. Hence the actual shape of that profile is, in this simulation, irrelevant. But, in the real world, then yes, the way the brakes are applied is critical in extremis!

[AJB]

PaulD86 said:

Just to add, an assumption that is being made is that a car will decelerate in a linear fashion losing speed at a constant rate. It won't.

Not sure - I wouldn't be surprised if it wasn't fairly linear.

PaulD86 said:

This effects the calculations on whether the cars will impact.

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.

PaulD86 said:

But the consensus is clear that a minimum of 2 is sensible.

Very much agreed!


Edit: beaten to it!
Edit again: I meant linear with fixed brake pressure. In the real world humans get involved with the brake pedal and it'll be nowhere near linear!

Edited by AJB on Friday 29th November 12:55

[anonymous-user]

Finally, as brakes become better, the reaction time becomes more important, because the total deceleration distance is shorter! 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.

Back in the day when cars too over twice as long to stop, the driver had a much wider window in which to modulate their cars brake force to avoid an impact..........

[Dodsy]

eldar said:

What if the car in front has EBS and you don't? It does work effectively if you remember to really stamp on the brake pedal....

I have EBA and once it takes over it feels like the world has slowed down, you stop faster than you would think possible.

I suspect if i ever have to use it for real i will end up being rear ended by whatever is behind even if they have left a decent gap.

[Smitters]

Maybe approach this in a different way? I assume that you're normal and not a PH Director Type with unlimited funds to throw at your handsome trust-funded offspring, thus cost is the carrot you bait him with/stick you beat him with.

Using the maths above, it's been calculated that an x second gap means y distance behind a vehicle at a given speed. So by back calculation, we can say that by adding three seconds to the gap, he's going to arrive three seconds later to his destination. Cost? Nothing.

Cost of a bumper, radiator, insurance, pride (incalculable in a young male - enough money to assuage dented young male ego simply doesn't exist)?

Then show him this: http://www.youtube.com/watch?v=6dI5ewOmHPQ - Fifth Gear Crash Test.

Then sit back and watch him act in exactly the same manner for the next few years anyway, because he knows better and hope realisation dawns before Darwinism does.

[PaulD86]

AJB said:

PaulD86 said:

This effects the calculations on whether the cars will impact.

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.

Edited by AJB on Friday 29th November 12:55

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.

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.

[Toltec]

Max_Torque said:

Finally, as brakes become better, the reaction time becomes more important, because the total deceleration distance is shorter! 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.

Back in the day when cars too over twice as long to stop, the driver had a much wider window in which to modulate their cars brake force to avoid an impact..........

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.

Excellent posts of yours in thread Max_Torque.

• Braking, cornering, stability controls and suspension etc. not just power.

[panholio]

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.

[anonymous-user]

PaulD86 said:

AJB said:

PaulD86 said:

This effects the calculations on whether the cars will impact.

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.

Edited by AJB on Friday 29th November 12:55

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.

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.

The decceleration profile is irrelevant as long as both cars do the same thing. Take the in-extremis case of cars that can stop instantaneously in zero time.

Car A & B, both travelling at 60mph, Car B is 1sec behind Car A in time.

At time = zero, car A stops dead, car B has a reaction time, say 1sec. In that 1 sec it has cross the 1sec "distance" between it and the now stationary car A. But, after exactly 1 sec, it too stops instanteously, and comes to a halt with it's front bumper touching the rear one of car A, but there had been no overlap and hence no crash!!

If the reaction time is less than the "following time" then you always avoid a crash, what ever the speed (because speed "cancels" out, as a following "time" is not a fixed distance but depends on speed (grows with speed)