Stopping distances - unexpected conclusions
Discussion
The following was worked out in my head whilst walking the dog this morning so I could have made an error with the physics or maths....
The maxim I've heard many times is "only a fool breaks the two second rule". 70 mph is roughly 30 ms-1. Assume braking at 1g = -10ms-2 then it takes 3 seconds to stop. Say half a second of reaction time (at 30 ms-1) that gives 15 m plus 45m (average speed of 15 ms-1 for 3 seconds) = 60m. Which is how far you travel in 2 seconds at 70 mph.
However the above is really for the case where something stationary appears in your path. If you are following someone then they are unlikely to come to an instantaneous stop. More likely they will brake hard is they see something. So how close can you drive to them?
If you are gong the same velocity as someone in front then your relative speed is zero. When they brake it is really like they are accelerating towards you. As above that would be at 10ms-2. So if they brake hard whilst you are both doing 70 mph it is no different from if they are behind you, you are both stationary and they accelerate towards you at 1g. Assuming you can also accelerate at 1g and (say) can react within 1 second how far in front do you need to be to ensure that don't hit you? After 1 second they are doing 10ms-1 so average speed over 1 second is 5ms-1 so distance is 5 metres. That means that if you are following someone at just over a car's length then you should have time to avoid hitting them is they slam on the brakes. It also means that this is not dependent upon how fast they are going (as long they are going more than 10ms-1 / 25 mph).
That conclusion surprised me as I would have thought going faster you would need a bigger gap. Probably explains why there are so many low speed shunts as reaction times in real life can be much more than 1 second (and people rarely brake fully).
The maxim I've heard many times is "only a fool breaks the two second rule". 70 mph is roughly 30 ms-1. Assume braking at 1g = -10ms-2 then it takes 3 seconds to stop. Say half a second of reaction time (at 30 ms-1) that gives 15 m plus 45m (average speed of 15 ms-1 for 3 seconds) = 60m. Which is how far you travel in 2 seconds at 70 mph.
However the above is really for the case where something stationary appears in your path. If you are following someone then they are unlikely to come to an instantaneous stop. More likely they will brake hard is they see something. So how close can you drive to them?
If you are gong the same velocity as someone in front then your relative speed is zero. When they brake it is really like they are accelerating towards you. As above that would be at 10ms-2. So if they brake hard whilst you are both doing 70 mph it is no different from if they are behind you, you are both stationary and they accelerate towards you at 1g. Assuming you can also accelerate at 1g and (say) can react within 1 second how far in front do you need to be to ensure that don't hit you? After 1 second they are doing 10ms-1 so average speed over 1 second is 5ms-1 so distance is 5 metres. That means that if you are following someone at just over a car's length then you should have time to avoid hitting them is they slam on the brakes. It also means that this is not dependent upon how fast they are going (as long they are going more than 10ms-1 / 25 mph).
That conclusion surprised me as I would have thought going faster you would need a bigger gap. Probably explains why there are so many low speed shunts as reaction times in real life can be much more than 1 second (and people rarely brake fully).
Esceptico said:
The following was worked out in my head whilst walking the dog this morning so I could have made an error with the physics or maths....
The maxim I've heard many times is "only a fool breaks the two second rule". 70 mph is roughly 30 ms-1. Assume braking at 1g = -10ms-2 then it takes 3 seconds to stop. Say half a second of reaction time (at 30 ms-1) that gives 15 m plus 45m (average speed of 15 ms-1 for 3 seconds) = 60m. Which is how far you travel in 2 seconds at 70 mph.
However the above is really for the case where something stationary appears in your path. If you are following someone then they are unlikely to come to an instantaneous stop. More likely they will brake hard is they see something. So how close can you drive to them?
If you are gong the same velocity as someone in front then your relative speed is zero. When they brake it is really like they are accelerating towards you. As above that would be at 10ms-2. So if they brake hard whilst you are both doing 70 mph it is no different from if they are behind you, you are both stationary and they accelerate towards you at 1g. Assuming you can also accelerate at 1g and (say) can react within 1 second how far in front do you need to be to ensure that don't hit you? After 1 second they are doing 10ms-1 so average speed over 1 second is 5ms-1 so distance is 5 metres. That means that if you are following someone at just over a car's length then you should have time to avoid hitting them is they slam on the brakes. It also means that this is not dependent upon how fast they are going (as long they are going more than 10ms-1 / 25 mph).
That conclusion surprised me as I would have thought going faster you would need a bigger gap. Probably explains why there are so many low speed shunts as reaction times in real life can be much more than 1 second (and people rarely brake fully).
You have a 2s gap at 70mph, when they slam the anchors on.The maxim I've heard many times is "only a fool breaks the two second rule". 70 mph is roughly 30 ms-1. Assume braking at 1g = -10ms-2 then it takes 3 seconds to stop. Say half a second of reaction time (at 30 ms-1) that gives 15 m plus 45m (average speed of 15 ms-1 for 3 seconds) = 60m. Which is how far you travel in 2 seconds at 70 mph.
However the above is really for the case where something stationary appears in your path. If you are following someone then they are unlikely to come to an instantaneous stop. More likely they will brake hard is they see something. So how close can you drive to them?
If you are gong the same velocity as someone in front then your relative speed is zero. When they brake it is really like they are accelerating towards you. As above that would be at 10ms-2. So if they brake hard whilst you are both doing 70 mph it is no different from if they are behind you, you are both stationary and they accelerate towards you at 1g. Assuming you can also accelerate at 1g and (say) can react within 1 second how far in front do you need to be to ensure that don't hit you? After 1 second they are doing 10ms-1 so average speed over 1 second is 5ms-1 so distance is 5 metres. That means that if you are following someone at just over a car's length then you should have time to avoid hitting them is they slam on the brakes. It also means that this is not dependent upon how fast they are going (as long they are going more than 10ms-1 / 25 mph).
That conclusion surprised me as I would have thought going faster you would need a bigger gap. Probably explains why there are so many low speed shunts as reaction times in real life can be much more than 1 second (and people rarely brake fully).
You react 1s later, and slam yours on.
Both of you are braking at 10m/s/s... (and let's assume that's linear)
Except after that first second, when you hit your anchors, they're already down to 45mph, while you're still doing 70.
One second later, you're still doing 45mph, but they're down to 22mph.
One second later, you're still doing 22mph, but they're already stationary.
One second after they stop, you stop. Have you already hit them?
It is of course also down to how hard each vehicle brakes, and the ability of its brakes to actually work. A Ferrari and a Morris Minor will have totally difference braking characteristics and one can imagine that if a Ferrari does a full emergency stop at 60 a Morris Minor behind them is likely to hit them if the driver brakes a second after the Ferrari driver.
Esceptico said:
If you have a 2 second gap and brake after 1 second then you will still be 1 second = 30 metres away from the point they started to brake. So you will stop 30m before they stop. So no, not even close to hitting them.
You need to sit down and try the maths again. How on Earth do you think that you can be 30 metres from a car doing 30mph less than you and not close at all on them?Read what was posted, it takes you a second to react to a car in front of you doing an emergency stop; what is the speed difference and separation once you start to react?
Good following distance shouldn't be about leaving just about enough room to react & brake to avoid something. Poor drivers do that.
Good drivers don't have to react to emergencies, they have a following distance that allows them to free their eyes from the vehicle immediately in front in order to see not only far ahead of that vehicle but to consistently update with vision scans all around them. That vision allows them to anticipate & plan for problems, not suddenly have to react to them.
Good drivers don't have to react to emergencies, they have a following distance that allows them to free their eyes from the vehicle immediately in front in order to see not only far ahead of that vehicle but to consistently update with vision scans all around them. That vision allows them to anticipate & plan for problems, not suddenly have to react to them.
James_B said:
Esceptico said:
If you have a 2 second gap and brake after 1 second then you will still be 1 second = 30 metres away from the point they started to brake. So you will stop 30m before they stop. So no, not even close to hitting them.
You need to sit down and try the maths again. How on Earth do you think that you can be 30 metres from a car doing 30mph less than you and not close at all on them?Read what was posted, it takes you a second to react to a car in front of you doing an emergency stop; what is the speed difference and separation once you start to react?
After one second Car A has gone 25 m (decelerated from 30 to 20 so average speed 25 for one second). So Car A is 85m from origin. Car B has not yet braked so did 30ms for a second so has moved 30m. The distance between the two cars has closed to 55m (from 60m).
After another second Car A will have travelled another 15m. Total distance from origin is 100m. Car B travels 25m. Total distance 55m. Distance between the two has closed to 45m.
After another second Car A comes to a stop. It travelled 5m in the last second. Total distance 105m. Car B travels 15m. Total distance 70m. Distance between them 35m.
After another second Car A is still at 105m as it has come to a stop already. Car B travels 5 m. Total distance 75m. Distance between them 30 m.
The distance between the two cars reduces from 60m to 30m but they don't crash.
vonhosen said:
Good following distance shouldn't be about leaving just about enough room to react & brake to avoid something. Poor drivers do that.
Good drivers don't have to react to emergencies, they have a following distance that allows them to free their eyes from the vehicle immediately in front in order to see not only far ahead of that vehicle but to consistently update with vision scans all around them. That vision allows them to anticipate & plan for problems, not suddenly have to react to them.
I'm not advocating driving close. I always leave a big gap. Good drivers don't have to react to emergencies, they have a following distance that allows them to free their eyes from the vehicle immediately in front in order to see not only far ahead of that vehicle but to consistently update with vision scans all around them. That vision allows them to anticipate & plan for problems, not suddenly have to react to them.
Esceptico's quite right.
The two-second gap doesn't have much to do with stopping distances, but rather allows a safety margin for varying decelerations and reaction times.
Assuming equal deceleration and reaction time, if you start reacting at exactly the same time as the driver in front, you'll end up stopped exactly the same distance from the car ahead as you were before the driver ahead started to react, and if you start reacting 2 seconds after the driver in front, you will end up stopped right on their bumper.
But varying reaction times and braking deceleration cuts into that two-second gap.
Add to that the fact that people generally only start reacting when they see the brake lights in front - at which point the driver in front has already begun braking, so their reaction time is already spent.
And, as Rich1973 indicated, it could happen that the car in front has run into a stationary object, in which case you're probably not going to be able to stop within the distance given you by the two-second gap.
But, given that it's not very likely that anyone will run into a stationary object without reacting at all, such occasions are mercifully rare.
This is more likely to happen when the driver ahead thinks that they have all the time in the world - the traffic in front is miles ahead, so they can take their eyes off the road and admire the scenery, write posts to PH, etc - and then something happens ahead which means those huge gaps suddenly close up in no time at all.
As vonhosen indicated, it's best to keep a good distance to anticipate and plan, and this is indeed sound advice.
But, while this is easy to do when the traffic's light, as the traffic becomes heavier it becomes increasingly difficult to maintain a good gap, because it becomes a trade-off between good gaps and maintaining the rate of traffic flow. A two-second gap equates to 1800 cars passing per hour, whereas increasing the gap to three seconds reduces the flow rate to 1200 cars per hour.
(This all assumes that speeds are high enough to render negligible the difference between gap time and headway)
Also, opening up a gap ahead of you may well mean closing up the gap behind you.
On motorways during busy times, you regularly see the gap times reducing to around one second. Not necessarily saying it's right, but that's what happens.
ETA: Deceleration is normally less than 1g - a good, dry road surface normally has a friction coefficient of 0.8 to 0.9, which equates to 7.85ms-2 to 8.83ms-2.
This, coupled with the fact that drivers seldom employ the full braking capability of their vehicle, even in an emergency.
But we're talking about relative distances rather than absolute stopping distances here, so this is just added for the sake of interest (and completeness)
The two-second gap doesn't have much to do with stopping distances, but rather allows a safety margin for varying decelerations and reaction times.
Assuming equal deceleration and reaction time, if you start reacting at exactly the same time as the driver in front, you'll end up stopped exactly the same distance from the car ahead as you were before the driver ahead started to react, and if you start reacting 2 seconds after the driver in front, you will end up stopped right on their bumper.
But varying reaction times and braking deceleration cuts into that two-second gap.
Add to that the fact that people generally only start reacting when they see the brake lights in front - at which point the driver in front has already begun braking, so their reaction time is already spent.
And, as Rich1973 indicated, it could happen that the car in front has run into a stationary object, in which case you're probably not going to be able to stop within the distance given you by the two-second gap.
But, given that it's not very likely that anyone will run into a stationary object without reacting at all, such occasions are mercifully rare.
This is more likely to happen when the driver ahead thinks that they have all the time in the world - the traffic in front is miles ahead, so they can take their eyes off the road and admire the scenery, write posts to PH, etc - and then something happens ahead which means those huge gaps suddenly close up in no time at all.
As vonhosen indicated, it's best to keep a good distance to anticipate and plan, and this is indeed sound advice.
But, while this is easy to do when the traffic's light, as the traffic becomes heavier it becomes increasingly difficult to maintain a good gap, because it becomes a trade-off between good gaps and maintaining the rate of traffic flow. A two-second gap equates to 1800 cars passing per hour, whereas increasing the gap to three seconds reduces the flow rate to 1200 cars per hour.
(This all assumes that speeds are high enough to render negligible the difference between gap time and headway)
Also, opening up a gap ahead of you may well mean closing up the gap behind you.
On motorways during busy times, you regularly see the gap times reducing to around one second. Not necessarily saying it's right, but that's what happens.
ETA: Deceleration is normally less than 1g - a good, dry road surface normally has a friction coefficient of 0.8 to 0.9, which equates to 7.85ms-2 to 8.83ms-2.
This, coupled with the fact that drivers seldom employ the full braking capability of their vehicle, even in an emergency.
But we're talking about relative distances rather than absolute stopping distances here, so this is just added for the sake of interest (and completeness)
Edited by Engineer792 on Sunday 1st October 14:50
If you were a cars length behind someone at 70, and they did an emergency stop for seemingly no reason, without emergency brake assist kicking in you are almost guaranteed to hit them.
The number may be true that a person with reasonable reactions and modern brakes could stop, but the the simple fact is they wouldn't.
If you personally could not see a situation ahead requiring an immediate stop your natural reaction to the car in front jumping on the brakes would be to brake firmly, but nowhere near maximum braking, and your reaction time to brake more firmly would almost certainly be too late.
The number may be true that a person with reasonable reactions and modern brakes could stop, but the the simple fact is they wouldn't.
If you personally could not see a situation ahead requiring an immediate stop your natural reaction to the car in front jumping on the brakes would be to brake firmly, but nowhere near maximum braking, and your reaction time to brake more firmly would almost certainly be too late.
ging84 said:
If you were a cars length behind someone at 70, and they did an emergency stop for seemingly no reason, without emergency brake assist kicking in you are almost guaranteed to hit them.
The number may be true that a person with reasonable reactions and modern brakes could stop, but the the simple fact is they wouldn't.
If you personally could not see a situation ahead requiring an immediate stop your natural reaction to the car in front jumping on the brakes would be to brake firmly, but nowhere near maximum braking, and your reaction time to brake more firmly would almost certainly be too late.
Eh?The number may be true that a person with reasonable reactions and modern brakes could stop, but the the simple fact is they wouldn't.
If you personally could not see a situation ahead requiring an immediate stop your natural reaction to the car in front jumping on the brakes would be to brake firmly, but nowhere near maximum braking, and your reaction time to brake more firmly would almost certainly be too late.
If they merely took their foot off the throttle you'd probably nudge them up the arse in your scenario, which is irrelevant to the one the OP was posting about, however pointless his initial post was.
cmaguire said:
Eh?
If they merely took their foot off the throttle you'd probably nudge them up the arse in your scenario, which is irrelevant to the one the OP was posting about, however pointless his initial post was.
I have no idea what you are on aboutIf they merely took their foot off the throttle you'd probably nudge them up the arse in your scenario, which is irrelevant to the one the OP was posting about, however pointless his initial post was.
Did you not read his unexpected conclusion ?
Esceptico said:
That means that if you are following someone at just over a car's length then you should have time to avoid hitting them is they slam on the brakes
ging84 said:
cmaguire said:
Eh?
If they merely took their foot off the throttle you'd probably nudge them up the arse in your scenario, which is irrelevant to the one the OP was posting about, however pointless his initial post was.
I have no idea what you are on aboutIf they merely took their foot off the throttle you'd probably nudge them up the arse in your scenario, which is irrelevant to the one the OP was posting about, however pointless his initial post was.
Did you not read his unexpected conclusion ?
Esceptico said:
That means that if you are following someone at just over a car's length then you should have time to avoid hitting them is they slam on the brakes
His own calculations show that the first second closes the gap between the two cars by 5m, by the time the rear car starts to brake.
The second second closes the gap by another 10m.
The third second takes another 10m off the gap.
By the time they've both stopped, four seconds after the lead car hit the brakes, the gap has closed from 60m to 30m.
So, if there's only 5m ("a car length") between the two cars to start with, then the second car is still going to be doing 70mph when they hit the first car at a closing speed around 25mph.
To not hit at all, there'd need to be a minimum of 30m gap.
Edited by TooMany2cvs on Sunday 1st October 17:26
All well & good but visibilty up ahead is massively reduced the closer you are. Hence the train of brake lights you see on motorways & the phantom stoppage. This is when you get cars rear ending & the sudden stops of more than 1g 
Leaving a gap large enough to overcome the stoppage time is how its broken back into flowing traffic.
https://www.youtube.com/watch?v=Suugn-p5C1M
I'll agree with Vonhosen on this subject.
Leaving a gap large enough to overcome the stoppage time is how its broken back into flowing traffic.
https://www.youtube.com/watch?v=Suugn-p5C1M
I'll agree with Vonhosen on this subject.
ging84 said:
cmaguire said:
Eh?
If they merely took their foot off the throttle you'd probably nudge them up the arse in your scenario, which is irrelevant to the one the OP was posting about, however pointless his initial post was.
I have no idea what you are on aboutIf they merely took their foot off the throttle you'd probably nudge them up the arse in your scenario, which is irrelevant to the one the OP was posting about, however pointless his initial post was.
Did you not read his unexpected conclusion ?
Esceptico said:
That means that if you are following someone at just over a car's length then you should have time to avoid hitting them is they slam on the brakes
I've reread his first post and that bit is so ridiculous I must have blanked it out the first time.
cmaguire said:
You were referring directly to his post rather than making stuff up. My mistake.
I've reread his first post and that bit is so ridiculous I must have blanked it out the first time.
I glossed over that bit myself, and, on re-reading it it doesn't make sense.I've reread his first post and that bit is so ridiculous I must have blanked it out the first time.
Esceptico, would you please clarify
Do as many calculations as you like but the reality usually is the first car brakes the second might stop in time the third one hits at about 35mph and the fourth hits at 60 pushing everybody together .
Most get fixated on just the one in front and only react to indicators and brake lights .
If you look further up the road then you notice situations developing and react before the car in front has even noticed the changing situation .
Some can process unfolding situations ahead far better than most on the road .
Many only react to whats 20mtrs ahead by then it's too late .
Most get fixated on just the one in front and only react to indicators and brake lights .
If you look further up the road then you notice situations developing and react before the car in front has even noticed the changing situation .
Some can process unfolding situations ahead far better than most on the road .
Many only react to whats 20mtrs ahead by then it's too late .
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