Although I can’t put my hands on it at the moment, we’re often told that the capacity of a motorway lane is 2,000 vehicles per hour. And this may well be the case. But it doesn’t quite gel with one of the other things that we’re told is important: the two second rule.
It’s important to maintain a safe following distance behind other vehicles, and, in good conditions, two seconds seems to be about right. Hence the marketing campaigns: “only a fool breaks the two second rule”. In bad conditions – wet weather, poor lighting, hurricanes – a longer gap is needed, perhaps equivalent to three, four or even more seconds. It’s worth pointing out that the rule is just a “rule of thumb”, a guideline – it’s not legally enforcable.
The High School Science Section
So what’s this got to do with road capacity? Well, the number of vehicles that can move past a fixed point in an hour depends on their speed, and the following distance. Imagine a line of cars which are literally bumper-to-bumper. The average car is about 4.2 metres long, so to work out the road capacity, you just need to know the speed (50 km/hr, say). That’s 50,000 metres that each car travels in an hour, so if you divide by 4.2, you’ll get about 11,900 cars past a fixed point in an hour.
What about if the cars are maintaining a 2-second following distance? In this case, you need to convert the distance to metres – high school science stylez. At 50 km/hr, the following distance is 27.8 metres, and this increases linearly with vehicle speed (so for 100 km/hr, you need 55.6 metres). Let’s stick with 50 km/hr for now.
The distance between the front bumper of a car and the the front bumper of the car behind it is 27.8 + 4.2 metres, or 32 metres. At a speed of 50 km/hr, cars take around 2.3 seconds to cover this distance. Over an hour, 1,564 cars will pass a fixed point, and that’s the road capacity.
Road lane capacity increases with speed, but it’s not linear – the difference in capacity between 50 km/hr and 100 km/hr is actually pretty small (1,564 vs. 1,673). This is shown in the graph below – which is labelled with the capacity for common speed limits (30, 50, 80 and 100 km/hr).
So, what happens if we use a shorter following distance? I’ve made an Excel file to demonstrate this, available here if you want to have a play around. If the cars have a 1.5-second following distance, capacity is quite a bit higher, as shown in the graph below:
With a 1.5-second following distance, it’s easy to get to 2,000 vehicles per hour, and even a local road with a 50 km/hr speed limit pretty much makes it. The flip side, of course, is an increased chance of crashes. After just coming back from a Labour Weekend holiday, it’s fresh in my mind that most people don’t keep a 2-second gap!
The Real World Section
The situations above assume ideal driving conditions, with no traffic lights, stops or other disruptions, or inclement weather, and free-flowing traffic at constant speed – of course, as soon as somone slams on the brakes, cars get backed up and things change. At their best, motorways can come fairly close to these idealised conditions: onramps are the only real intersections which can be a problem, and these days most of them are controlled with traffic lights. Distractions, or poor conditions, can interfere, but for large parts of the time it can be relatively smooth sailing, until someone gets spooked and starts using their brakes.
However, a lane capacity of 2,000 vehicles per hour is not compatible with everyone observing a 2-second following distance. That’s not the end of the world, and traffic engineers and other professionals who work with motorways will be well aware of it, but I think it’s an important point to note.
For other roads, it’s quite a different story. Speed limits are lower, but more important are all the other things that impede smooth vehicle movement. Intersections, traffic lights, congestion and so on. The average speed in Auckland is around 32 km/hr, lower than pretty much any speed limit you’ll see posted, and these are the factors that bring it down.
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Im not sure if you included in those addition for each lane. 2 lane road double the 1500 @ 50km/hr and you make the capacity with spare
Sorry, I should have made it clearer – I’m talking about capacity per lane, not for an overall road.
Thought I’d read somewhere that the speed to maximise throughput was about 70 kph?
I read on here it was in the 40’s, I assume this is to do with people having longer following times at greater distances.
Normally I love the back of a serviette calculation, but in this case not. Many people have done very long theses on this subject, and it is a lot more complicated than your quick high school maths reveals
The two second rule is a marketing gimmick. Actual stopping time/distance can be more or less depending on car, tyres, road surface, road conditions, driver reaction and sleep level etc
Most people tend to drive within a safe envelope they discover for themselves. In high speed braking, this usually results in a safe stop or a slight but non-injurious crash. If this was not the case we would have a significantly higher nose-tail crash rate (currently around 1.6% of all crashes, and not usually fatal)
Lane capacity is also influenced by road shape, width, gradient, intersections, driver behaviour, and road surface. Most roads would be lucky to get more than 1,000 vph over any length of time
Roll on technology. The world is awash with simple technology to track and control cars to make them very safe, but does any road authority implement them, yeah…na
We have car ID by RFID (a better licence plate, it’s a public privilege to use a road, cars should be tracked). Car to car communication for safe braking distance given road surface and weather etc (this could be retrofitted to nearly all existing cars using current off the shelf tech)
Vehicle braking performance is not a constant, even in the same car. Tyre pressure, tread wear, suspension wear, weight (and weight distribution), temperature of brake surfaces, to name a handful of variables off the top of my head. It’s actually not straightforward to retrofit the necessary sensors to a car to allow proper calculations of braking distance, and I’m not entirely sure that these could all be measured with sufficient accuracy to allow no-margin following distances.
You’ve seen the adverts about how a single worn shock-absorber increases braking distance significantly, right? How the heck do you accurately and constantly measure that wear without replacement of all the struts with ones that have embedded self-check equipment (with the enormous associated cost)? What about tyre condition? And even if there were tiny cameras that tracked tread depth, what about the pressure and temperature of the tyre? How about weight distribution? Even a small adult affects weight distribution, which affects braking and handling.
Yes, everything affects braking performance, even the air temperature. It doesn’t take too much processing power to log all previous braking attempts by the car in question, and to get the average of similar cars braking performance for the section of road you are on for the current conditions, to get a pretty good safe estimation of minimum safe distance, if it’s off by a bit you may cause panel damage, but not major injury
Either way you only have to remove reaction time to take 45% of the distance out!
A computer based following system will continuously monitor following distance, road conditions, weather, crash reports etc to keep a safe distance. Realistic tests by some car makers have reduced following distances to just a few meters at 100 kh/m. This requires all vehicles following to be on the same system and data network of course. Hence a road authority needs to do it
Given the technological changes in just the last ten years, why would anyone think this couldn’t be done?
@Pete “Most people tend to drive within a safe envelope they discover for themselves”
Since when did “most” become an acceptable safety standard? Would you fly on an airline where most of the pilots were sober?
Seeing you’re calling others out on their assumptions, saying that nose-to-tails represent 1.6% of all crashes does not prove your point- it only says something about the proportion of this type of crash to all others.
We have a hundred years of driving history to prove that humans are incredibly unsafe as drivers.
@NCD, fully agree, humans are incredibly unsafe as drivers. Roll on some decent crash avoidance automation one day, all we have after a hundred years of driving is ABS and TCS
under wet or foggy conditions, the road code specifies a 4 second following period
The “two-second” rule is a good rule of thumb for keeping a safe following distance but it’s not routinely followed and it’s not the law. The legal minimum stopping distances are at http://www.legislation.govt.nz/regulation/public/2004/0427/latest/DLM303092.html – the minimum following time is never longer than 1.44 seconds at any speed.
What exactly would be the point in deliberately underestimating road capacity? That would lead to roads being widened or otherwise having their capacity increased when in reality they were well below capacity.
Hi Steve, the main point of this post was simply to highlight the disconnect between the 2-second rule (of thumb) and road capacity, and illustrate the way in which capacity varies with following distance. I don’t have any background in this subject and I can’t really take it any further than the ‘high school science’ examples mentioned in the post – I will defer to the engineering community on any more complicated matters 🙂
We would normally say that the capacity of a lane is somewhere between 1700 and 1800 depending on a number of factors. However as we know roads often operate between 2000 and 2200 vph per lane.
More proof that we dont design roads with the intention providing for all peak hour traffic.
“More proof that we dont design roads with the intention providing for all peak hour traffic.”
What?
How on earth is that proof?
Decreased distances increase congestion. That’s because people are not computers or racing drivers, and tend to slow up as soon as they get ‘too close’, and overcompensate. This sends the famed slow wave through the system, and in practical terms causes traffic jams.
The way to increase throughput is to slow everyone to ~45km/h, and prevent overcapacity on motorways. The latter has been done, but there’s a way til we get the former.
Most traffic engineers simplify the maths even further and think in terms of headways rather than gaps so the length of car is included. With that assumption you can convert time per vehicle and vehicles per time unit by taking an inverse (and correcting units). For example there are 3600 seconds in an hour so 3600/2 sec per vehicle = 1800 veh/hour. (which is close to saturation flow in ideal urban conditions). As for the two second rule there isn’t any real science to it that I know of but it is better for marketing than the 1.800 second rule. Like the difference between “finger lickin good” and “warm dead bird”
Just a thought, you said “It’s worth pointing out that the rule is just a “rule of thumb”, a guideline – it’s not legally enforcable”. I got a ticket today for “following too closely” so had a dig around the various websites and found that the minimum following distance at 90kph or more is 36m (following distances are different at lower speeds) so it is in fact ‘legally enforcable’ and IN THEORY no one on NZ roads should be following at 1.5seconds or less. This was according to the “Land Transport (Road User) Rule 2004”
Great blog. Can you tell me where you got the statistic for the average car being being 4.2 metres? I want to use that statistic but I want to be able to justify its accuracy. Thanks, Will.
Hi Will, don’t have anything formal sorry… I’d suggest flicking a quick email to MOT (who have stats on the current vehicle fleet) or NZTA (who have stats vehicles entering the fleet). I think I might have just based 4.2 metres off a Corolla or similar.
Holy necropost Batman! I was doing some of my own calculations about this topic today, well, the inverse, and found your article. If an at-capacity lane was operating at 100km/h carrying 1673 cars, then it would only take 29 extra cars per hour to slow this at-capacity lane to 80km/h. Or 110 cars per hour to slow it to 50km/h. This is an amazing statistic to promote public transport. So few people need to be using the service to have an incredible impact on traffic flow for the rest of the community. Hold on, is that maths right?