Sections
Vehicle Flux in a Road or Highway
as function of speed and safety distance
Recently it has been proposed a raise of the speed limit for some italian highways. This idea has been criticized, so I decided to make some calculations to understand whether it is really useful or not and to know the exact effects on the car flux a road can sustain.
The Italian Road Regulation
The law states (italian version) that every car has to keep a distance from the previous one such that they would never touch each other, no matter the circumstance. This law is pretty clear: the distance must be the sum of the space covered during the reaction time plus the space needed to completely stop. However, not everyone follow this rule: my own drive instructor told me to keep only the space ceovered during the response time, since it's not enough to avoid the collision in only two situations: either there is a cross where you cannot see at all the other roads or a wall suddenly appears in the middle of the road, in front of the previous car. In a highway there aren't crosses and wall probably will never appear, so the formula was given to me was ok.
In the real world, things go slightly different... most people don't even know that the distance among cars should be adapted to actual speed, they keep a constant distance most of the time... it only depends on the road: urban road, 2 meters is ok, country road 8-10 meters (well, even less)m highway... about 15 meters. Just think about it.
The Formulas
When studying the problem, we can make some assumptions, to simplify thinkgs a bit. Let's assume that all the cars drive at the maximum allowed speed, 130 km/h in almost every italian highway, fogetting about people that run slower than they could or faster than they should. Let's also assume that the chosen distance (whatever it is) is always kept, so we ignore overtakings and every other event that could perturb the traffic.
Speaking about safety distance, these are the formulas involved (all the units are from SI: m/s for speeds, obtained with km/h divided by 3.6, and meters for distances):
where the first formula is everything many people know about, the second one represents the distance covered during the response time (1 second), the third one is proportional to the braking distance (proportional to the square of the speed: it's related to kinetic energy) and the last one is the safety distance as defined in the italian regulation. Some formulas have constants, I chose those that have been taught to me, they should work well with today's cars, they are experimental and not computed from theory.
The formula that gives the cars flux is the following:
The flux is represented by the greek letter Φ, the cars density with ρ, the car length with l and the distance among them with d (it must be substituted with one of the previous formulas).
Once we have the formulas, all we have to do is to plot graphs whith speed as independent variable and flux as dependent variable. In the following two graphs I show the results, in the first one I set l to 4 meters, while in the second one 10 meters (a truck instead of a car). Axis scales are kept constant, in order to allow easy comparisons. The ordinates axis represents how many cars transit per second (imagine to stop on a side of the road and to count how many cars every second you see). All the graphs have been plotted with MathPad, a free utility for Mac OS. The source code used is available here.
The color shows how the distance has been calculated: black means constant distance (20 meters should be realistic), red means distance proportional to the speed, green means proportional to braking distance, blue is the distance according to the official regulation. The X axis ranges from 0 to 50 m/s, in other words from 0 km/h to 180 km/h. The Y axis has been limited to 1.25 cars/second, to show with more details the useful curves.
Final Results
It's pretty clear that a generalized speed limit increase cannot give an overall benefit (if everyone followed the rules), however, if applied only to highways without traffic jams, it can help to reduce travel time (time = space / speed).
I think the idea of raising the speed limit from 130 km/h to 150 km/h (41.7 m/s) in selected stretchs of road is a good choice to reduce travel time, while in city streets the limit must not be touched, since a lower speed limit allows for greater fluxes, much more useful (the maximum flux is obtained with a speed equal to about 5,5 m/s, or 19,8 km/h if you prefer).
A thought about safety distances: it's interesting to think what a great benefit could an integrated electronic traffic control could give... with much shorter response times we could raise a lot the speed limit and keep a constant distance between vehicles to achieve the fluxes of the black line in the graphs, about 10 times greater than the blue line at 50 m/s!
If you'd like to have more details and infos, feel free to contact me by electonic mail.
Author: Olaf Marzocchi
First revision: March 5th, 2006.