When I was learning how to drive, the one thing that got drilled into my head over and over again is to maintain proper following distance. For rear-end collision prevention, following distance is a lot more important than pure speed.
The rule I was taught was pretty simple: there should be three second gap between you and the car in front of you. When the car in front of you passes a fixed object like a road sign, you should be able to count three seconds before you pass it.
I do my best to abide by this and have never been in a collision, so I think it’s been a useful rule.
What it is not, however, is popular.
I’ve done my fair share of highway driving along some very busy corridors (the most notorious among them being the Jersey Turnpike). The norm on every single one is a following distance much, much shorter than what is safe.
This discrepancy between safe following distance and observed following distance is acknowledged by many states’ Department of Transportation. When teaching drivers, they promote safety. When building highways, they rely on unsafe driving. Here is an excerpt from an assessment by the Virginia DOT:
Excerpt from VDOT
Many state departments of motor vehicles (DMVs), in their driving training courses, recommend headways [following distances] of 2 to 4 seconds. The Virginia DMV recommends a varying headway of between 2 and 4 seconds depending on the speed of the vehicle, as shown in Table 1.
| Table 1. Recommended Time Headways in Dry Conditions | |
|---|---|
| Recommended Headway | Vehicle Speed |
| 2 seconds | Under 35 mph |
| 3 seconds | 35-45 mph |
| 4 seconds | 46-70 mph |
These headways are directly related to capacity: 2-second headways yield 1,800 vehicles/lane/hour; 3-second headways yield 1,200 veh/ln/hr; and 4-second headways yield 900 veh/ln/hr.
Observed and theoretical capacities are higher than those suggested by the Virginia DMV following distances. The Highway Capacity Manual assumes a theoretical maximum capacity of a freeway under ideal conditions of 2,250 to 2,400 passenger cars per hour per lane depending on free-flow speed. These equate to headways of 1.5 to 1.6 seconds.
A Necessary Evil?
Highway designers and driving norms implicitly command drivers to drive criminally fast and far too close to each other. Three lanes of 50 MPH, 1.5-second headway traffic would take eight lanes of 50 MPH, 4-second headway traffic to match the flow safely, per VDOT.
Because law enforcement cannot enforce speed limits and safe following distance without severely hampering the efficiency of highways, the speeding tickets that are awarded feel random and unfairly punitive.
Challenging the Contradiction
In 2002, Gord Thompson received a speeding ticket for going 117 KPH (73 MPH) on 100 KPH (62 MPH) Ontario Highway 401. After a judge told him going even one KPH over the posted speed limit was breaking the law, he and a friend drove exactly 100 KPH on the two lanes of the highway. This caused a 4 KM (2.5 mi.) traffic jam. He was then ticketed for impeding traffic.
Five years later, after being fed up with the 75 MPH traffic flow on 55 MPH I-285 in Georgia resulting in exorbitant speeding fines for minor deviations, a group of college students organized “an extraordinary act of public obedience.” They would simply drive four cars on the four lanes of the highway at the speed limit. This resulted in an enormous backup and drivers illegally passing in the shoulder. They made a video of it, but please forgive the 2007 editing.
What should the headway actually be?
You’ll notice that in the VDOT excerpt, they mention that there is no consensus among states on what is a safe headway, and their own recommendation for headway has a couple discrete breaks. Instead, we are going to take a formulaic approach and see how the different recommendations stack up.
How much road does it take to come to a complete stop?
For this, we’re going to take a page from kinematics, the study of bodies in motion. By using some of distance (x), time (t), initial velocity (v0), final velocity (v), and acceleration (a), we can figure out the other desired values.
We know final velocity will be 0, since we’re attempting to stop. For acceleration (deceleration), we will see what experts think a good estimate is. The consensus estimate of a safe braking rate for an average driver is 15 feet per second per second. We’ll flip this to negative since we’re slowing down and keep everything in FPS for now. We can convert to MPH later when appropriate.
We want to find the distance traveled, which will be a function of the initial velocity. To do this, we can use the following kinematics equation:
v2 = v02 + 2ax
Plugging in what we know:
(0)2 = v02 + 2(-15)x
Then rearranging to solve for x:
x = v02 / 30
However, this is the time to come to a stop once breaking begins. We also need to consider the reaction time. Experts place the average driver’s reaction time at 1.5 seconds — 3/4 seconds to decide to brake and 3/4 seconds to hit the brake. During this time, the car still travels at the initial velocity. So our true distance traveled before coming to a stop is:
x = ( v02 / 30 ) + 1.5v0
That graph looks like this:
Since we’re later going to be determining traffic flow, we need to account for the length of the average car, 14.8 feet. Combined with the distance to brake, this tells us the length of road dedicated to each car:
x = ( v02 / 30 ) + 1.5v0 + 14.8
What headway does this translate to?
To determine headway, we will take our stopping distance and see how long it takes us to travel that far at our initial velocity:
t = ( ( v2 / 30 ) + 1.5v ) / v
And simplifying:
t = ( v / 30 ) + 1.5
Graphing this, we get a simple straight line:
Because 15 FPS is just over 10 MPH, we can determine a very simple headway rule from the back of the car you’re following to the front of your car:
1.5 seconds plus 0.5 seconds for every 10 MPH
How does this stack up?
Comparing DMV recommendations to these results, it seems they may be overly optimistic. The Virginia recommended headways actually hold up for 0-10 and 46-50 MPH. The flat 3-second rule holds for up to 30 MPH. But I don’t think anyone is recommending a 5-second headway when driving 70 MPH.
I will concede that the likelihood of the car in front of you immediately coming to a full stop is low. A comfortable headway can also give you enough time to find your way into another lane or the shoulder, depending on the obstacle. And in the case that the car you’re following begins braking, the road they cover during their deceleration is extra room for you to come to a stop.
Another consideration is that the more gradually you come to a stop, the less probable it is that you will be hit by the car behind you, which in all likelihood is not observing a safe following distance.
Where things really deviate is comparing our formula to observed headways. The Virginia Highway Capacity Manual observes real-world headways of 1.5 seconds. This is barely enough time to touch the brake pedal. A modest highway speed of 60MPH would require triple that to safely come to a stop.
The headways used by that manual and for calculating traffic flow are measured from the front of one car to the front of the next. The formula for that headway is:
t = ( ( v2 / 30 ) + 1.5v + 14.8 ) / v
Simplified:
t = ( v / 30 ) + 1.5 + ( 14.8 / v )
What is the maximum safe traffic flow at each speed?
Now that we can find our front-to-front headway, we can determine how many cars will pass a fixed point in an hour. To do so, we will take the 3,600 seconds in an hour (60 seconds per minute, 60 minutes per hour) and divide by the front-to-front headway, like so:
f = 3,600 / ( ( v / 30 ) + 1.5 + ( 14.8 / v ) )
This formula translates to this graph:
There are a lot of interesting takeaways from this graph. There’s a pretty clear trend that the slower you go, the greater the flow. That is until the time it takes to travel the length of a car becomes non-negligible.
It’s also fascinating to see how comparatively low flow is at highway speeds. More vehicles can flow safely at 5 MPH than they can at 50 MPH. That means that despite going ten times faster, each car needs to take up more than ten times as much road.
And the answer to our original question can be found at the local maximum.
The greatest safe traffic flow is 1,239 vehicles per lane per hour at 14.4 MPH.
If you’ll excuse me, I have some fellow drivers to persuade on the Jersey Turnpike.
They’ll cooperate, right?