A mathematical analysis of the "speed kills" arguments
Jan 3, 1996
by Alex Kuznetsov,
ABSTRACT:
There seems to be a universal belief, reflected in the current speed limits, that "speed kills", and that, as someone in this newsgroup said recently, "it does not take a rocket scientist" to figure out why. Ever since I got my first speeding ticket, I was trying to to just that - understand in a rational way what are the speed-related driving risks, and is there an optimal driving strategy that minimizes those and other driving risks. Perhaps my "rocket scientist" background (I do theoretical physics for a living) has made this analysis more complicated than it should be, but I hope the conclusions would be of interest to everybody here. In short, I argue that while increased speed is obviously a risk factor, there are other factors that can compensate that risk and are almost always more important. I propose a quantitative model to assess the speed-dependent risks, and use it to discuss the optimum driving strategy.
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To analyze the "speed kills" argument quantitatively, let us consider the speed dependence of your chances to get killed or seriously injured in an accident. Let us call this number R. As in many other areas of science, this probability can be represented as a product of several factors:
R = S*E*A*K
S stands for "skills" and includes factors such as driving skills, vehicle capability, how much you concentrate on your driving, etc - everything that is speed-independent. It is just a constant number that is different for different drivers and different cars, but is not very important for our analysis of speed-dependent risks.
E means "exposure". This is an extensive (means it accumulates as you keep on driving) factor that should reflect your exposure to various driving hazards. For example, it can be simply proportional to the number of miles driven. We will discuss a model for the exposure factor below.
A stands for the probability of getting in an accident at any given time. It is an "intensive" quantity (does not accumulate as you keep on driving). It is obviously dependent on speed as well as some other factors that we discuss below. It can be thought of as "risk rate": total risk is your risk rate times exposure. Let's call it "accident rate".
Finally, K is a "kill factor": the conditional probability of getting killed or seriously injured provided you got in an accident. It will depend on speed.
There are of course many unknowns involved in each of those factors, so it is impossible to give an absolute number for R (like "you will get killed every 125,000 miles on average"). However, it is possible to analyze how the risk depends on speed by making some reasonable assumptions about the risk and exposure factors.
Let us start with the simplest case: no traffic. You have an empty highway in front of you, and you need to cover L miles going from point a to point b - what can we assume about the above risk factors?
E: exposure will be simply proportional to L (miles driven). Think of it this way: there is a certain chance to encounter a road hazard (a pothole, a deer, a slick spot) per every mile, so the more you drive, the more likely you are to encounter something you'll have to avoid. Exposure here is speed-independent.
What is speed-dependent is the accident risk, A. Your ability to avoid a hazard will be reduced at higher speeds. To a good approximation, A will simply be proportional to the speed, v: A = c*v, where c incorporates road conditions. The argument is simple: if you go twice as fast, you will have twice less time to react to a hazard that doubles the chances of an accident (same goes for veering off the road in a turn - the risk is also proportional to v). More realistic models of the accident risk should allow for rapid increase in A at speeds above the mechanical limit of the vehicle. I will disregard this effect here because I believe most cars still behave quite competently at 80 mph which is the highest traffic speed I will dare to consider.
Now the "kill factor", K. It is a chance to get killed or gravely injured in an accident when we already know the accident occurs. This is the quantity crash tests attempt to measure. It is a number that varies between zero and one, like all probabilities. It obviously grows with speed, but the important thing here is that it cannot exceed one: to put it another way, if you crash at 200 mph, you will be just as dead as if you crashed at 100 - doubling the speed does not double the kill factor. This factor should grow with speed at low speeds, but once you are over a speed where almost any accident results in severe injury or death, the kill factor levels off and gradually approaches one. For the purposes of this discussion, I will use the following functional form for K:
K = 1 - exp(-v/30 mph)
This function grows linearly with v at low speeds (below 30 mph), and approaches one as you go above 30 mph. It DOES NOT mean you get killed if you crash at 30 mph: it gives a 63% chance of injury or death for a 30 mph crash, a 73% chance for 40 mph, an 86% if you crash at 60, and a 95% chance to get killed in a 100 mph crash. I think this is reasonable, but there is room for debate here.
So, what does it give us for the risks of driving down an empty highway?
R is proportional to : E*A*K = (exposure proportional to L)*
*(accident rate proportional to v)* the kill factor =
= L*v*(1 - exp(-v/30)
per mile driven: R/L ~ v*(1 - exp(-v/30))
It is a growing function of v and, I think, the cornerstone of the "speed kills" ideology. It tells you that, if you take the risk at 50 mph as a reference of 100, your risk at 10 mph is 7, at 30 mph it is 48, at 60 mph it is 130, and at 90 mph it is a whooping 212. So, you are half as likely to get killed if you go 30 than if you go 50, and you are more than twice as likely to get killed if you go 90.
The conclusion that risk is a monotonously increasing function of speed is, however, valid only for the specific conditions (empty highway) which very few of us actually encounter in real life. The presence of traffic makes a major difference here. Let us try to incorporate the effects of traffic on the driving risks in our model.
E: exposure to driving hazards should grow with traffic density. You still have the above-discussed road-hazard component of exposure that is simply proportional to miles driven (L), but in the presence of traffic we should also add exposure to traffic hazards. This is proportional to traffic density which we will characterize by a factor, d (e.g. the number of cars per mile of highway), and to TIME you spend in the traffic: E = L + d*T . L and T are related, L = vT, so if we want to calculate risk per mile driven we should rewrite it as: E = L( 1 + d/v). It is very important to note that now exposure is speed-dependent, in fact it DECREASES with speed. The reason it decreases is simple: if you go faster, it takes you less time to go from a to b so you have less time to get in trouble (though the accident rate may increase with speed). I have never seen a discussion of this point but it is very important: every second you spend on the highway with traffic around you adds to your risk exposure, if you spend less time there by going faster, you DECREASE the exposure.
A: accident rate will still have a component that is simply proportional to v. However, the presence of traffic means that it should also depend on your speed relative to other cars, and on the traffic density. The simplest way to incorporate such dependence is to add a "traffic" component to the accident rate that has a minimum at the average speed of traffic, u (although Rahul may disagree, there is such a thing as the flow of traffic: u can be defined as the average speed of cars in your immediate vicinity). The simplest function with a minimum is the parabola, so let's write:
A = c*v + d*(v - u)^2
the relative balance between the first (obstacle avoidance) and second (traffic) components of the accident rate can depend on road conditions which can again be taken into account by varying c . This functional form provides for a rapid increase in accident rate when you deviate from flow of traffic, just as it is in real life. I am tempted to make this adjustment asymmetric by making it more dangerous to go slightly below the flow of traffic than to go slightly above, but that would be beyond the accuracy of the model.
K: it seems reasonable to assume that the kill factor does not depend much on the presence of traffic: once you got in an accident, you will or will not get killed according to your speed. This may be an oversimplification, but let's leave it at that.
So, in the presence of traffic our risk becomes:
R [per mile driven] ~ (exposure )* (accident rate) * ( kill factor)
= (1 + d/v) * (c*v + d*(v - u)^2)* (1 - exp(-v/30))
This now depends not only on your speed, but also on the density and speed of traffic. Anyone with a graphing calculator can have some fun plotting this function for various values of the parameters. Let me just verbally summarize the main features of such plots:
1) even in a modest traffic (low d), it becomes extremely dangerous to deviate from the flow speed, either above or below.
2) in moderate and heavy traffic, if you go with the flow (v = u), the increase in accident rate due to higher speeed (c*v in second term) is largely offset by the DECREASE in exposure (d/v in the first term), so the product (the total risk) would remain independent of speed.
3) The increase in the flow speed (which is what speed limits attempt to regulate) does increase the risk even if you go with the flow. E.g. if the flow speed increases from 50 to 90 (for the sake of argument), the risk increases by 17% due to the last term, the kill factor. However, the increase is much less than what we had on an open highway, where the same increase in speed led to risk increase of 112%. This is again because of decreasing exposure at higher speeds.
4) Any attempts to obey the speed limit when the flow is substantially faster are suicidal, according to this model. Doing 55 when everybody else is doing 70 can increase your risk by more than a factor of 100!
5) It also appears that "traffic kills" rather than "speed kills": traffic density d is the single most important factor that affects the total risk. Doubling the traffic density approximately doubles the risk and makes it twice as dangerous to deviate from the flow.
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In conclusion, this analysis corroborates what most of us already know from experience: the safest thing to do is to go with the flow and screw the speed limit. Contrary to what the insurance industry wants us to believe, the increase in flow speed DOES NOT lead to proportional increases in death risk. The decrease in exposure to traffic hazards resulting from spending less time on the road (that's why everybody wants to go faster in the first place) largely offsets the increase in accident rate at higher speeds. Finally, it appears that a good way to decrease your risk is to try reducing the local density of traffic around you - avoid traveling in "platoons" even if it means momentarily increasing your speed to get away from the pack.
I fully realize that many assumptions made here are debatable, and I would appreciate suggestions and criticisms, especially statistical data that could help improve this model. I believe that trying to understand the complex phenomenon of auto accidents on the basis of rational analysis is a better way to deal with it than just cry "speed kills" every time a drunk ends up wrapped around a pole. I hope this contribution has been constructive, inspite of its length.
