We use figures from official sources and well respected research to show that we could reduce all UK speed limits to just 12 mph and still have the same numbers killed on the road. This isn't supposed to be about what would happen if we reduced speed limits to 12 mph.

This page is specifically about the contribution of speed to accident outcomes, set against the contribution of other factors (notably driver response).

This page replaces a previous page of the same title with identical logic and intentions. 

Working entirely from official figures this chart (Fig 1) shows the proportion of UK car drivers, injured, seriously injured, and killed on UK roads in 2002 (the last year on record). See the source figures here (table 5c). We chose car drivers because they are typical of relevant trends, they are affected by legislation, and also because they fit the categories below. There's nothing special or clever about choosing car drivers. In 2002 in GB, 116,994 were slightly injured, 10,884 were seriously injured and 1,146 were killed.

Notice how we can use these figures to deduce the probability of death in a collision. If you add up all the figures and compare the total to the number killed you can deduce that drivers in injury accidents have an average 1 in 113 chance of being killed.

1993 research by Hans Joksch determined the probability curve in this graph (Fig 2). It shows for example that in a 60 mph crash a car driver is 50% likely to die.

The equation is properly applied to "change in speed", sometimes known as "delta V". If there was a crash at 90 mph and after the crash due (perhaps) to "glancing off" the vehicle was travelling at 30 mph in the same direction then the crash had a delta v of 60 mph. Similarly hitting a heavy fixed object at 40 mph and ending up at 0 mph would have a delta v of 40 mph. Note that in the vast majority of crash situations the "delta V" is significantly less than the pre accident speed of the vehicles involved.

We accept the findings of the research. They determined that risk of death varied with the fourth power of speed according approximately to the following "rule of thumb" equation:

• probability = (speed /71)⁴

Like the first chart above (fig 1) this graph and equation deals with the probability of dying in an accident.

ref: Joksch, H.C. Velocity Change and Fatality Risk in a Crash -- A Rule of Thumb. (summary)

We realised that these two pieces of information could be combined. We can calculate the average crash delta v from the proportion of drivers killed. We know the proportion of drivers who are killed, and we can use the equation from fig 2 to calculate an average impact speed.

From the first graph (Fig 1) we know the real probability of death to a car driver from an injury accident. 

• probability of death = 1,146  /  (116,994+10,884+1,146) = 0.00888207 = 1:113 chance of death 

• speed = 71 * (fourth root of) 0.00888207 = 21.8 mph 

Perhaps you're worried that Joksch's equation is unreliable below about 30 mph? Let's see how many would have died at 30 mph:

• probability = (30/71)⁴ = 0.031875

• 0.031875 * (116,994+10,884+1,146) = 4,113.

This is about 4 times the number who do die, so we can see clearly that the average impact speed is significantly less than 30 mph.

So we could reduce the speed limit to 30 mph over the entire country, enforce it rigidly and still kill 4 times more car drivers than we do at present.

So far we've only looked at injury accidents. Perhaps we should make things a little more realistic by looking at all accidents? One problem is that damage only accident figures are not gathered nationally. There's some data from the insurance industry, but we'll need to do a little intelligent guesswork. There were about 4,000,000 motor insurance claims for the last year on record (2,000). We'll assume that 75% were vehicle accidents, and 50% of those applied to private motor cars. So we have 1.5 million accidents (give or take)

• probability = 1,146 / 1,500,000 = 0.000764 (probability of fatal for driver in UK car accident)

• speed = 71 * (fourth root of) 0.000764 = 11.80 mph That's the calculated average impact speed of all GB car accidents affecting car drivers to give the right proportion of fatality. Let's call it 12 mph.

Perhaps you are still worried that the Joksch equation lacks resolution below 30 mph? So calculate 0.031875 * 1,500,000 = 47,813. That's the number of drivers we would have expected to die in all accidents at an average crash delta v of 30 mph. It's almost 42 times more than the number who do die. And that's at just 30 mph.

But there's more...

• What if we start talking about near misses? Perhaps we could say that 7.5 million accidents and near misses resulted in 1,146 fatals and push the average impact speed down lower?
• What if we start talking about excluding reckless drivers, joyriders, police drivers and drunks (any that wouldn't be affected by the 12 mph speed limit)? Then we could reduce the 1,146 fatals to much less than 1,000.
• What if we exclude all those fatals where the driver crashed at above 50 mph impact speed? They weight the average massively against the rest of us don't they?

The following factors result in our headline conclusions being high estimates of average crash delta v.

Averaging across the curve.

Refer to figure 2 above. The green average line from the vertical axis maps to a higher than correct average on the horizontal axis. This occurs because of the "concave" nature of the curve. Calculating from an average probability to a speed will always give a high estimation of the average speed due to this effect irrespective of the distribution of the probabilities contributing to the average probability. Conversly, reading the other way, from average speed to fatality probability would give rise to an overestimate. However (and we've been careful to do this) reading from a fixed speed to a probability causes no error at all - the error only arises from reading an average speed across the curve. 

Choosing car drivers.

We chose car drivers for this exercise specifically because they suit the defined category of the Joksch equation. Had we chosen a less well protected road user group the average impact speed would obviously have been lower. 

Near misses excluded.

There's nothing in the physics of the situation that separates a near miss from a deadly crash. Both may start with similar vehicles speeds, similar opportunities to avoid and similar time to react for the participants. But in calculating our average crash impact speed of 11.85 mph we didn't look at the many millions of near misses. 
 

Other estimation errors

The Joksch equation probably lacks accuracy below 30 mph.

The Joksch equation predicts that 1 in 24 million car drivers would die in 1 mph impacts. It's not entirely impossible, and 24 million is a very large number, but we expect that the equation is pessimistic at very low speeds. However the results here are clearly far more remarkable than could be explained by a lack of accuracy in the equation at low speeds. One useful test is to re-compare the results at 30 mph - 30 mph is surely one of the lowest free travelling speeds in regular use on UK roads, and it's into the range where we could expect the Joksch equation to yield reasonable results. We calculated above that if the delta v of our (estimated) 1,500,000 annual accidents was just 30 mph we'd have 42 times the present number of deaths.

The Joksch equation was derived from observations of early 80's American cars

Modern European cars are clearly more crash worthy. But what happens if we go back ten years? Using 1992 figures from table 5c in RAGB 1999 we find 1,146 killed, 14,260 serious and 97,946 slight injuries affecting car drivers. Traffic has risen by about 18% between 1992 and 2002. Slight injuries have risen by 19%, which is close to the rise in traffic. So as a fair guess we'll suggest that damage only accidents have probably also risen by 18%. This leads us to an estimate of 1.27 million damage only accidents in 1992. This would lead to an average impact speed in an injury accident in 1992 of 12.3 mph. Since that's not materially different from the more modern estimate, we can reasonably discount improvements in vehicle design as an important factor.

Accident delta v is normally less than pre accident speed.

And that, of course, is the entire point. Most potential accident delta v (and hence kinetic energy) is mitigated by appropriate driving practice.

The purpose here has been to show the relative importance of vehicle speed and driver response in accident outcomes.

Taking a simple case we saw that making existing accidents into impacts at just 30 mph would kill 42 times more car drivers than die at present. This is a way of illustrating that driver response is more than 42 times more important than pre accident speed in the real world.

We've also seen that if we take driver response out of the equation (but leave them with the same number of accidents), we could kill just as many car drivers with no vehicle exceeding 12 mph. And yet we have countless thousands of vehicles exceeding 100 mph daily.

We have not considered the role of driver response in avoiding accidents at present speeds. Remarkably drivers go something like 7 years on average between accidents. How many accidents do they avoid in those 7 years? How many accidents would they have in 7 years if they drove - literally - with their eyes shut? 10,000? 100,000? What if they just shut their eyes for 20 seconds once a day while driving? In 7 years there are over 2,500 days. How many of those days wouldn't result in an accident? Not very many!

So what we really want is an estimate for the relative contribution to real world crashes of real world free travelling speed and "other factors". The principle "other factor" is of course driver response. It's a bit of a wild guess, but we'd say the driver response is at least 500 times more important to real world incident outcomes than free travelling speed. We derived a similar figures from analysis of pedestrian collisions (here).

We have also touched upon the fact that only about 1 crash in 1,300 kills a driver. Would that 1 in 1,300 event be distinguished by pre crash speeding? With 70% of drivers exceeding the speed limit at some sample sites, it would be a completely unreasonable conclusion. 

It isn't speed that kills. We can reduce the speed limits endlessly or enforce them perfectly without ever hoping to get close to the thresholds where free travelling speed will play a larger part in the outcome than driver based factors like skill, attention, attitude and training level. In fact, small variations in these factors will have far more effect on accident rates and outcomes than big variations in limited or enforced speed. See elsewhere on this web site.

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