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.

Key point: Smith acknowledges that his model bears no relation to real life.

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. 

Interesting use of the word logic! But the intentions are indeed identical: to take data which proves that speed kills and twist it to assert that speed doesn't kill. The apparent purpose of Smith's page is either to try to prove that speed doesn't kill, in which case its many logical fallacies ensure that it fails; or to attempt to prove that the arguments used against speed are invalid, in which case the many errors of application make it invalid. As an insight into the mental gymnastics of a speedophile it is, however, illuminating.

Fatal flaw. As in the original page, Smith deliverately excludes the groups who are most affected by speeding: users of benign modes. That's pedestrians, cyclists, horse riders.

To help understand why excluding these groups is a fatal flaw, consider the case of a car crash at 30mph. At 30mph the driver of a car hitting a pedestrian has approximately 0% chance of dying. The pedestrian has approximately 45% chance of dying. Now speed up to 40mph. The driver still has a vanishingly small percentage chance of dying, but the pedestrian now has a 95% chance of being killed. This is important because two thirds of those who died on the roads in the period under consideration were not drivers.

Nor are drivers "typical of all trends." Over a period of decades the risk of death or serious injury per mile travelled for cyclists rose steadily, while it fell for drivers. When compulsory wearing of seat belts was introduced the driver fatality level stayed the same but there was a steep rise in pedestrian and cyclist fatalities.

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.

Note: this is the probability of fatality in a recorded crash causing injury, not the probability of fatality in any crash - the figures for recorded crashes are sufficiently prone to error that no serious researcher uses them for anything other than indicative purposes and there are somewhere between five and ten times the number of crashes reported to insurance companies as to the Police.

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.

Fatal flaw. Joksch's equation relates the probability of fatality in a crash to the mean speed of a road, at highway speeds. It is a rule of thumb only, although other research supports it reasonably well. Joksch explicitly states that it is not valid for speeds below 30mph, so extending the graph down to 0 as Smith does is clearly a misrepresentation. It was misrepresentations of this kind which led Joksch to ask Smith to remove his name from the original 12mph page.

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.

Fatal flaw: The largest component in differentiating delta-V from V is actions by the driver. Braking. If we are confronted with a hazard at a free travelling speed of 30mph we will undoubtedly apply the brakes, and the impact speed will be much less. This is an important point - the underlying message of this page is that if we all drove slowly we could kill as many people, but that requires a radical change in behaviour, abandoning braking in favour of ploughing straight into things. Hardly a sound basis for argument. But as Smith says at the beginning, this argument bears no relation to real life.

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 

Fatal flaw. Joksch's rule of thumb relates probability of fatality given a crash to mean free travelling speed on the section of road, at highway speeds. Smith is using this empirical value to infer impact speed from the proportion of fatalities to injury crashes, for all road types. The answer is outside the applicable limits of the Joksch rule of thumb. In other words, the approach Smith is using is fundamentally wrong. Oh, and the KSI figure includes the fatality figure, so he's double-counted the dead. Oh, and he's only including recorded injury crashes, so the ratio is wrong, possibly by an order of magnitude.

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.

Fatal flaw. This implies that free travelling speed and impact speed are the same. Had all those drivers been travelling at 30mph prior to the hazard arising, then it is obvious that by the time they had braked in reaction to the hazard not only would their impact speed have reduced substantially, in most cases, but the drivers themselves, protected as they are by safety cages, seat belts, airbags, crumple zones and the like, would be able to walk away from most such crashes. So, this assertion is invalid because it confuses v with delta v, because it is outside the limits of applicability of Joksch's rule of thumb, because Joksch relates mean speed to probability of death but Smith uses it to infer something else entirely, and because it ignores the lack of any mechanism whereby driver fatality could result at speeds substantially below 30mph. That's four fundamental errors in one argument - not bad going.

Any reasonable person would, on realising that the figures completely fail to add up, realise that their approach is fundamentally wrong. But full marks to Smith for dogged persistence in the face of overwhelming contrary evidence.

Might one suggest that getting a great deal more realistic would probably be a better bet?

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.

Fatal flaw. Smith advances no reasons for assuming that 50% of motor insurance claims apply to private cars, or indeed for assuming that 75% of insurance claims are due to motor crashes (other figures suggest that the number of motor claims alone could be as high as five million annually). He also applies a rule of thumb which relates solely to fatalities at speeds above 30mph, applies it to crashes a large proportion of which happen at speeds well below 30mph (parking damage etc.), and extends it to infer that if speeds were limited to the speed of a trundling cyclist the same number of drivers would die.

How? I have seen a car which was involved in a head-on crash with a closing speed estimated by firefighters in excess of 70mph, where the driver suffered no serious injuries. I know at least one driver who fell asleep at the wheel, ran off the road at 45mph as estimated by Police, rolled twice and landed upside down, and still walked away with bruises only. These are extremes, but the point remains that at 12mph in a modern car one would be extremely aggrieved if one suffered a broken bone, let alone death.

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.

Fatal flaw. Smith has just proved that Joksch is right in saying his equation cannot be used to predict fatality rates in this way. Well done. At this point (if not before) we have a strong clue that the equations are being mis-applied and it's time to think again. Or rather, time to start thinking.

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?

Fatal flaw. The nature of whole population statistics (including those on which Joksch based the equation) is that they include whole populations. Seeking to exclude the individuals said to be most likely to crash is invalid. By this point Smith has started with a rough rule of thumb for the probaility of death in crashes above 30mph, used it to infer speed by comparing death and serious injury (not part of Joksch's equation), extended this to make wild guesses about near misses, and only then started talking about excluding precisely those crashes to which the Joksch rule of thumb would apply, crashes above 50mph impact speed.

What Joksch tells us is that the probability of fatality increases with the fourth power of speed. Smith tells us that what Joksch really means is that speed doesn't kill. The mental gymnastics used are pretty impressibe, but the arguments aren't. Go back to Smith's original source - it is a synthesis of research showing the link between speed and death.

In addition, by reference to vague guesses about near misses and conveniently ignoring a mere two thirds of the dead, Smith asserts that being involved in a crash is not something we should be concerned about. Here at least he addresses a key issue responsible for many deaths. By assuming that our safety as drivers is a measure of all safety even though most of those who die are not drivers, we endorse the view that only driver safety matters. This is the mindset which gives us dangerous roads, dangerous trees and dangerous children instead of dangerous drivers. In reality the risk of bad driving to the driver, well protected in his car, is reducing constantly; and this very fact can lead to increased risk for others through risk compensation - as happened when seat belt wearing was mandated.

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

As detailed above, these are not "high estimates" but "invalid estimates." The numbers are the same as the ones from which Joksch asked for his name to be removed, and they are invalid for the same reasons.

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. 

Fatal flaw. This is a really odd one. First, the coloured lines don't respect the published applicable speed range of the rule of thumb. Second, the exponential nature of the curve makes no difference if one is genuinely calculating speed from probability - that would be the entire point. Engineers don't refer to kinetic energy measurements as being inaccurate because E=1/2mv^2 is not a straight line. Third if the exponential curve makes for inaccurate averaging, clearly the entire argument is specious in the first place! But just look at the numbers of decimals Smith uses. Clearly he has never encountered the concept of meaningless precision.

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. 

Fatal flaw. There is no relationship between Joksch and pedestrian fatalities. The pedestrian fatality rate curve is a completely different shape, flattening off at 100% above about 40mph.

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. 

Fatal flaw, and it's a doozy. The figures arrived at by invalid application of a rule of thumb are an underestimate because they don't include crashes which never happened. Right. So, including the proportion of near misses Smith invents above and using his methods I calculate that the average fatal crash happens at just about walking pace. Nice one!

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.

It doesn't "probably lack accuracy," it's simply inapplicable. As Joksch says in his papers.

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.

Now account for the fatality mechanism. Drive a ten-year-old car into a brick wall at 30. Now drive a brand-new car into a wall at 30. Smith is trying to cover up the inapplicability of the Joksch equation and the unrealistic values he has arrived at by reference to something for which Joksch has not controlled, secondary safety, which would in any case play a decreasing role as impact speed increases. Smith has, however, accidentally hit upon one of the primary reasons why applying Joksch's rule of thumb outside its context is a pointless exercise. As pointed out above, what is the mechanism for causation of fatality in drivers in a 12mph collision in a modern car? Surprise? Given the supposed rarity of crashes, only a few millions a year, perhaps that's it.

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.

Right. So we don't crash at free travelling speeds because we brake, adn this exercise set about proving that - what? if we didn't brake we would have the same impact speed form a lower free travelling speed? I think we could have worked that out for ourselves without the pseudo-scientific bullshit. But we wouldn't have been so idiotic as to apply a rule of thumb for the probability of fatality in freeway crashes in American cars in order to put a figure on it.

We must also remember that speed plays a determinant role in how effective braking will be, and in what happens when we brake. Consider for a moment the chart of stopping distances in the Highway Code, in which we can infer that the thinking time is reckoned to be about 2/3s and (if they are assuming linear deceleration) the braking force is about 2/3g. Reaching for the O Level physics book and plotting a curve of v²=u²+2as we get the following:

This crude chart gives an indication of how speed changes with distance as we brake. You can see that using this unsophisticated model, if we were travelling at 40mph before the hazard was perceived, then we will still be doing around 30mph at the point where, if we'd been doing 30mph, we would have stopped. It takes a third longer to stop from 40mph than it does from 30mph. As long as the hazard didn't occur in that zone, then there's no problem - but if it did, and the limit is 30, then speeding alone "caused" the crash. All hypothetical and not at all sophisticated, but as you see it is impossible to divorce absolute speed from crash causation and outcome in the way Smith is seeking to do.

In point of fact speed is seen as a primary factory in around one third of crashes (amusingly coincident with the one third of fatalities made up by drivers, but not in any way related). This is often disputed by drivers' organisations but the science behind it is robust and repeatable. And even if one were to discount this entirely it is impossible to argue that speed is not a prime determinant of the seriousness of outcome of any crash.

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

Fatal flaw. This issues is alluded to by assertions in the summaries, but not addressed in the body of the document.

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.

Fatal flaw. Joksch's rule of thumb is inaplicable at that speed. One can draw no valid inferences from it.

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.

Fatal flaw. There is no credible mechanism for causation of fatality at that speed.

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!

Self-congratulatory waffle of no relevance to Smith's argument.

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).

"A bit of a wild guess." Uh-huh.

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. 

Unsupported assertion. The relationship of fatality to speeding is well documented in the summary of research which Smith references, but this is not considered in his arguments. Smith also apparently supposes that most drivers don't care that they are more likely to kill a pedestrian than die themselves. I hope he is as wrong about this as he is about everything else.

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.

Unsupported assertion. Smith advances no evidence to support this statement, it is a "conclusion" not based on any of the figures or arguments on this page.