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In a recent answer I asserted

Statistics show that the probability of a collision rises rapidly after the speed difference between two road users exceeds 20 kph (12 mph). So choose paths without vehicle traffic, or roads with lower speed traffic.

User ebrohman requested a source for this. Since I remembered this "fact" from years back (er ... not only was it last millenium, it was from before the internet) I had to scrabble about to try to find an online source.

I found weak support for my thesis

... this US pdf Synthesis of Safety Research Related to Speed and Speed Management; it doesn't explicitly support the 20 kph claim. This Dutch pdf The relation between speed and crashes says at a collision speed of 20 km/h nearly all pedestrians survive a crash with a passenger car; again not the same as my claim ...

Intuitively, my claim makes sense. The greater the speed difference between a vehicle and me, the more likely a random driver is to make an error and collide with me.

But can we factually quantify how the risk varies with the speed difference?

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    Your two quotes are addressing different things. Obviously an accident, when it happens, is more likely to produce serious injuries if the closing speed is greater. This is (mostly) unrelated to the probability of the collision in the first place.
    – Daniel R Hicks
    Commented Jun 28, 2015 at 13:16
  • More relevant is this answer: bicycles.stackexchange.com/a/9035/1584
    – Daniel R Hicks
    Commented Jun 28, 2015 at 13:17
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    How many collisions really happen head-on? I think the cases where a driver overlooks a cyclists driving right in front of him are quite rare. Subjectively it’s mostly cars "cutting" cyclists on right turns, cars overlooking cyclists when they’d have to yield or parked cars suddenly opening doors.
    – Michael
    Commented Jun 28, 2015 at 14:34
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    @Michael - It should be noted that a substantial fraction of car-bicycle collisions (perhaps half) are technically the fault of the cyclist. And a substantial fraction of serious bicycle injuries (again, I'm recalling that it's about half) don't involve an automobile at all.
    – Daniel R Hicks
    Commented Jun 28, 2015 at 18:59
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    @DanielRHicks There must be a lot of bad cyclists in (where i assume you are from) USA, as over here in the UK the figure is around 17-25% where the blame can be apportioned to the cyclist (as per a study carried out by the Department of Transport). Just over 15% are where the cyclist crashes on their own.
    – Cearon O'Flynn
    Commented Jun 29, 2015 at 13:01

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As some have pointed out in the comments, the question is somewhat misleading, i.e. speed difference is always going to be greater in a head on collision, all things being equal, versus if the collision had been a rear ended collision or a t-bone, but, as @Daniel notes:

"This is (mostly) unrelated to the probability of the collision in the first place"

According to The League of American Bicyclists:

In 40 percent of the cases (628 total), the victim was struck from behind. The second-most common category was T-bone crashes, which accounted for 10 percent of fatal crashes. Head-on collisions (8 percent) and right-hook crashes, where the driver makes a right turn into a traveling straight (6 percent), were relatively less common.

enter image description here

Getting rear-ended is obviously the most common type of accident. These numbers do not say the percentage of fatalities from rear-end collisions (they mention it for T-Bones). However, if one holds to certain basic assumptions regarding the topic at hand:

  1. Most cyclist will try to avoid a collision at all costs
  2. A collision is prerequisite for fatality
  3. Speed difference is always greater head-on than same direction, rider speed and vehicle speed held equal

The conclusion one can draw from this is that all things being equal, following these stats (and other sources below) one should always ride head on, because statistically, one has the least chance of collision, even if the closing speed is the greatest, which itself, would then seem not to have any correlation with risk of collision, or possibly an inverse relationship if assumption 3 is held.

Head on collisions may have a higher incidence of fatality than others, I don't know. If someone finds data on that please share.

In summary, riding at a lower relative speed (to other vehicles) alongside traffic is more dangerous but probably less fatal, whereas riding at a higher relative speed against traffic is statistically less dangerous but may prove to be more fatal in the event of an accident.

Other relevant sources: http://www.trafficsafetycoalition.com/bike_studies

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    Rear end collisions are probably most common for same reason that home is the most common place to have an accident: Most cyclists ride with the flow of the traffic most of the time and getting in to head on collision would require riding against the flow. There's just not so many opportunities for head on collisions as rear end ones.
    – OttoK
    Commented Jul 24, 2015 at 21:54
  • Actually all it would require is for a vehicle to swerve into oncoming traffic.
    – ebrohman
    Commented Jul 24, 2015 at 22:21
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    And cross over the car lanes, and the cyclist not to notice this car and move. And cars don't swerve into oncoming traffic anywhere near enough to compete with the rear-end statistic.
    – LastStar007
    Commented Jul 24, 2015 at 22:24
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    Firstly, I appreciate your effort in answering; +1 this answer is useful. But I am asking how the risk varies with the speed difference. I see that people don't understand what I'm asking; from my point of view that is because of confusion between the concepts of speed, closing speed, and velocity. I am asking about speed difference. If two cyclists approach head on, each at 20 kph, the speed difference is zero, but the closing speed is 40 kph. If you think this makes it clear, I'll add this to the OP.
    – andy256
    Commented Jul 25, 2015 at 4:03
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    @andy256 thanks for clearing up the question. I understand the difference b/t speed, velocity, etc. but the problem is that if one thinks about it for a minute, comparing velocity makes much more sense if what you are trying to get at is the risk of collision and its possible effects. Speed difference only really makes sense in the case when the driver and cyclist are always traveling in the same direction. Consider a bike flying downhill at 50mph and a driver trying to avoid him at the same speed coming up the hill.
    – ebrohman
    Commented Jul 25, 2015 at 13:01

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