Using math to get out of a traffic ticket

We've talked about arXiv here before. It's a pre-print server for scientific papers in the fields of physics, mathematics, and computer sciences. Basically, what that means is that scientists can post papers to the site without first putting that research through the process of peer review. And that's not a bad thing. ArXiv is a great way for scientists and mathematicians to critique each other's work and do a little bit of vetting before submitting the paper to peer review. That's why the faster-than-light neutrino reports were published on arXiv—the results looked so crazy that the researchers wanted their colleagues to figure out what had gone wrong before a prestigious journal got involved. It's a way of collaborating.

The other nice thing about arXiv: It's a great home for interesting data that doesn't necessarily have a place in a formal, peer-reviewed journal.

Case in point: "The Proof of Innocence", a paper in which physicist Dmitri Krioukov uses math to explain why the cop who stopped him for running a stop sign was clearly seeing things. Physics Central summarizes the first step in this defense:

When Krioukov drove toward the stop sign the police officer was approximating Krioukov's angular velocity instead of his linear velocity. This happens when we try to estimate the speed of a passing object, and the effect is more pronounced for faster objects.

Trains, for instance, appear to be moving very slowly when they are far away, but they speed past when they finally reach us. Despite these two different observations at different distances, the train maintains a roughly constant velocity throughout its trip.

In Krioukov's case, the police cruiser was situated about 100 feet away from a perpendicular intersection with a stop sign. Consequently, a car approaching the intersection with constant linear velocity will rapidly increase in angular velocity from the police officer's perspective.

Krioukov's basic argument: The officer thought he saw Krioukov speed right through the sign. But he was wrong. Instead, Krioukov stopped at the sign, but stopped very quickly and sped up quickly, both of which happened out of the cop's direct line of sight.

It's worth noting that this argument was good enough to get Krioukov out of a $400 fine.

Read Krioukov's paper.

Read the summary on Physics Central.

Image: Stop, a Creative Commons Attribution (2.0) image from misteraitch's photostream



  1. Ha! My calculus teacher told us about this way out of tickets in high school  – but I could never really remember the details. (Shocking.) Great to be reminded. 

  2. It’s worth noting that this argument was good enough to get Krioukov out of a $400 fine.

    Really? Because while I follow the summary you provided (and haven’t clicked through to look at the original), it seems to me that the driver indeed didn’t accelerate towards the intersection, contrary to the cop’s possible perception, but on the other hand the driver didn’t stop at the sign long enough for the cop to notice, and their “rolling stop” probably was a moving violation.

  3. I’ve always wondered if radar guns have a way to compensate for the off angle axis to which many are used..

    1.  Surely a radar gun could do the basic trig required to work out the vehicle’s actual rate of movement?

      1. Actually looking at Wikipedia it basically states they don’t take angle offsets into account:

        Mobile traffic enforcement radar must occupy a location above or to the side of the road, except when the roadway is occupied by only one vehicle. The user must understand trigonometry to “guess” vehicle speed as the direction changes while a single vehicle moves within the field of view when positioned adjacent to the roadway. Vehicle speed and radar measurement are rarely the same for this reason.

        I guess that explains the old 5mph over rule.

  4. If you didn’t stop long enough to scan the whole intersection (about 2 seconds), you might as well have not stopped at all. Stop signs, at least, are one traffic rule that is almost purely about safety for drivers and pedestrians. You don’t treat them as an arbitrary rule that you obey just enough to not get caught. They notify you that you need to check to make sure the way is clear, and that someone else isn’t about to run a stop sign and hit you.

    1. Here in the UK, we don’t have very many stop signs. We have “Give Way” (yield) signs where in my experience there would be a stop sign in the US. Our roads are still relatively safe (I think safer than in the US).
      Checking that the way is clear does not always require you to bring the car to a complete stop. Especially if you’re turning left (or right in countries that drive on the right), you only need to check for traffic in one lane, and it is often safe to do that while moving slowly.

  5. … he had a terrible cold that day, and one sneaky sneeze caused him to slam on the brakes hard as he approached the stop sign

    Given that he claims that he was stationary for a very short time this means that without the sneeze, he’d have gone through the stop sign.  Perhaps he should have been fined for driving without due care and attention?

  6. On the original forum one commenter notes that no evidence is provided that this incident ever happened and another notes that based on his own testimony he didn’t actually stop long enough for a legal stop.

  7. It can’t be ignored that this paper was published on April 1. Although there doesn’t seem to be any indication of shenanigans. I was curious enough to email Krioukov and he indicated that the court case was real. That’s good enough for me.

  8. Something I expect to show up in a case before long involves the validity of video tape evidence shot as MPEG. Only the I frames are real complete snapshot like images. The B frames and the P frames are concocted from information in I frames. An attorney could blow this up and claim the entire image can’t be used as evidence. I’m surprised it hasn’t happened yet.

  9. I’m showing that to my algebra 1 class tomorrow! They won’t get the trig, but they get $400.

  10. I suspect that  “happened out of the cop’s direct line of sight.” has more bearing on this dismissal than fancy physics. How the heck does a cop testify to something he didn’t see?

  11. Using AutoCAD, I created a 4-panel PDF poster showing how a radar gun can double a speed reading a you near a stationary flat surface facing  your vehicle.  I used the published radar gun’s operations and basic physics to show what actually happened to me while driving one night. Outside court, when the cop saw the rolled up drawings  (I had a detailed area contour map, too) he asked what they were.  I chuckled,  said “Just wait til we see the judge “, and walked away. After a few minutes in the courtroom, the cop walked in, whispered to the judge and left. He had told the judge to just drop the charges.

  12. Stop signs are one of the most annoying things about (the very limited set of which I have experience of) American roads. On the corner of *every* grid is a stop sign (or lights, which are even worse) and you *have* to stop. And this experience was primarily on a bike, which apparently are still required to stop. One really feels the energy wastage when cycling, so I can only guess at the total increase in the carbon footprint of the US of having soo many junctions and requiring every vehicle to stop at every one.

    I learned 2 things from this – (1) grid systems are stupid for if you don’t want the world to be ruled by cars (every branch will get used, making every street a busy street) and (2) roundabouts are fantastic.

    1.  I agree with you that roundabouts are better than stop signs, but you are wrong about grid systems.  If you have a grid, you are correct that every branch is used for cut through traffic.  But because that traffic is spread out throughout the whole grid, no street is busy.  In fact, you can set low speed limits on each street, and still move cars more quickly than if you have separate developments connected by huge arteries.  Spend any time in a modern US suburb, and you will see how terribly congested roads can get when there is no grid and all cars must go out to the main road to go anywhere.

      1.  I would argue the problem is a more related to poor transport planning. Too many cars is the *real* problem. I absolutely accept that cars distributed over the grid means no roads are stupidly busy, but I never was in a position where I felt I was away from cars. Even in London I feel that there is always a route I can take on a bike or foot that is largely away from cars, wherever I want to go.

  13. When I first saw this story, I thought it was the more usual case of traffic-cop maths- “I had to do 80 mph to catch up with you, therefore you must have been speeding”.

  14. … and I was expecting the old Doppler  Effect idea that “the light looked green to me” excuse. It’s always fun to calculate the speed required for that kind of blue-shift.

  15. This is mostly BS wrapped in a veneer of respectability.  The only part that may have worked in his favor is saying his car was not visible to the cop at the time he stopped.  I especially liked his description for estimating his rate of deceleration, saying that he sneezed at the moment he applied the brakes, therefore he assumed he stopped as quickly as possible.  Which also means he was not able to control where he stopped, and very likely did not stop at the stop line.

  16. Yes, the assumption of symmetric deceleration and acceleration is just lazy, but still pretty much guaranteed to get past a non-mathematically savvy court.  I suspect they let him off simply because of the obscuring of the cop’s field of view by the other car – I also suspect Dr K did coast through the intersection, given the rather extreme deceleration, short pause, and symmetrical extreme acceleration required for the illusion to hold …

  17. (Almost) every big rig driver knows about angular velocity &  linear velocity. No, not in those exact terms, but when idiots pull out or step in front of you (I drove and articulated bus), they think you’re going 20 when you’re really going 40. We call ’em suicide attempts. 

    Or “hey, that’s a big bus, it must have BIG breaks, therefor it can stop faster.”

    Or the person pushing a stroller, which dangles out in traffic. All they know is they are safe on the sidewalk, but have no idea something is sticking four feet out in front of them. 

    The city of Seattle’s Metro Transit has an excellent class B licensing and training course.

  18. That has actually been debunked. The court said that the ticked was dismissed not because of the paper but because the court determined that the cop was not positioned properly and did not have an adequate view of the intersection.

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