“We show that if a car stops at a stop sign, an observer, e.g., a police ocer, located at a certain distance perpendicular to the car trajectory, must have an illusion that the car does not stop, if the following three conditions are satised: (1) the observer measures not the linear but angular speed of the car; (2) the car decelerates and subsequently accelerates relatively fast; and (3) there is a short-time obstruction of the observer’s view of the car by an external object, e.g., another car, at the moment when both cars are near the stop sign.” (pdf)
In tonight’s Moment of Geek, Chris Hayes describes the mathematical proof presented by Dmitri Krioukov, a UC San Diego physicist, to explain why the police officer who ticketed him was wrong in his perception that Krioukov did not stop at a stop sign. The math in the proof is pretty heavy, but the reasoning isn’t too hard to follow.
The chart above on the left is graph of the angular velocity of an object passing an observer. Angular velocity, as opposed to linear velocity, matters in this case because it creates an optical illusion. As Krioukov explains:
For example, if we stay not far away from a railroad, watching a train approaching us from far away at a constant speed, we perceive the train not moving at all, when it is really far, but when the train comes closer, it appears to us moving faster and faster, and when it actually passes us, its visual speed is maximized.
So the peak of the line in the left chart is point when the object passes the observer and seems to be going the fastest even though its speed is constant.
On the right is the graph of angular velocities when the object comes to a complete stop and then accelerates again at a rate matching the rate of deceleration. The chart shows a few different rates of acceleration, but point is that if your view was blocked, by, say, a Subaru Outback, for the time between the two blue peaks, the passing object would appear to have behaved a lot like the object in the left graph that didn’t stop.
It may be, as io9 suggests, that it was sufficient to argue that the police officer didn’t have a clear view of the stop, but it’s nice that he offers this explanation of the officer’s misperception.
The one question I’m left wondering is whether the defendant’s car, a Toyota Yaris, is capable of the rates of deceleration and acceleration described by his math. It seems like enough data exists to find out, but unfortunately the angular velocity of my physics lessons receding into history makes figuring it out impractical.
(How’d I do with this explanation? Let me know if I’m not getting it.)
UPDATE: Here’s the video: