Physics proof for traffic ticket

[12markkram34]

So I read about a scientist who wrote a physics proof to prove that he did not run a stop sign, basing his argument on the fact that angular speed of a car moving near a stop sign )as observed by a distant perpendicular observer) peaks shortly before and after the stop, where the observed angular speed is zero. During that brief moment that the speed is zero, another car was blocking the sight of the officer. It's probably easier to understand if you read the actual proof.

The proof is here: http://arxiv.org/pdf/1204.0162v1.pdf

So I understand everything up to part IV, where he calculates the times that obstruction starts and ends. What I don't understand is why he added the two lengths together for one and subtracted them for the other. I think the problem is that he doesn't really specify where the obstructing car is at a particular time, though it could just be me because I'm not really good at mechanics.

Of course, his overall argument is pretty unrealistic since it contends that acceleration starts immediately after deceleration, but I still want to understand the basis of the "proof."

EDIT: I also see an issue with how he defines where his car is. From his part about speed, we can assume that he is defining x as the distance from his front bumper to the stop line. If that is the case, then it would not make sense to calculate when his front bumper is the sum of the lengths away from the line, since if partial obstruction began when he was at that point (that would make the front bumper of the obstructing car be l1 away from the line), his car would be exposed at the stop line (since the obstructing car does not reach that far). Could it be just that that part of his proof is totally bad?

 

[12markkram34]

Bumping since it's been two days without a response.

 

[A.T.]

What I don't understand is why he added the two lengths together for one and subtracted them for the other.

That makes sense for calculating total and partial obstruction. But I didn't go through his math.

 

[Jasso]

What I don't understand is why he added the two lengths together for one and subtracted them for the other.

His definition of the distance of partial obstruction spans when the front bumper is the same distance from the stop sign as the other car's rear bumper, to when his rear bumper is the same distance as the other car's front bumper.

Assuming the other car is stationary at the stop sign, this is a distance from when the front bumper is at $x_1 = -l_2$ (front bumper is even with the other car's rear bumper) to when the front bumper is at $x_2 = +l_1$ (rear bumper is even with the other car's front bumper). Using $\Delta{x} = x_2 - x_1$ gives $x_p = (+l_1) - (-l_2) = l_1 + l_2$.

His definition of full obstuction occurs when the rear bumper has passed the other car's rear bumper, yet his front bumper has not passed the other car's front bumper. So at the point when his rear bumper is even with the other car's rear bumper, his front bumper has a distance of $l_2 - l_1$ to travel before it goes past.

I think the problem is that he doesn't really specify where the obstructing car is at a particular time, though it could just be me because I'm not really good at mechanics.

He is using small angle approximations. Actually, he uses a lot of simplifying assumptions, approximations, and worst case scenarios, probably for an audience which does not have a thorough background in physics.