# Can height of a curb be determined by the angles of scratches on the perimeter of a wheel that struck a curb?

At least to me, this turned into an interesting math question. All jokes aside.

With any level of certainty, can the angles of scratches on the outermost edge of a wheel of known diameter be used to calculate the height of a curb, which the wheel struck at low velocity?

The wheel didn't strike the curb head on, but rather scraped against the curb while approaching (So, imagine rolling forwards or backwards beside the curb and then attacking it at an angle less than perpendicular).

It comes to mind that the type of curb is unknown, but suspected to be either semi-circular or perpendicular to the ground.

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I recommend you do some experiments. I enjoy TV forensics shows, but am often disappointed with their sense of what constitutes truth in geometry, especially when there is some movement involved. Same for the one trial where I was on the jury. – Will Jagy Apr 30 '12 at 23:05
Thanks Will. I have been tinkering with circles of paper and a pen and patterns are emerging. In doing so, I realized that it could likely be solved more quickly with math, and handling the paper wheel is difficult. – Michael Prescott Apr 30 '12 at 23:16
I see. I meant an actual car with the wheel and tire you have in mind, tried on a few selected curbs. If this is not for a courthouse trial, I suppose that is lots of work. – Will Jagy Apr 30 '12 at 23:31
Can you explain why you've tagged this "probability" (or else remove that tag)? – Gerry Myerson May 1 '12 at 0:17
Sorry Gerry, not my specialty so I just guessed on how to categorize it. Perhaps, you could recommend the most accurate tags to get this more attention. The only reason I tagged it "probability" is it seems to me that there isn't an absolute way to determine the answer, but that it might be calculated with some amount of certainty. – Michael Prescott May 1 '12 at 1:23

Let us imagine that only one point of the curb scratches the wheel. If we let the wheel radius be r, the height of the curb h, and the horizontal distance of the contact point of the wheel to the point below the point scratching, and measure $\theta$ as wheel rotation from the point where the contact point of thwe wheel is just below the scratcher, we have $d=r \theta$ and the distance from the scratcher to the center is $\sqrt{r^2\theta^2+(r-h)^2}$. Then the distance in from the circumference is $r-\sqrt{r^2\theta^2+(r-h)^2}$. The angular position of the momentary scratch is not $\theta$, however because the scratcher moves only at horizontal velocity $\frac {r-h}rv$. So the angular poistion on the wheel is $\phi=\arccos \frac {r-h}{\sqrt{r^2\theta^2+(r-h)^2}}$If the scratch goes from one side of the wheel to the other, you can just take $h$ to be the maximum distance in from the circumference the scratch attains. But without that, if you have two ends of a scratch the angular distance between them will give $h$ or you can use the angle which the scratch meets any given radius.