## 22 February 2016

### Daytona 500 Finish Line Math

I was not able to see yesterday's Daytona 500, in which Denny Hamlin beat Martin Truex, Jr by the smallest time since electronic scoring came into NASCAR back in 1993.  But headlines and stories this morning got my attention.  Most stories I read reported a winning time of 0.010 s, i.e. three digits past the decimal were stated.  I then read that Hamlin "won by inches" in several articles, some of which gave winning distances in the range 4 in - 6 in.  Before I saw the photo finish, the numbers weren't making sense to me.

A NASCAR race car travels about 200 mph, which converts to 3520 in/s.  Winning by 0.010 s means winning by (3520 in/s)(0.010 s) = 35.2 in, which is nearly 3 ft.  That's why the numbers I read didn't make sense to me.  I'm assuming a constant speed here and in what follows, a reasonable approximation for such a short time interval.

I then watched a replay of the final lap.  Speedometers on the lead cars showed they were just over 190 mph in the final turn.  I grabbed a screen capture of the finish (click the image for a larger view).
That winning distance looks larger than 6 in to me.  I checked out one web site (click here) that told me that the Toyota Camry Hamlin was driving is 189.2 in long.  Another web site (click here) told me the tire diameter is 28 in.  Using either of those reference lengths as my calibration distance, Tracker told me that the winning distance was about 20 in.  If Hamlin's tires really have a diameter of 28 in, a 20-in winning distance looks reasonable.  The gap between Truex's car and the finish line looks smaller than a tire diameter, but not by much.

So take the winning distance to be 20 in.  If the winning time really was 0.010 s, that means a speed of 2000 in/s = 114 mph.  That can't be right!  If the cars were going 200 mph and the winning distance was 20 in, the winning time would have been (20 in)/(3520 in/s) = 0.00568 s.   Crank up the speed to 220 mph and the winning time drops to 0.00517 s.  Drop the speed to 180 mph and the winning time increases to 0.00631 s.  All three of those times round to 0.01 s, but the winning time was reported as 0.010 s, i.e. three digits past the decimal.  I actually found a couple of articles that had the winning time as 0.011 s, but that makes what's being reported even worse!

I didn't see the race and thus may have missed commentary about the winning time.  It seems to me that the winning time that was reported was too big by a factor of 1.8 or so.  If there are any NASCAR fans out there who can provide me with a missing detail, I would love to hear from you!