Long Walk After the Sun – Challenge

This question popped into my head the other day out of nowhere but sounded really interesting.  It may involve, but is not limited to, programming, geometry, 3d data visualization and astronomy.

Setup: At 12:00 AM, January 1st, 2009 a man is standing in his front yard.  That morning, as soon as the sun breaches the horizon, he begins walking at a constant rate in the direction of the sun.  He walked all day, always in the direction of the sun, until nightfall when he stops and sleeps.  He does this every day for an entire year.  As the sun climbs higher in the sky each day, he always walks in the direction of the sun relative to directly above his head (assume he has a magical sundial that is very accurate).  Oh, and he can walk on water and through mountains (terrain doesn’t slow him down nor stop his motion).

Challenge: Given he lives in downtown St. Louis, Missouri, and that he walks at a constant rate of 3 miles/hour, where will the man be sleeping on December 31st, 2009 after the sun has set?

Bonus points for solutions that can compute his final location based on an arbitrary starting location and/or walking speed.  Averaged sun rise and sun set times can be used, but using some historical monthly/daily averages would be even better.  Also the changing location of the sun’s arc during the year could/should be taken into consideration.  I can imagine this is a fairly hard problem, but it seems doable with some effort.  I will make an honest attempt at solving it myself, but if anyone is up to the challege, please post your thoughts and findings.  I found the problem interesting b/c several “easy” answers jump out initially but without the math, I can’t be sure which one if any are true (will he end up in the same place?  somewhere near the equator? at a pole?  go in circles?