How Fast Must Santa Travel

Today is Christmas and along with Christmas comes Santa Claus.  Depending on the age of your kids, they may or may not still believe in Santa, but that's not the point of this post.  Each year on the first day of the second semester of my physics I give my students a thought problem where they have to calculate an approximate speed that Santa must travel to visit every house in the world in 24 hours (ignore time zone changes).  Let's work through it.

To start this problem I assume that Santa begins at the North Pole and slowly moves in circles about a given latitude toward the equator and then to the South Pole.  Let's also assume Santa has some magical powers and doesn't have to physical stop at each house.  He just flies through the air throwing presents left and right.  The presents magically work their way into the houses.  Let's also assume Santa has incredible arm strength and has a throwing reach of 500 meters in each direction.  Thus each swipe around the Earth covers a width of 1 km.

The circumference of the Earth is 40,075 km.  Santa makes circles going down one side of the Earth, so the distance Santa travels down the Earth is half this...20,037 km.  If each swipe has a width of 1 km, Santa must circle the Earth 20,037 times.  So how long is each of these swipes?  At the equator the swipe is 40,075 long, but at the North Pole the swipe is 0 km long.  Let's assume the average swipe is half the circumference of the Earth.  Thus we end up with:

20,037*20,037 = 401,491,387 km traveled.  Add on to that a one way trip back from the South Pole to the North pole and we get:

401,491,387 + 20,037 = 401,511,424 km traveled

Santa must do this in a 24 hour period.  To get Santa's speed we divide distance by time (converting to seconds):

401,511,424/86,400 = 4,647 km/s

That's blazing fast!  But is it physically possible?  The speed of light, the fastest anything with mass can travel is 300,000 km/s, so Santa is traveling less than this.  In our estimate, Santa travels at a speed of 1.5% the speed of light.  Well within the realm of possibility!

Too much geek for one day?  :-)