December 24th, 2013

eyes black and white

Christmas Warning!

Let's assume Santa visits all NYC households with children at midnight, delivering a measly average of .5 kg of toys at each place. With 30% of 3 million households having children, that's 9e5 stops. Assuming the households more or less equidistributed over the 7.8e8 m^2 area of city, a circuit to cover them with 9e5 stops, using either a linear scan or a fractal scan a la Hilbert Curve or Polya sweep has a length of the order of sqrt(surface*households) = 2.6e7 m. Let's assume one order of magnitude savings because of non equidistribution, so only 2.6e6 m, to cover in 60s. Therefore the average speed of Santa, assuming instant stops, is 4.3e5 m/s. If he carries all his undelivered toys all the time, the average weight of toys he has to carry is 9e5*.5/2 kg = 2.2e5 kg. So his kinetic energy while traveling is .5 * 2.2e5 * (4.3e5)^2 J = 2e16 J, which is that of an atomic bomb hundreds of time more powerful than the Hiroshima bomb. But since he has to stop and resume 9e5 times, this energy is released in the environment at every stop while decelerating and then again while accelerating, so the net result of the visit is a dissipation of energy of 2 * 9e5 * 2e16 J = 3.6e22 J, which is hundreds of thousands of times worse than the greatest nuclear weapon of all times, the Tsar Bomba, except directly delivered to households with children. Of course, all these numbers are lower bounds and constitute a very optimistic scenario. To be even more optimistic, let's suppose that Santa is even more clever, and actually makes intermediate stashes of toys that he manages to instantly load and offload, and arrange his circuit so that on each segment of his fractal path he only carries the toys for that destination; this way, he might reduce the average weight of toys to something of the order 1kg or so. Assuming he is himself a weightless elf and that so are his sleigh and reindeers, so that only the toys are physical objects subject to the law of physics, that's a hundred-thousand fold saving in energy, which brings us back to only a few Tsar Bombas. Again, delivered directly to households with children. I recommend you not be in town or anywhere near children at midnight on this (or any) Christmas.