Numbers Wednesday #1: Santa Claus
So, preceding the actual content here, what exactly is Number Wednesday? Well, while I would love to dedicate a day of the week (explanation on that I am saving for Sunday) to Maths and another to Physics and another to some other number-based thing, the fields tend to overlap, especially as soon as application starts to happen. So in the midst of the other days, I’m putting the “cool shit with numbers” day right in the middle.
Anywho, many people have likely already seen some maths regarding Santa Claus and his improbable journey around the world. The numbers have been floating around the web for quite some time in various forms; I recall e-mails from years ago describing them, a website I copied onto my Facebook about a month ago, and lately some Facebook pictures being shared. So the concept is hardly new, but I would like to go ahead and take a harder look.
If we take a look at this post at PayDirt, a pretty common version of the maths is found. If we break the paragraph into basic data, we get:
-2 billion children
-Santa only delivers to Christian children
-Christian children comprise 15% of children, about 300 million
-Household average is 3 children, resulting in 100 million houses
-31 hours of Christmas, thus 900 stops/s
-0.001s to do each home
-Even distribution of homes yields 70 million miles total
-Santa thus moves at 650 miles/s
-Voyager 2 moves about 9 miles/s
-Suggests Santa has a warp drive
-If every child asked for small LEGO set, results in 380,000 ton sleigh
-Requires 250,000 magic reindeer, increasing weight to 460,000 tons
-Titanic weighed 46,000 tons
-Santa experiences 14 quintillion J/s, enough to vaporize reindeer and sleigh
So now for the fun part: analysis and criticism!
…and addition.
The claims about quantity of children come first. The number of Christians is hard to pin down, but for the sake of this data, the number 2 billion (2,000,000,000) is close enough with a whopping one significant figure as well as uncertainty due to non-Christians who celebrate Christmas for other reasons. Narrowing down to children should cut down the number, but not as low as 300 million. WolframAlpha gives a 26.8% of the world population being 0-15 years old. Multiplication of 2 billion by 0.268 yields 536,000,000. Smudging upward to 600,000,000 (600 million) because the 16-18 year olds can feasibly account for enough to set the number high enough to round up.
Working from here, the value of 3 children per household works for an average as most sources seem to aim around there. Using division, this resolves in 600,000,000/3=200,000,000 houses for Santa to stop at. Where the hard and sketchy part lies now is distribution. My first guess is the original maths was (earth circumference)/(households). This ignores the issues of things like apartments and non-linear housing arrangements. After all, not every house is in a line around the equator; a lot have houses in all directions. Making the issue more complex is Santa having knowledge of efficiency. For example, when calculating how far the United States can launch a nuclear missile, distance east or west is ignored; instead the path chosen to aim for, say, Russia, is over the North Pole. Considering the distance over the Pacific or Atlantic, Santa can benefit from heading over the Arctic if he hits Canada or Greenland and then Russia. This path also runs the benefit of having multiple loads from the North Pole, where he might be based. We know he can’t be in Norway because the oil drills haven’t found him; I’m knocking out Sweden on a hunch; and Canada is pretty damn close anyway.
What, then, is the best route? A straight line isn’t going to hit everything and zig-zagging hard is going to involve a lot of backtracking. A feasible good route would start at the North Pole, go through Greenland west-southwest, arc through Canada and southward, heading south steadily until leaving the tip of Chile at which point flooring it to Australia over the South Pole (which could be pre-loaded with gifts due to low population), arcing through Asia, Europe, and finishing with Africa. Notably, a lot of ocean is ignored this way, making Antarctica the main time-waster and letting the highest velocity of the sleigh be useful.
So what is Santa’s average velocity? There’s 24 timezones. Assume Santa can work from midnight to 05:00. After all, children can wake up early and Christmas Eve can end late (and in some countries this will vary, but for the sake of simplicity, this is good enough for now). The real issue here is confliction with a good route. While the route I outlined in the previous paragraph is rather efficient, Santa need to start in Western Alaska and head east in order to stay with the timezones (or else go from Greenland to Alaska instantly). So he has 29 hours (24 timezones and 5 hour timespan) and needs to be slow enough in every timezone to not end up coming too early. The latter constraint shouldn’t come into play, but keeping all constraints in mind is good practice anyhow.
So how much distance is Santa covering in 29 hours? He still has a lot of population density to cover. The US has about 4 million miles of road. They criss-cross to give an idea of 2-dimensional inefficiency, but flight still saves time, as does ignoring commercial districts. The US also has a population of 300 million, which would yield about 60 million households. Cutting the roads down a bit due to the savings mentioned by, say, ¼ (or multiplying by a factor of ¾) yields .5 miles/house. How much the separation of Christmas-celebrating children matters is also hard to calculate due to lack of raw data. Nonetheless, at .5 miles traveled to every house, we get 200,000,000*.5=100,000,000 (100 million) miles traveled by Santa in 29 hours. (And, since I did not explain earlier, the roads in the US were used as a sample of distance from family to family because the US has a large population but for the most part is connected by roads, giving some data to work with.
The amount traveled regardless of extra distances (oceans) is going to be very minimal. The distance around the Earth is roughly 25 thousand. That number is nothing at this point. 3,500,000 miles/hour is Santa’s resulting average speed assuming zero time spent inside. Presume 9 hours are spent inside, roughly a third of Santa’s time, as he has a fair bit to do (watch The Grinch. There’s a lot to do.). That results in an average speed of 100,000,000 miles/20 hours=5,000,000 miles/hour. Switching to metric for some easier conversions means he’s at 2,235,000 m/s.
If we take a moment to look back at the original attempt at this problem,
the original claimed 650 miles per second. How does that compare? 650 miles per second is the same as 2,340,000 miles per hour, or roughly half. In other words, if we used the original’s population, the numbers would match at this point. I do wonder what methods were originally used.
I digress.
The Earth itself only travels 66,000 miles per hour. Santa’s going well beyond fast enough to escape the gravity of the Milky Way itself. An incredible amount of force is going to be needed just to stay near Earth. Nonetheless, considering the multiple orders of magnitude greater than the speed of air molecules, some collisions are going to occur and blow things up. So if Santa isn’t killed instantly, everything else is. Also, time dialation.
So, if we ignore the weird physics for a moment and return to Newton (because, really, who actually understands Einstein?), we find a Santa moving at 2,235,000 m/s. Let’s assume he also has to accelerate. Since 2,235,000 m/s is his average, let’s go for 3 million m/s as the peak velocity and .5 miles the average distance, covered in (.5 miles) / (5 million miles/hour) = 1/1 million hours. If he arcs his acceleration and decceleration to peak in the exact middle, he gets to 3 million m/s in .0000005 seconds. The only really comparable things are machine cycles on older computers. In other words, Santa is approaching a gigahertz.
So using change in velocity (3 million m/s) over change in time (.0000005 s), we get Santa’s average acceleration while accelerating of 6,000,000,000,000 m/s^2, or 6 trillion meters per second faster per second (do remember he isn’t traveling for an entire second at any point). There’s not much in the physical world to compare him to here. This is faster than acceleration in free-fall on Earth by a long shot. This is actually about free-fall on a neutron star. The Large Hadron Collider that was going to doom us all? Santa passed that 3 orders of magnitude ago.
The next step here is how much in presents we have. How much the presents weigh is another toughie to guess. Poverty-stricken children aren’t going to be getting a very massive load, but then some rich children may end up with a very massive amount. Also, anyone who gets a car screws up the average. (Actually, I would not be shocked in some of the ultra-wealthy got entire houses, which is about the definition of outlier.) Also, toys over time have changed and video games are very light. So lets say ten pounds. Ten pounds, six hundred million children, six billion total pounds. I’m not adding Santa’s fat ass nor the reindeer. Despite the popular post that’s sent around, magic reindeer are fucking magic and do not have weight limits. There are ten reindeer, approximately. So we will divide from there.
So what’s six billion pounds? Larger than any mobile man-made object. Think “Great Pyramid”. Also think “really heavy”. We’re talking roughly 3,000,000,000 kilograms. Hopefully the transition to force is played out here. Mass (3 billion kg) time acceleration (6 trillion m/s^2) equals 18 billion trillion Newtons. (Written out with zeroes, that’s 18,000,000,000,000,000,000,000 N or 1.8*10^22 N.) That’s about (really close, actually) to half the force between the Earth and the Sun. That’s not even meaningful in this context; the biggest trains are 16 orders of magnitude less (or, ten million billion times less) forceful.
The energy doesn’t really make sense to calculate. I know it’d be fun to point out the 3 billion kg * (3 million m/s)^2 *.5 = 13,500,000,000,000,000,000,000 J of kinetic energy (which, that Joulage is actually only about the amount of energy that the Sun would exert on the Earth that day), Santa is going too damn fast for such things to work nicely.
So, in sum, Santa is a relativistic being that by the laws of physics ought to decimate life on his journey of exploding fucking air. So nevermind flying reindeer. The real magic of Santa is not causing mass radiating explosions.
Nichi 