# Christmas Kinematics

In the finest traditions of christmas, how about a timely blog post meant to cynically cash in on a blogosphere craving seasonal articles about nothing much in particular (see previous, sadly failed, attempt). What are the implications of Santa flying around the UK in a single night?

Santa Claus is said to fly around in the sky to each house in the world delivering presents. How physically plausible is this? The answer, given some massively simplifying assumptions, is surprisingly so.

Let’s assume that Santa, as the world’s preeminent expert in mass package delivery (sorry Amazon), has developed in his elf-based R&D labs a wide-area present dispersal unit such that he only needs to stop once per city.

Plotting the 300 most populous cities in the UK, these are the population centres he must visit:

What’s the quickest route through these cities? As I’ve covered before on this blog, this is the famous Travelling Salesman Problem (TSP). There exists an efficient TSP solver in widespread use called Concorde. I’ve used it here to solve this relatively small TSP problem – you can see the solution path plotted below:

Using this shortest path, the total distance is 5,993 kilometers, or approximately the radius of the Earth. If we assume that Santa, being a utilitarian sort, divides his time over a 12 hour night amongst the world’s countries in proportion to their population, the UK is allocated a princely 374 seconds to intensive chimney stuffing. Santa’s average velocity is therefore 16,000 metres per second, or nearly 36,000 miles per hour. This is significantly larger than escape velocity, so a single mistake could cause the sleigh to fly off into space.

However, Santa is stopping during this trip, so his velocity is constantly changing. Assuming that he can accelerate at a constant maximum rate given by his finite reindeer power, he must accelerate and decelerate equally between stops. The total time to travel a distance $d$ under acceleration and deceleration of magnitude $a$ is

$\Delta t = 2\sqrt{d/a}$

Adding up these time intervals along the solution path, and allocating 50% of Santa’s time to radially blast presents like a festive clusterbomb over the metropolitan area, his acceleration must be at least 4,000g. This sounds unfortunately wearing for poor Santa, indicating his jolly red costume is actually an ultra high-spec compression suit designed for crash decelerations. Assuming an unladen sleigh mass of a ton, his 12 reindeer must be kicking out over 3 MN of thrust apiece to achieve these numbers.

I’ve read that a horse can accelerate horizontally at approximately 2g, indicating that Santa’s reindeer are around 5 orders of magnitude more efficient by mass. Clearly Santa has a world-leading breeding programme.

I was looking forward to digging into some relativistic kinematics for this post, but Santa appears keep his Lorentz factor stubbornly near 1 over the UK. Perhaps over less densely-populated countries his velocity would necessarily increase.

An exercise for the reader would be to

1. Generate a worldwide population map
2. Create a Voronoi stippling of some arbitrary density of this map
3. Solve the resulting TSP problem
4. Increase the stippling density and extrapolate to a converged result

Any takers? There’s a bottle of sherry with my name on it, so I will have to wait until next time.

Thanks for reading, and Merry Christmas!