Tuesday, 24 December 2013

The Gift-Bomb: Could Santa Really Deliver Presents to Every Kid on Earth in One Night?

This week's special Christmas article was written by guest author and University of Waterloo mechatronics engineering PhD student Arun Das.

The age old tale of Santa Claus is well known to many across the world.  At some point, we all probably believed  the story of how jolly old St. Nick, with the help of his elves, reindeer and some good ol’ fashioned magic, zipped across the earth delivering presents to all the the good little boys and girls on Christmas eve.  As we grew up, we became all-knowing adults and threw the notion of this fat, old man from the North Pole, and his infinite generosity straight into the preposterous category, right beside bad-ass wizards, talking dogs, and the questionably fashionable sweater-vest. But what if this wasn’t the case? What if Santa was real?

After all, the story of Santa Claus dates back to well before the 19th century, and we’ve figured out a lot of stuff since then. We no longer die from the common cold, have access to unlimited information at the touch of a button, and even have cars that can drive themselves around. So, it stands to reason that Santa has caught up to the times as well.  With this in mind, the Sketchy Science team set out to answer the question, “Could Santa feasibly deliver presents to the children of the world using current day technology?”. Authors have attempted to answer this question before, but we feel their approach was overly pessimistic and focused too strongly on the single Santa angle.       

When we initially started to tackle the problem, we thought of many ways to accomplish the task, including strapping presents to ICBM’s, outsourcing the entire operation to FedEx, or just getting everything shipped with Amazon Prime. Clearly, Amazon has been looking at this delivery problem, as recently they announced plans to use unmanned aerial vehicles (UAVs) to deliver packages. However the UAV technology is still at least a 5 years away from deployment, as the solution is currently limited by battery technology, small payload capacities, and government regulations.

To that end, we quickly disregarded the existing methods, as we felt they were too far removed from the spirit of Santa and definitely could not be accomplished in one night. We wanted to know what it would take for Santa to deliver the presents all in one go, on Christmas Eve.

After many long nights, we finally cracked it, and so, we present to you…

What would it take? Santa in 2013.

For the analysis, we make a few simplifying assumptions:

1) Each child shall receive a gift with dimensions of approximately 12 cm x 25 cm x 6 cm, and a weight of 1.5lbs.  A box of Lego easily fits this description.

2) Santa is still able to use his magic, just in moderate doses.  We assume that once the present reaches the roof or doorstep of a home, Santa’s magic is used to get the present into the house and under the tree.

3) There are approximately 1.8 billion children in the world.  We can take 32% of them to be Christian, thus yielding 608 million gifts which Santa has to deliver.  For the sake of simplicity, assume a uniform distribution of children over the world. Note that this approximation is actually the worst-case-scenario in terms of area coverage.   

4) Santa purchases all the presents ahead of time and stores them at the North Pole.  

Although the classic story of Santa has him travelling from rooftop to rooftop delivering gifts, we deemed that grossly beyond unfeasible for a science article.  Instead, we’re going to drop the presents from airplanes travelling at close to the speed of sound from 35,000 feet above the surface of the earth. Yeah, that’ll do.

How does a present get to your house?  Thanks to many recent advances in technology, we can drop the presents from the plane and autonomously glide them into the rooftops.  In fact, DARPA worked on a similar autonomous glider system... except their’s travelled at the edge of space and at 13000 mph. Engineers can build simplified, scaled down versions of these, outfitted with a simple GPS, an inertial measurement unit, and lightweight processing and battery.   The required technology is pretty much exactly like what you have in your smart phone, though I bet you didn’t know it could fly a plane.  With modern day carbon fibre and 3D printing techniques, we could build a simple glider like this weighing about 1lb.  This brings our total gift payload to 2.5 lbs, with dimensions of 15 cm x 30 cm 10cm.  

In order to carry all the presents, we will use the largest cargo plane ever engineered, the Antonov 225, which has a maximum payload of 551,000 lbs, and an average range of 10,000 km, which takes into account a decreasing payload while in flight. Given the payload, each plane could carry 220,400 gift units. Taking into account the volume, 220,400 gift units is about 1000 m3, and with a cargo hold size of 1300 m3, we should be able to fit the maximum payload with bit of room to spare.  

In the most optimal sense, we would like all the airplanes operating in parallel.  This means to deliver 608 million gifts, we need at least 2759 airplanes flying at once.  If you think that’s a lot, it’s not. These days, about 6000 planes are flying over the world at any one time.  Now the only question is if Santa could use these 2700 odd planes to deliver the presents on time.  Let’s assume a maximum deflection angle of 6 degrees from the vertical (12 degrees total), which results in a cone through which the gliders can navigate. From 35,000 ft (or about 10km), the 12 degree cone yields a 2 km diameter footprint over the ground that covers the houses which the gliders can reach for a given airplane location.  

This now becomes an optimal coverage problem, where we want to sweep the 2 km footprint over the landmass of the earth in order to deploy the gliders.  Given that the earth is 150 million km2 in area, and about 50% of that is habitable, we have to cover a total of 75 million km2 with the planes.  Using a rectilinear approximation for the sweeping area, the total distance to be covered by all the planes is 37.5 million km.   

The maximum distance each plane can travel is 10,000 km. Although the aircraft is capable of a maximum speed of 850 km/h, we will assume an average cruising speed of 500 km/h to account for the payload. If we wanted to deploy all the planes in parallel, we would need 3750 planes total, and they would be able to accomplish the task in about 20 hours. Including time zone changes, Santa actually has about 31 hours to work with, leaving some wiggle room in the schedule in case things fall behind. 

So there you have it. With a mere 3750 cargo planes, Santa could deliver gifts to all the children of the earth. But how much would all this cost? Using a safety factor of 1.5 to account for any errors in the analysis, we end up with about 5625 airplanes. Assuming a price tag on the aircraft of $150 million each, we would need $843.8 billion for the planes (though these wouldn't have to be repurchased every year).  Assuming Santa gets a bulk discount for the Lego, we’re looking at another $9.2 billion for the gifts. At about $3/gallon, the jet fuel for our fleet of planes would be another $120 million. Adding some extra for the gift gliders, airplane maintenance, and other misc costs, for a measly $1 trillion, Santa could deploy the gift giving mission and still have enough time and money left over to throw a kick ass party for the elves. To put that into perspective, in 2011, the top 10 military powers of the world spent $1.2 trillion on defense spending.

And you thought your holiday shopping was expensive.

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