Monday, 2 February 2015

Sum to infinity: refuelling aircraft

I recently purchased a copy of, "The Rainbow Book of BASIC Programs", a hardback book from 1984 featuring the BASIC text for a number of programs for readers to type into their home computers.  I'll forego the trip down memory lane to the time when I owned an Acorn Electron, and move directly to one of the interesting maths problems in the book.

Quoting from the book: "You are an air force general called upon to plan how to ferry emergency supplies to teams of men in trouble at various distances from your home base.  However, one of the conditions is that your planes do not have the capability to teach the destination directly, which is always outside their maximum range.  Nor are they able to land and refuel en route.  Ace pilot Rickenbacker suggests that mid-air refuelling might provide the solution.

"'Just give me a squadron of identical planes,' he tells you. 'During the flight the point will come when the entire fuel supply in one plane will be just enough to fully refill all the others.  The empty plane then drops away and the rest continue.  At the next refuelling point another plane tops up all the others leaving the full planes to continue.  The squadron keeps going in this way until only one plane remains. It uses its last drop of fuel to get to the destination with the emergency supplies.'"

So, having provided his ingenious solution, you are left with your home computer to solve the problem:  how many aircraft will it take to double or triple the maximum range of one aircraft?  And how many aircraft will it take to extend the maximum range of an airbase by six times?

To tackle this problem, I'll start with a few simple examples and look for any patterns.

Let's assume that the maximum range of the aircraft is r, and let's say that it's 100 miles.

With two aircraft: when both aircraft reach half empty (50 miles) the second aircraft refuels the prime aircraft, which then travels a further 100 miles, so the total is 150 miles (r + r/2).
Second aircraft refuels prime aircraft when it has used up 1/2 fuel

With three aircraft: the third aircraft will have enough fuel to fully refuel the others when each aircraft has used up a third of its fuel. It will transfer a third of its capacity to the second aircraft, and a third to the prime aircraft.  We will then follow on with the two-aircraft case shown above. Total distance covered is 33 miles (to the first refuelling point), then 50 miles (with two aircraft) and then the prime aircraft travels the last 100 miles alone.  Total is 183 miles, (r + r/2 + r/3).
Third aircraft refuels second and prime aircraft when all have used up 1/3 fuel
And the final example, four aircraft.  In this case, the fourth aircraft will transfer its fuel to the other three after they've all used up a quarter of their fuel.  This fourth plane will add 25 miles to the overall total, (r/4).

Fourth aircraft refuels three aircraft when all aircraft have used up 1/4 fuel

So we can see that each nth plane adds on r/n to the total distance. The first plane adds r/1, the second added on r/2, then r/3, r/4 ... r/n.



This series is known as the harmonic series, and is a well-studied mathematical series, and its properties are well-known.

The most surprising property (to me, and apparently many other people) of the sum of the harmonic series is that it doesn't converge: it doesn't get closer and closer to a fixed total. Instead, it keeps growing and growing, just more and more slowly.  If the squadron had enough aircraft, it could reach any distance necessary. Each additional aircraft adds less and less to the overall total, but the total continues to increase.

It may be counter-intuitive to find that the total doesn't reach a limt, but there are a few proofs that show this is true, the first of them discovered by Nicole d'Oresme (circa 1323-1382).

So, to answer the original question: how many aircraft will it take to double the range of one aircraft? 
1 + 1/2 + 1/3 + 1/4 = 2.083

Which means it will take four aircraft (including the prime aircraft) to double the range of the prime aircraft.

And to triple the range of one aircraft?  


1+ 1/2 +1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 = 3.0199
Eleven aircraft!

And to extend the range to six times the initial range?
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 ... ... + 1/226 + 1/227 = 6.0044

Scarily, to increase the range of a 100 mile aircraft to 600 miles would take 227 aircraft (including the prime aircraft).  This also gets ridiculous, as the distance between subsequent refuels gets smaller and smaller, 0.00001 x r in the first few instances.

So this is clearly a theoretical exercise:  the instantaneous refuelling is tricky enough to believe, but the rapid usage of aircraft (and the 'falling away' to the ground) is just wasteful!

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