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On Puzzling.SE I found this question:

You have a 1 mile long x 1 mile wide private island that you wish to turn into a resort. A plane requires a 2-miles long runway to take off. What do you do?

There are plenty of answers, among them some irrational ones, but also some good approaches to solving this puzzle.

From an aeronautical point of view, what would be a reasonable solution?

I am thinking about a vertically curved runway. To meet the 2 mile requirement it would need to have an angle of (estimated) 25° at the edges of the curved runway. It would go diagonally across the island and by its curvature add the extra length needed to the √ 2 length of the diagonal length.

I am sure there must be a better solution though (extensions of the island shouldn't count!)

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    $\begingroup$ provide helicopters. $\endgroup$ – Erich Mar 22 '15 at 5:59
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    $\begingroup$ @erich Some of the comments on Puzzling also mention that there's a perfectly serviceable 2-mile runway adjacent to the island, provided you have a seaplane. $\endgroup$ – cpast Mar 22 '15 at 6:02
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    $\begingroup$ I repeat my comment from Puzzling SE: an island of that size simply doesn't need a 2-mile runway. The smaller Boeing 737s can take off in less than a mile and an island that's only a mile square doesn't have the capacity for planes of even that size. $\endgroup$ – David Richerby Mar 22 '15 at 12:26
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    $\begingroup$ @Maverick283 Sure. And that is the aeronautical point of view: a 2-mile runway is ridiculously over-specced for the kinds of plane that would need to visit that island. If answers along the lines of "Don't bother: use seaplanes instead" are acceptable, then so is "Don't bother: build a runway long enough for the kinds of plane that might be needed for a location of that size." $\endgroup$ – David Richerby Mar 22 '15 at 15:37
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    $\begingroup$ @AE One problem with banked curves is that being in a turn increases the load factor, such that lifting off in a turn requires a greater airspeed than lifting off from a straight takeoff roll. As far as I can immediately see (very back of the envelope), this would tend to cancel out the extra length you can get from a curved runway of constant radius. A banked turn just before the point where the plane reaches $v_1$ followed by a straight section to actually lift off from might have a better chance of working (on paper!). $\endgroup$ – Henning Makholm Mar 22 '15 at 22:40
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Anyone who has ever flown at Zar (EPZR) in Poland knows how much a runway slope can reduce the needed field length. For aircraft which need a 2 miles long level runway, a 1 mile long one with a 15% slope at its end will do just fine. Make the slope shallow at the bottom and increase it the further up the runway goes, just like the hillside runway at Zar. If the end point is 183.5 m higher than the touch-down point, the height difference is equivalent to a speed reduction from 60 m/s to zero.

Take-offs work in the opposite way, giving the same speed boost on the way down. A 1 mile long runway will do just fine. You do not even need to run it diagonally across the island.

Yes, this does not exactly answer the riddle, but is an engineering solution to a simple problem.

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    $\begingroup$ This actually appears to be correct answer to the original question as well. The question specifically mentions 2 miles of flat runway and being on Earth, which leads me to think that the effect of slope on acceleration/deceleration should be included. $\endgroup$ – Jan Hudec Mar 22 '15 at 21:51
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    $\begingroup$ A 15% slope would put your mile long runway at 15% of a mile up, 792 feet. Assume 200 feet wide, this would require 5280*792*200/2 = 418 million cubic feet of earth. Given this island is only 5280x5280 feet already, (28 million square feet) this would lower the average height of the island by 15 feet. I wouldn't be surprised if this island wasn't even that high at its maximum point! There literally may not be enough earth to do this locally, unless perhaps you can find boulders in the sea to act as filler. $\endgroup$ – corsiKa Mar 23 '15 at 16:49
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    $\begingroup$ @corsiKa: By no means I intended to say that the runway slope is 15% over its total length! It starts horizontally (That is the meaning of "shallow at the bottom"), so aircraft have a level touchdown zone. Only at the rollout end will it have more slope. Like the runway in Zar. $\endgroup$ – Peter Kämpf Mar 23 '15 at 23:19
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    $\begingroup$ @corsiKa - there is no requirement that all material for the runway be sourced locally (I doubt said island has the material for making the concrete/tarmac for the runway surface), so there's no issue with shipping in all the fill dirt or dredging it from the water surrounding the island. Dredging that much dirt would have the added bonus of lowering the ocean's surface enough to expose more island for you to play on! (that's a joke...) $\endgroup$ – FreeMan Jan 18 '17 at 21:58
  • $\begingroup$ As an aside (but a potentially relevant one), it is a certificate limit for many commercial airliners that the runways they operate on must have only up to 2% slope (up or down). More than that would be prohibited for these aircraft. $\endgroup$ – Cpt Reynolds Jan 20 '17 at 22:39
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A seaplane would be a suitable solution. Then the only infrastructure required on the island is a jetty and fuel/servicing. This also enables a much larger aircraft and so easier cargo resupply and a longer range on the aircraft.

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    $\begingroup$ It certainly enables much larger aircraft, but most sea planes these days are on the smaller side. Still a very viable option. $\endgroup$ – fooot Mar 22 '15 at 16:57
  • $\begingroup$ @foot There are a number of large seaplanes still being manufactured. The dominant use is as fire bombers and as air sea rescue but they are being made. There is even an unmanned cargo seaplane in development link $\endgroup$ – Joni Mar 23 '15 at 9:04
  • $\begingroup$ Joni, at best the flying boats currently in production (Beriev, Bombarier) are medium, not large. Large is something like the Martin Mars or BV 222. $\endgroup$ – egid Mar 24 '15 at 5:16
  • $\begingroup$ @egid How large does large have to be and how big is medium? They certainly aren't building any on the scale of the Hercules HK-1 but large seaplanes are still being built. The Question did not specify any cargo requirement. My original point was merely that the use of a seaplane enabled a larger aircraft that was not limited by the sqrt(2) mile long runway possible on the island. $\endgroup$ – Joni Mar 24 '15 at 14:53
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    $\begingroup$ I would say that aircraft sizes for landplanes and seaplanes should be interchangeable. By that definition the CL-415 is not large! $\endgroup$ – egid Mar 24 '15 at 15:21
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There really isn't a better answer than "extending the island." To function, runways must be straight (or very nearly so, a half-mile radius of curvature is unacceptable at takeoff speeds), and level (or very nearly so, gradients are measured in percent and 11% is an extreme gradient, which still gets you very little extra length). However, aircraft carriers in a lot of countries have ski-jump ramps which seem to go up to 12 degrees (20% gradient or so, let's be generous and say 15 degrees), which means that's probably a good absolute limit on the slope of the runway anywhere (it doesn't work on the whole runway, but this is irrelevant). Let's look at some numbers. We'll start with the longest possible flat runway, and then put it at the highest possible slope, to get an upper bound on length.

The largest radius of curvature you can have horizontally is 1 mile; this is also a very high radius of curvature (around what the Endless Runway people have; a comment on the curved runways question calculates that at takeoff speed, you need the equivalent of a 20-degree bank to stay on this runway), so anything tighter won't be acceptable. This gives you $2\pi/4\approx1.57$ miles; this is the longest possible flat runway.

You might also curve up while doing this; the maximum slope at any point is 15 degrees. The longest way to do this is to have a constant 15-degree slope; this means that the max possible length, with tight horizontal curvature and a ridiculous slope, is $1.57/\cos 15^\circ\approx1.63$ miles.

So there you have it. The maximum length available, with extremely generous slope and curvature constraints, is well under 2 miles.

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I will go through some answers of the puzzle, some rational, some not:

Add a third dimension (i.e. sloped diagonal runway):

Your plane will be going downslope at a 45 degree angle. It will pick up speed pretty fast, but requires a skillful pilot to pull it out of the dive before hitting water. Not very practical.

If you go upslope, your pitch up will be 45 degrees and you will likely have insufficient engine power to maintain airspeed (not to mention accelerate). Not feasible either.

Do we really need 2 miles?

If the plane needs 2 miles due to the low acceleration of the engine, then we can use a bit of slope (e.g. 5 or 10 degrees) instead of 45. A fully vertical acceleration gives 9.8 ms-2. A 10 degrees slope will give the plane an additional acceleration of $9.8 sin(10^\circ)$ ms-2 on top of the acceleration given by the engine. This has the hope of significantly shortening the takeoff distance if the engine is weak.

Aircraft Catapult

This one has been done practically, given a strong enough catapult and reinforced aircraft frame.

My personal answer...

Wait for enough head wind to takeoff!

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The answer is to extend the island, the main issue I see here is that you really cant change any of the other variables.

A catapult wont work because (as far as I know) modern airlines do not have the airframe necessary to support this. Then again I may be wrong but I would think they could not take that kind of force.

I have no idea how a ski jump like slope would affect an aircraft. I know they have them on aircraft carriers so maybe someone could weigh in on how that works both for plane size and the effect it has on the airframe from a force on the gear perspective.

Sea planes would still require people to come into a local airpot and take a smaller seaplane to the island assuming it was near a larger airport. Or take a sea plane of a certain vintage, capable of flying long distances.

Extending the runway (depending on the surrounding sea bed) is not very difficult and has been done. KIX in hong kong is build on a completely man made island so adding some land to the end of the island it not a big deal. I also add this since I assume you want the ability to fly direct to the island and not have to fly to a larger airport then take a small plane into the island.

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Curved runways have many issues. Especially in bad weather, aircraft sometimes run off of straight runways, so a curved one would be even harder to stay on. The curve would also increase the distance needed for both takeoff and landing, meaning that even more than 2 miles would be needed.

Sloped runways do exist. However, sloped runways like EPZR (mentioned by Peter Kämpf) or Lukla tend to be small (both are less than 2000 feet, EPZR is not paved). The physics limit the runway to operating in one direction only. Though wind becomes less of a factor, the issues still prevent this from being used for large airports.

Another issue is the height of the high end of the runway. Zar and Lukla are both located in terrain that requires one end of the runway to be raised. A 1-mile square island would probably not be shaped this way. Even if it is, if an airplane goes off the runway, or needs to abort a takeoff or landing, the slope will make this much harder. The pilots would also have to adjust their takeoff and landing technique to account for the slope, which will probably result in many more hard landings and tail strikes. A sloped runway may be acceptable for small aircraft and short runways, but typical aircraft needing 2 miles of runway will be much more difficult to operate this way.

Some aircraft carriers solve the runway length issue with a slope only one the very end. However, this is most helpful to aircraft like the Harrier that already can provide some vertical thrust, and the aircraft must still perform an arrested landing. Other carriers use a steam catapult to assist takeoff, which requires much more maintenance but is capable of launching heavier aircraft.

Using a sea plane to land on the water is another option. However, aircraft on the scale of needing a 2 mile runway are typically not sea planes. If you are going to design a new plane, you might as well design one that can operate to the runway length requirement.

Another fun option is jet-assisted takeoff (JATO). Rocket engines are strapped to the plane to accelerate it to takeoff speed more quickly. These can be used for landing as well, but that is a bit more difficult.

Do to the issues with these options, large airports in need of land tend to just make more land. You can put an entire airport with runways >2 miles long on a completely man-made island. There is typically enough room around an island that creating an extension is not an issue.

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Maybe unconventional, but it has been done: Make a ramp out over the water that is a runway extension.

Funchal (Madeira) is an example where that was done (though not nearly as long as your requirements). http://www.getalookatthis.com/2008/01/17/new-airport-runway-in-portugal-part-3/ has photos of the runway extension under construction.

Another solution, if you're at low altitude and the water is shallow, may be to not build on pillars but landfill/reclaimed land. This is being suggested for New York La Guardia, and of course Osaka Kansai airport takes this to the extreme by being built on an artificial island, offshore from the city because there was no space there.

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