# Building a 2 mile runway on a 1x1 mile island [closed]

You have a 1-mile long x 1-mile wide island.

When taking off from a flat runway, a plane requires a runway of minimum length 2 miles in order for takeoff to be physically possible, achieving minimum takeoff speed at the very end of the runway.

Remaining within the boundaries of the island, what shape and size do you build a runway to make it physically possible for the plane to take off?

The island is on the planet Earth and the usual laws of physics apply. No other form of transport is available. The plane cannot take off from water or perform any part of its takeoff run on water. Flight safety legislation does not apply.

## closed as too broad by AzaJul 17 '15 at 19:03

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• Since you got there with the infinite rope you could just... oops, wrong Google interview question. – GOTO 0 Mar 21 '15 at 12:56
• What I do is point out that there's no way such a tiny island requires such a long runway. The smaller Boeing 737 models can take off from a runway less than a mile long and it's not like you need to be landing Jumbos at your resort. – David Richerby Mar 21 '15 at 20:34
• Are we talking real-world airplanes, or spherical-cow airplanes? – Mark Mar 21 '15 at 21:03
• Build a dock that extends past the shore. I feel like this puzzle is too vague to be interesting. – Ian MacDonald Mar 22 '15 at 2:53
• I have designed an ingenious solution involving the use of a giant treadmill which this comment is too short to contain. – Adam Davis Mar 22 '15 at 19:53

The solution is simple: Build an elevated two mile long runway and anchor all of its supports on the island land.

Everything that touches the ground is within the boundaries of the island. Since we are talking about a 3D structure and the island boundaries is something only valid on 2D. So, this is arguably valid since there is no clear definition of island boundary in airspace.

Lets see if it passes all the requirements:

• The runway must be flat. - $\color{green}{\text{OK}}$
• The runway must be at least 2 miles long. - $\color{green}{\text{OK}}$
• You should remain within the boundaries of the island. - $\color{orange}{\text{I am abusing this, but it passes as explained above}}$
• The island is on planet Earth. - $\color{green}{\text{OK}}$
• Usual laws of physics applies. - $\color{green}{\text{OK}}$
• No other form of transport. - $\color{green}{\text{OK}}$
• The plane cannot take off from water or perform any part of its takeoff run on water. - $\color{green}{\text{It is not on water, it is on over-water airspace, so it's OK}}$
• Flight safety legislation does not apply. - $\color{green}{\text{OK}}$

About the picture, I know that it would be easier to build it along the diagonal, but this way it is more beautiful and the island size and shape almost do not matter in this answer.

• "Everything that touches the ground is within the boundaries of the island." Ah, yes. The "I'm not touching you!" provision of international law. – KSmarts Mar 23 '15 at 20:38
• That's it. This should be the accepted answer. – Rand al'Thor Mar 23 '15 at 23:59
• you could do the same but over the diagonal, and make it more robust ;) – gorgi93 Apr 7 '15 at 0:28
• @Victor Stafusa Great Answer.. I would build the bridge diagonally instead of the one shown in the diagram. – kanchirk Apr 16 '15 at 17:57

Build two straight runways side to side, 1 mile long each.

Then, put a portal-conducting surface at the end of the first runway and another one at the beginning of the second one. Finally, use your portal gun to place a yellow portal on one of those surfaces and a blue portal in the other one.

• Now you're thinking with portals! – dmg Mar 22 '15 at 17:13
• +1 for realistic physics: this is one of the few answers that doesn't require aircraft to turn impossible corners or go up impossibly steep hills on their take-off roll. – David Richerby Mar 22 '15 at 19:16
• You are wasting material; just have a portal at the beginning and the end of the runway, and you'll have an infinite long runway. Start in the middle and during the third lap you'll take off and not hit the wall with the portal. – Sumurai8 Mar 22 '15 at 21:30
• @DavidRicherby Realistic physics?! :-o – Rand al'Thor Mar 22 '15 at 23:14
• In fact, forget the planes. Just have a portal to each place planes go to, if answers are allowed to use portals. – immibis Mar 23 '15 at 6:13

• "The island is on the planet Earth and the usual laws of physics apply." Either you have really strong engines, or that crane is absurdly tall. – mdc32 Mar 22 '15 at 21:01
• @mdc32 - What do the engines have to do with anything? You could launch a glider like this. Wing load during the dive is the limiting factor here. – Steve V. Mar 22 '15 at 21:15
• Creative, although in some ways just a variation on @leoll2's catapult idea. WP: Assisted takeoff – A E Mar 22 '15 at 22:47
• @SteveV, the power of the engines makes a difference to the necessary crane-height. A gilder would require a higher crane than a powered aircraft, because the powered aircraft's engines allow it to achieve greater speed in the same vertical drop, although I agree that in either case you're screwed if the wings fall off. – A E Mar 22 '15 at 22:48
• If you spin the crane up you wouldn't have to worry about engine power :) – user2027202827 Mar 24 '15 at 17:03

You need to:

build a runway that goes diagonally through the island and is sqrt(2) miles higher in one corner than it's diagonally opposite corner.

Why?

The diagonal of the island is sqrt(2). You need a straight line of length 2, so we add in the third dimension. Using a height of sqrt(2), we get a hypotenuse of length exactly 2 miles.

Bonus:

Add additional height to both sides so that the plane does no hit the water at the end of the runway.

More boni inspired by the coments:

Make the runway curved or make a horizontal segment at the very end of the runway (in that case we'll need a the other segment to be a bit steeper).

Finally:

Use a floatplane.

• This sort of defeats the purpose of a runway, though, since your plane will either be climbing up the runway which means it won't have enough speed to take off, or climbing down the runway which means it won't have enough angle to avoid hitting the ground. – Joe Z. Mar 21 '15 at 16:24
• You'd have to add quite a lot of height in order to be able to counteract a 45° downward slope. – Joe Z. Mar 21 '15 at 16:25
• @Relaxed This is definitely the case in Lukla (Nepal - see youtube.com/watch?v=v_NFd9nsjEw ) and in La Paz (Bolivia). I have landed and taken off from both and they can be fairly 'exciting'. – Penguino Mar 22 '15 at 1:37
• @Relaxed This may be a puzzle, but it's not "build a 2-mile strip," it's "build a 2-mile runway". A 100% gradient fails to be a runway (as "planes can use it" is a necessary component of that). – cpast Mar 22 '15 at 15:47
• Related research: Aircraft Operations From Runways With Inclined Ramps (Ski-Jump) – A E Mar 22 '15 at 17:49

Build a runway that goes along one side of the island, turns in a banked quarter circle with a small but sufficiently large radius, and then goes along the adjacent side. If it were horizontal the length would be slightly less than 2 miles, so raise it at the end where the aeroplane will be stationary. The key word is banked.

• Excellent. This would be cheaper than a 'ski jump' runway because much smaller amount of earthworks involved. Would need a lot of skill to stay on the runway in the event of an aborted takeoff (because plane would be going at a different speed than the banking was designed for so couldn't just go straight down the centre-line), but pilots would probably enjoy it. Sorts the men from the boys. – A E Mar 22 '15 at 17:23
• @raptortech97 - the question specifically states "Flight safety legislation does not apply.". – Bob Jarvis Mar 23 '15 at 21:51

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?

The question says the plane needs a 2 mile runway to take off, but says nothing about landing. So have people fly in, land on a short runway, then dump the plane in the ocean to make room for the next one. If people insist on leaving, they can swim for it.

• Rofl ... I can see this question is going to be the new puzzling.stackexchange.com/questions/4304/… – Rand al'Thor Mar 22 '15 at 13:55
• Generally speaking, planes take more runway to land than to take off. – raptortech97 Mar 23 '15 at 14:58
• Tuck and roll! You'll be fine! – Duncan Mar 23 '15 at 14:59
• @reptortech97 Not necessarily. Most planes I've flown are capable of stopping shorter than taking off. Especially if they have more than just brakes to help. The bigger question is, how do you define a takeoff? Is it off the ground or able to climb away from the surface? :-) – Shawn Mar 23 '15 at 22:25
• @Shawn: all aircraft are capable of stopping in a shorter distance than is required for takeoff. No brakes, thrust reversers, or etc required... :-> – Bob Jarvis Mar 26 '15 at 13:57

Your longest horizontal path on the island is $\sqrt{2}$ miles (approx 1.4 miles), so there's no straight, horizontal solution on the island proper. (That is, what looks like the intended approach seems to have no valid answer.)

Use alternate transport (e.g. seaplane) or start a land reclamation project. Or make arrangements with a neighbouring island that happens to be 2 miles long :) .

• I thought to make runway on water. actually its an interview question I found on Google but don't know the answer. – Himanshu Mar 21 '15 at 12:28
• Yup, I considered that also, but it's effectively land reclamation. – Lawrence Mar 21 '15 at 12:32
• Interview questions have had some very creative solutions on this site and elsewhere - I'm looking forward to what others come up with. – Lawrence Mar 21 '15 at 12:33
• yes you are right. lets see who will give best solution. – Himanshu Mar 21 '15 at 12:36

Build a giant bowl and let the plane land initially at the top (albeit sideways). The centripetal force will keep the plane stuck to the inclined surface until it can dissipate enough speed doing loops around the bowl to settle at the bottom.

Aircraft normally experience a max of around 2g during takeoff/landing so that's a nice benchmark to shoot for. Going to neglect gravity itself here since optimizing the roll angle to account for what a 747 could actually handle would take a bit of digging (especially the rudder as I doubt a jetliner could fly at 90 deg roll sustainably). With those simplifications though, a = (v^2)/r ---> r = 83^2/(9.81*2) = 354 meters. So with roughly half a mile of square space, you could land a 747 in your bowl. So long as the plane doesn't fall to pieces.

All that being said, taking off is probably even harder since you now have to overcome that centripetal force to break away from the bowl...

EDIT: Could also double as amusement park ride.

• I like this answer very much, although I have a few concerns about practicality. An airliner can do a barrel roll but I'm not confident that it can land while sideways. Perhaps we should ask on Aviation.SE? – A E Mar 21 '15 at 20:53
• Psh... ok what if it didn't land at all? You could just pack everyone into some kind of glorified hamster ball and release them into the edge of the bowl as the pilot makes a pass. That way the ball does a couple laps around the bowl before settling to the bottom. – user2027202827 Mar 21 '15 at 21:07
• And here are some answers from an aeronautical point of view: aviation.stackexchange.com/q/13385/5592 – Maverick283 Mar 22 '15 at 10:43
• Thanks for link to the aviation.se version of this! I realize this solution involves putting the plane into what's probably the most unstable position in can be and then trying to land in that position, so obviously not for the faint of heart ;) As a side note, would you mind pointing me in the direction you took to calculate the bowl incline? I'd be interested to know how you approached that since I have basically zero experience in aircraft aerodynamics :) – user2027202827 Mar 22 '15 at 19:29
• An early prototype – A E Mar 22 '15 at 20:16

There are several solution:

1. Aircraft Catapult

2. Build an artificial hill on the corner of the island so that you can build a downhill runway, which can be longer than 2 miles if the hill is high enough.

3. Build a circular runway (max length is π) and let the airplane run on the track until it can take off.
• Your circular runway has the length: 2$\pi$r = $\pi$ > 2 miles when r=1/2. You could also use 2 straight sides of the square connected by a curved piece. However, all runways I know are straight. – Lawrence Mar 21 '15 at 12:47
• I made a mistake, the max length of the circular runway is π (not π/2). I chose a circle as shape because it isn't much different from a straight line when the object moving on it is much smaller than radius. The example is earth's surface, it's spherical (ok, I know it isn't), but you see it flat when you walk. – leoll2 Mar 21 '15 at 12:55
• It doesn't work, but endlessrunway-project.eu – David Mulder Mar 21 '15 at 19:40
• Why not? The runway in your link has a 3 km diameter, which isn't much different from the 1,6 km of this problem. Also, I don't think that you want a Boeing 747 land on this little island, a small aircraft is very agile and has enough turn rate to take off from a circular runway. – leoll2 Mar 21 '15 at 19:52
• "The runway in your link has a 3 km diameter, which isn't much different from the 1,6 km of this problem." It's a factor of two! That's a huge difference. And I'm not sure where you got the idea that small aircraft can take off on a corner. – David Richerby Mar 22 '15 at 19:20

Build a runway:

That starts in one corner of the island, goes straight for half a mile, then makes a half circle turn until it hits the other side of the island, then continues straight for half a mile. The half circle will have a circumference of 1/2 * 2pi*r, where r = 1/2.

This gives it a length of:

1/2 + 1/2 * 2*pi*1/2 + 1/2 = 1 + pi/2. This is greater than 2 (about 2.57), so it gives you some leeway.

Leaving most of the space in the middle for the resort.

and

Having straightaways at the beginning/end of the runway for when the plane has the most speed (takeoff and landing) and can't turn as quickly.

You better hope:

That the wind is blowing in the right direction, though.

• If your runway has a 180-degree corner in it, it isn't a runway. Having the corner in the middle isn't an advantage: it's the worst possible place because it minimizes the length of the longest straight section and only the straight section is useful. – David Richerby Mar 21 '15 at 20:23
• Well, having a 1 mile by 1 mile island where you need to build a 2 mile runway also isn't an advantage. I'm going with most of this situation being unrealistic, so the minor issue of having a curved runway is something I'm willing to work with. – Duncan Mar 23 '15 at 13:38
• The situation is unrealistic in the sense that it's a bad business decision; your answer is unrealistic in that it's physically impossible. That's a rather different level of unrealism. – David Richerby Mar 23 '15 at 13:57
• Bank the runway where it curves. – h34 Mar 25 '15 at 8:39

Mobius Landing Strip FTW! :)

That or cut the island into fractions and stick it to itself.

Also you could have an extending drawbridge type runway that retracts or one that has buoys to float with in the water.

My favorite for the fancy resort factor would be a runway that is in a completely glass tube that goes underwater and takes you to the under ground grand entryway of the resort

• Or you can make it a James Bond/ Secret Agent themed resort where people have to parachute in :) – Neil Mar 22 '15 at 6:33

If I was making a resort on such a tiny island, there's no way in the universe that I'd put a runway on it. The space it would take up would ruin the resort atmosphere of the island. That land would be needed to make the resort.

require people to come by boat or, as DMG said, by floatplane

An entertaining option:

Leave the runway flat and level at length 1 mile (or sqrt(2) miles if you like) and use Rocket-Assisted Takeoff.

JATO (acronym for jet-assisted take off), is a type of assisted take-off for helping overloaded aircraft into the air by providing additional thrust in the form of small rockets. The term JATO is used interchangeably with the (more specific) term RATO, for Rocket-Assisted Take Off (or, in RAF parlance, RATOG for Rocket-Assisted Take Off Gear).

Like this:

As the take-off thrust of jet engines has grown, JATO has fallen from favor. It is still used, however, when heavily-laden aircraft need to take off from short runways or when operating in "Hot and high" conditions.

Rather surprisingly, rocket-assisted takeoff is actually flight-certified for the Boeing 727:

JATO:
6 Aerojet 15KS-1000-A1 thrust augmentation units, approved for use with Pratt and Whitney JT8D-11, JT8D-15, JT8D-15A, and JT8D-17, JT8D-17A Turbofans with 15° flaps.

A3WE The Boeing Company Model - 727 - 200

• Apparently, Mexicana experimented with fitting JATO to their 727s as a backup in case there was an engine failure during take-off at a high-altitude airport on a hot day (e.g., Mexico City). There's a photo of a test run about three-quarters of the way down this page. Discussion there says it was never used on a passenger flight. – David Richerby Mar 23 '15 at 7:46

Surprisingly I don't see the obvious answer:

Use a different plane, for goodness sake! One that can actually take off and land in 1.4 miles.

Reduce the problem to fitting a 2-mile length into a length of 1.414 miles. Let's say we don't want any bends. How do we do it? Use rollers! Say each roller has a diameter of 1 yard. Make them of a suitable material and construct the holding mechanism in such a way that when the plane travels 1 yard relative to the island its wheels have been in contact with the roller for 1.414 yards.

• This doesn't work. To take off, aircraft need speed relative to the air, not to the ground. It's not that you need to travel for two miles before take-off: it's that it takes two miles to reach sufficient airspeed to take off. (Except that it takes way less than that, unless you're in a very, very large plane.) – David Richerby Mar 22 '15 at 12:48
• It also doesn't work because planes are powered by the action of their engines against the air, not by their wheels on the ground. If you effectively moved the ground backwards under the plane, it would just make its wheels rotate faster, and slow the plane very slightly because of increased friction in its wheel bearings. – David Richerby Mar 22 '15 at 22:13
• @DavidRicherby so giant air blowers could help? – Paŭlo Ebermann Mar 23 '15 at 21:22
• @PaŭloEbermann Very much so, if they created a headwind. That's why planes take off into the wind wherever possible. – David Richerby Mar 23 '15 at 21:25
• @DavidRicherby - Many thanks for this! I should have recalled that when launching a kite after laying its string on the ground, you need to stand with your back to the wind, and if necessary run against the wind not away from it! – h34 Mar 25 '15 at 8:36

You could

dig the runway parallel to one of the sides of 1 x 1 island such that the first 0.5 mile of the runway has an angle of declination of 60 degrees and the next half has an angle of inclination of 60 degrees. The total length of the runway will be 2 miles then.

• Diagram not to scale. The angles you've marked as 60deg are closer to 30! In any case, this wouldn't work. No commercial plane can accelerate to take-off speed while going up a 60-degree slope. – David Richerby Mar 22 '15 at 19:14
• Obviously the diagram is not to scale. I'm hoping in future they can invent technologies that allow for an aircraft to take-off a 60 degree slope. – nsane Mar 23 '15 at 3:34

If the runway is approximately straight and level, as it should be for it to be "physically possible for the plane to take off," then there is no way to fit a 2-mile-long island on a 1-mile square island. Especially if you consider that a plane requiring a minimum of two miles to take off should probably have a runway that is longer than two miles, in case there are any sub-optimal conditions for takeoff.

So, how can we build such a runway while "remaining within the boundaries of the island"? Simple. Expand the island. Use land reclamation to build up the shores, so eventually the island gets bigger, and then build a runway on it. Some commenters have observed that there isn't much use for a one-square-mile island, so this fixes that problem by giving you more usable space, too.

Also, this is much less far-fetched than some of the other proposed constructions. In fact, the airport in Genoa, Italy is built on an artificial peninsula of reclaimed land, proving the feasability of this idea.

• "If the runway is approximately [...] level, as it should be for it to be 'physically possible for the plane to take off,'" Tell that to people who've flown to Courchevel. – David Richerby Mar 23 '15 at 21:32
• @DavidRicherby For one thing, the 18.6% grade there is a lot less than the 100% grade required to build a flat runway on the existing island. Also, if you achieve "minimum takeoff speed at the very end of the runway" and are traveling downward at that point... I don't think you're going to be taking off. – KSmarts Mar 23 '15 at 21:48
• That's why you put a flat bit (or even a 'ski jump' upwards ramp) at the bottom, going into a gradual curve upwards. See dmg's answer (above) and its comments or aviation.stackexchange.com/a/13388/4067 – A E Mar 23 '15 at 22:18

I can think of several approaches:
1. If you must use planes, you can extend the island out to meet the requirement.
2. You can use seaplanes/floatplanes, so there's no need for runway, since the ocean is your runway.
3. You can use vtols, so you just need a flat area large enough to fit your plane.

• The "requirement" for a two-mile runway is misleading. The smaller models of Boeing 737 can take off in under a mile and an island of this size isn't going to require planes even that big. So there's no need for float planes or VTOLs (though, in reality, float planes might be a good option, since they avoid paving over large amounts of your beautiful island). – David Richerby Mar 22 '15 at 12:57

Go down one edge of the island for .9 miles then turn and go along the diagonal of the island. Which gives a length >2.17 miles.

• This works for tron, why not planes? – JLee Mar 24 '15 at 11:49

You can use a rotating platform at the end of the runway (on a corner of the island). Thus when the plane enters onto the circular platform, it rotates by 180 degrees while your plane continues where it came from.

• You're proposing to turn the plane through 180 degrees in the middle of its take-off run? Conservation of momentum isn't going to let that happen. – David Richerby Mar 22 '15 at 12:54
• Depends. You can put 2 more wheels at the edges of the wings and put 2 rocket assisted engines on the nose of the plane positioned at an angle of 90 degrees relative to the aircraft. This should distribute the angular momentuum and ensure that it works. Clearly, much also depends on the skill of the pilots. – xxx Sep 30 '15 at 11:21
• If you're going to put rockets on your plane, just put them pointing backwards to generate more thrust so the aircraft can take off from a much shorter runway, without needing infeasible corners. It's called JATO. It's an actual thing. – David Richerby Sep 30 '15 at 18:30
• JATO rockets can be used to take-off only, but for landing that can't be used. With the help of the mobile platform you would have several alternatives. The first solution is to use the engines deflectors so as to help airplane to rotate (rather than to stop as normally used); – xxx Oct 2 '15 at 1:26
• if the previous was not enough, as mentioned before, may be implemented only two rockets side (instead of several more if used for the primary thrust) to make it take-off or land, or create a flow channel that uses the main turbines (as it happens with the deflectors) but leading the flow to the front of the plane and it does get out of side. – xxx Oct 2 '15 at 1:27

Ideally this runway should be a shallow "skateboard ramp" design. Lizard Island on the Great Barrier Reef is a good example. Aircraft need to be able to taxi uphill at both ends. Such a perfect half tube would be 2.22 miles. So I probably would increase the radius a tad. Or implement a tractor tow system.

Build a runway like this where the height of the run way is sqrt(3) m high. Or go through the digonal in that case you only need height of sqrt(2)m

I would probably try to go with an Euler spiral (Wikipedia explanation)

While indeed it becomes increasingly difficult to turn at higher speeds, this has the benefit that the curvature of the spiral would decrease with the increasing speed. Seems like a two mile length would easily be doable, IF:

1) The motors thrust can be 'unequal', to actually perform said turns

2) The tires don't blow out

3) You have little to no 'use' for the island. The shape really messes up what functionality the island can have.

It is a circular runway that is inscribed within the 1 mile x 1 mile square with an approach tangent to the circle.

• Never seen an aircraft land on a circular runway... – Marco Bonelli Mar 21 '15 at 14:57
• Never seen a run way 2 miles long! :).... Never mind, just checked and they exist. – Ghost Koi Mar 21 '15 at 16:33
• that's true actually... ahahah – Marco Bonelli Mar 21 '15 at 16:36
• @GhostKoi 2 miles is 10,560 feet. Major international airports tend to have runways in the 8,000-13,000ft range. – David Richerby Mar 21 '15 at 20:30
• I know... that's why I edited my comment. It was still a funny response though – Ghost Koi Mar 21 '15 at 22:27

How advanced are we in technology?

Just build a treadmill-like runway in a straight line across the Island (1 mile long) with the plane going against the direction it rotates. The plane pick up the speed required to takeoff, then you stop the treadmill runway.

Flight safety legislation really does not apply here*

• Since the airplane is not powered by traction with the runway, a treadmill will not significantly affect the distance required to take off. – KSmarts Mar 23 '15 at 21:03
• Oh ok, I was thinking the traction might give a boost to the initial speed which result in a much shorter runway is required. – Alex Mar 23 '15 at 22:32
• @KSmarts: Even if the plane doesn't gain enough speed to become airborne on the treadmill, couldn't it gain some tire-rotating speed, so then when it hits the post-treadmill section of the runway, it's like hitting the ground running... and doesn't need as much runway to build up the initial speed? – TOOGAM Mar 24 '15 at 5:29