Dave_Phillips wrote:
Of course it will
Either the winch and aircraft have no physical limitations (infinite torque, infinite cable strength, infinite strength of coupling and airframe) and then you will be in an “impossible” situation where the question is meaningless, or it is a “real” winch and a “real” aircraft in which case it will not.
One of the same three things I wrote about above will happen, or the cable will break or the winch stall.
I think Airborne_Again has interpreted the conditions slightly differently (he is envisioning a higher multiplier for the wheel bearing friction, or a faster multiplier for the reverse conveyor speed). This point is valid, but not under the conditions stated.
Under the conditions stated, plane will of course take off. If the take off speed was 100 knots, the wheels will be spinning 200 knots. Yes there will be additional friction (from the bearings and tire contact patch), but not enough to to prevent forward acceleration of the entire aircraft.
Now if the conveyor was going a multiple of speed fast enough that wheel friction was gigantic, then perhaps a steady state condition could arise where the plane could not take off. Say for example the conveyor was going 100 times as fast as the wheels in the opposite direction. This would imply that for an aircraft moving forwards at 10 knots, the wheels would be spinning equivalent to a speed of 1010 knots in the reverse direction. There might be enough friction to prevent further acceleration of the aircraft and steady state might be achieved against the engine thrust.
I have experienced this in an under powered float plane on a river. Travelling upstream I was unable to get enough airspeed due to the drag of the water (quite similar to the conveyor example actually!!). Similarly, I have experienced sufficient wheel drag to prevent take-off with a PA28 in a muddy grass runway that was 5000 feet long. Lots of motoring, but no airspeed.
No, of course the glider will get airborne! It doens’t require anything to be infinite.
A glider will get airborne usually around 30-35 knots. Given a conveyor belt running in the opposite direction at the equivalent ground speed, the wheel will only be rotating as if it were going 60-70 knots as the glider lifts off. The tyre on the main wheel on our glider is rated at 120mph so will be just fine, and there isn’t so much drag in the wheel bearing that it can bog down a Jaguar 4.2 litre engine on the winch because the wheel is turning as if it were doing 70 knots (the winch engine would hardly notice the extra drag from the wheel being turned at 2x its normal speed). If the wheel bearing was generating so much drag at a rotational speed at 70kt it could break a winch wire or bog down a 4.2 litre 6 cylinder petrol engine, then the wheel bearing is shagged and definitely not airworthy (and would probably get red hot in normal launches and landings without the conveyor belt).
Not quite. The conveyor moves at the speed of the wheel (tread, pretty much the only sensible interpretation)
When the aircraft starts moving, let’s say at only 1mm per second, the belt will then start moving backwards at 1mm per second, which accelerates the wheel to 2mm per second, which the belt then needs to match, which in turn acelerates the belt… and all that at the most miniscule movement of the aircraft. If the belt and wheel are not infinitely strong, somebody will get hurt by all the shrapnel.
If I find the time I might calculate a rough energy requirement to accelerate an infinitely strong belt (the mass of of the wheel is negligible) but given that this process will reach infinite speed, at some point all the electricity in the world will be insufficient to accelerate it further. It will go at a significant proportion of light speed, and I might then have to recalculate the additional effect of general relativity on the ILS beam, as I did somewhere before…
If the energy available is infinite, the whole thing will collapse into a black hole.
So if the belt matches the wheel speed, everybody dies.
If the belt only matches the aircraft speed, it will move backwards at ground speed, and as long as the wheels can take 2x ground speed at lift off, the aircraft is fine.
Agreed. I was wrong about the glider.
That shows that the real difficulty in solving this kind of problem is not at all understanding the underlying physics but straightening out the tricky and counterintuitive details of the problem statement.
Another very good example of this in another area (probability theory) is the Monty Hall problem. Many PhDs in the area who certainly understood the underlying principles extremely well and even Paul Erdős (one of the most prominent mathematicians of the 20th century) got it wrong — some aggressively so.
The problem is this line in the question
“The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction”.
This part of the premis is impossible.
Take an aircraft that is moving foward at 20kts (in zero wind) on a conveyor belt that is not moving. The wheels are now rotating at 20kts. So according to the question as worded, the conveyor belt should now start to move in reverse at 20kts.
But once the conveyor belt starts to move backwards at 20kts, if the aircraft maintains it’s 20kts IAS, then the wheels are rolling at a speed to 40kts (20 kts IAS and 20kts to compensate for the conveyor belt moving backwards). So the converyor belt must speed up to 40kts, but it still can’t match the speed the wheels are rotating at. So this is impossible to do. The only speed at which the aircraft can be moving (in still air) and which it’s possible for the speed rotation to match the speed of the conveyor belt is zero.
Becuase this is impossible you are left with two choices.
1. Follow it to its logical conclusion and realise that the coveyor belt has only two possible speeds. Zero and infinify. At zero it doesn’t take off and at infinity friction takes over and burns everything, leaving the logical conclusion that the aircraft can’t take off.
2. Ignore the part of the question that requires the speed of the wheels to match the conveyor belt in reverse, and then reach the conclusion that the aircraft will take off.
Those following the question based on pure theory will come up with 1, and those thinking of a practical example will ignore the logic problem and follow 2.
In Dave’s Myth Buster example, if the speed of the vehicle matched the aircraft airspeed (in still wind) then the rotation speed of the wheels would have been zero, and the ’conveyor belt was moving forwards not backwards)
XKCD has a good roundup of the various interpretations.
https://blog.xkcd.com/2008/09/09/the-goddamn-airplane-on-the-goddamn-treadmill/
I think most of the ambiguity comes from how you interpret wheel speed, and how the increase in angular momentum impacts the acceleration of the plane. Perhaps it would be less confusing if we thought of the plane as having skis or teflon skids or something of that nature.
Of course, the true answer is to use a helicopter, because then you can take-off without upsetting the treadmill at all :)
dublinpilot wrote:
Becuase this is impossible you are left with two choices.
Exactly so!
I can’t believe I’m allowing myself to get dragged in to this “debate” but:
Wheel ROTATION speed will be zero because every time it attempts to rotate the conveyor will counteract the rotation.
If engine thrust is sufficient tyre friction will be overcome and if enough conveyor belt length is available then flying speed may be achieved before fire consumes the undercarriage. Probably something like an EE lightning would be OK but a fully loaded 747 less so.
Then, Zaphod Beeblebrox would appear on the scene and remind us that the real answer to this question is “42”.
redRover wrote:
XKCD has a good roundup of the various interpretations.https://blog.xkcd.com/2008/09/09/the-goddamn-airplane-on-the-goddamn-treadmill/
Thanks a lot for posting this. I find myself belonging to the apparently rare group #1 when thinking about this
