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My understanding was that the engines were not used. If I took that one wrong, sorry.
But engines give forward thrust irrespective of the behavior of the wheels (assuming they can turn), so in that case the plane would go forward anyway, the wheels just having to turn faster in response to the belt.
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If the engines are not running, I would definitely not want the plane to take off!
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Only if you have a really big fan blowing at a speed at least that of the aircrafts V2 (take off) speed. 
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Then the aircraft will go and hit that fan 
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It obviously can since propulsion for airplanes is provided by i/e a jet engine. Airplane doesn't accelerate using wheels, like i/e a car. Conveyor belt would only make the wheels spin faster, and that's it.
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Actually ... no. The conveyor will have an effect and can increase lift - slightly - since it will also drag the air along with it and if that interacts with the wings it effectively act like the plane is going faster. (That's why aircraft carriers always launch steaming full ahead into the wind when launching and recovering aircraft - it adds a few MPH to their airspeed and reduces the chances of a stall.)
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Your answer is confusing. First, you say: actually no, then continue with a reasonable explanation on why some extra lift would be achieved. So, airplane WOULD take off, which I assume was the point of the q. Whether it'd have more lift or no due to conveyor belt is a minor detail.
There's also the matter of how long the conveyor belt is. If it's as long as the runway, then yea, but if it's as long as the airplane, airplane could fall off and probably crash in front of it.
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What a plane needs to take off is lift which is created by the pressure difference between the top and bottom sides of the airfoils (wings). To generate that difference, we use high-speed air flow aimed at the leading edge of the wing. The shape of the wing (an airfoil) makes air pile up in a high-pressure zone under the wing and zip over the top to create a low-pressure zone above the wing. This crazy magic lifts the plane with all of its weight into the air.
To generate the air flow, we generally use the easiest thing at hand, the velocity of the plane itself. On a conveyor belt, it won't move against the wind, but if it's facing into a gale strong enough, it could theoretically lift into the air and its jets would then be sufficient to make it go so long as the gale persists long enough for the jets to achieve enough speed through the air to maintain lift.
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The ONLY way the airplane can take off is if the speed of air respect of the airplane's speed is equal to the minimum speed the airplane needs to take off when the wind is absolutely calm. That is, because the conveyor makes the plane to be static respect to the ground, the only way the plane will take off is if there is a really hard hurricane that accelerates de wind to the plane's take off speed.
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The conveyor is not able to make the airplane stay stationary.
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gervacleto wrote: because the conveyor makes the plane to be static respect to the ground,
How does it do that, with free running wheels? The wheels are the conveyor belt's only contact with the plane, and I cannot see how you can enforce a thrust of the same magnitude (but opposite direction) as the plane engines, through free running wheels.
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Has anyone posted the Mythbusters episode about the question?
I’ve given up trying to be calm. However, I am open to feeling slightly less agitated.
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Yes. An airplane is not propelled forward by the wheels, but by the propeller(s) or jet engines. How fast the wheels are spinning is irrelevant. It may be a little trickier to steer but it'll take off.
If you think 'goto' is evil, try writing an Assembly program without JMP.
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wind speed is all that matters. Nothing else for take off. Not ground speed, not wheel speed. Only the speed with which the plane is moving in relation to the air. This is why planes prefer to take off into the wind. It reduces the amount of time on the ground before liftoff. But again ground speed doesn't matter. Air speed matters.
To err is human to really elephant it up you need a computer
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Yes. Unlike a car that uses it's engine to turn the wheels to propel it forward, on an airplane the wheels are not what propels the plane at all. Its the thrust being generated by moving air via propellers or jet engines that provides thrust for an airplane, and the medium they are moving in, and what that thrust is relative to, is a sea of air.
Put a boat on wheels, place it so it sits on a conveyor in water, and turn on the propeller. As long as the wheels on the boat that are in contact with the conveyor are free wheeling the boat will move forward.
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In general, no. The movement of the wheels has no effect on lift. To take flight, a craft needs either sufficient airflow across the wings or enough thrust to overcome gravity. A plane can take off from a stationary position, without a conveyor, if there is enough wind and the engine has enough thrust to counteract the drag; the plane would simply rise straight off the ground. Additionally, air craft with very powerful engines, like an F-16, can accelerate vertically. In this case, the wings do not generate lift; the engine itself provides all the lift. Theoretically, an F-16 could take off from a stationary but vertical position. Again, the wheels would not be used for this.
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Ignoring the possibility of a head wind strong enough to take of on the spot (or very close to it): No, it will not take off.
The engines will obviously apply a force to the aircraft. To match the speed of the wheels, the conveyer belt would have to keep the plane still - if the plane is moving forward, the wheels much move faster than the belt (ignoring maximum friction, so the wheels will not slip). This means the belt will have to apply enough reverse force on the tires that the increased tire rolling friction and bearing friction transferred to the landing gear is identical to the force applied by the engine. This would quickly require the wheels to spin so fast the centrifugal force will rip them apart - first the ties, then the wheel or bearings. Then anything remaining of the landing gear will be ripped off, and the aircraft will crash on its belly on top of a conveyer belt moving it rapidly backwards. Kind of hard to get in the air from that position.
Any limit to the available friction between belt and tires will allow the tires to slip over the belt - this means the aircraft could be moving forward while the belt is still matching the speed (but not position) of the wheels. But any friction available will be "used" to accelerate the wheels - so anything but the most minuscule friction would not allow the plane to reach takeoff speed before the wheels collapse. To take of, you should basically be able to do it with the breaks applied (ignoring the pesky nose or tail wheel without breaks).
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The conveyor stuff is just complication. The entire point of the conveyor is to prevent lateral movement. Mythbusters had no budget for a real conveyor and their facsimile wasn't a conveyor and didn't prevent lateral movement. The weight of the plane and the stretch of the material allowed for lateral movement. A dynamometer would have been better.
Sure, the wheels don't matter. So take the wheels off and jack the plane onto cinder blocks. Same concept, only without the fake-conveyor nonsense that lets people think you can have lift in no wind with no lateral movement.
Jet, plane, whatever... If you prevent it moving forward, it's not going to move upward just because you tilt a control surface. Tie a sea plane off by its rear to a dock. Throttle slightly to get all the slack/stretch of the line out, then push to full. It would remain more or less stationary.
"Prop wash" lift is a thing in R/C aircraft where thrust-to-weight blows pretty much all real planes out of the water. You can hover some R/C planes like a helicopter. You might be able to do that with some real sport planes, but I doubt it.
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jochance wrote: If you prevent it moving forward
But what would prevent it from moving forward? The engines, whether jet engines or propellers, thrust the plane in the forwards direction, and there is nothing from stopping it. The free running wheels will not stop it, even if they are spinning at a fairly high speed.
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That's the definition of an actual conveyor belt for purpose of this thought experiment.
If the conveyor moved backwards as fast as the plane would otherwise be moving forwards there would be 0 horizontal movement, and so, 0 air flow to generate lift.
Prop thrust is what gets the plane moving forward, it has nearly nothing to do with lift. So, simplify it further and take the plane bit out of the equation a moment. Make it a car instead.
If X RPM of a propeller will make it top out at 20 mph, use gearing to make that same RPM drive these car wheels at 20 mph. Put that car on conveyor moving the other way at 20 mph. You're telling me that car goes forward? No.
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jochance wrote: If the conveyor moved backwards as fast as the plane would otherwise be moving forwards there would be 0 horizontal movement, and so, 0 air flow to generate lift.
The only force from the conveyor belt on the plane to make it move backwards is the friction in the wheel bearings, which should be rather small. You have the same friction at a normal take of, so the engines are dimensioned to handle it. Compared to the force required to accelerate the plane to take off speed.
If the conveyor belt somehow has managed to get the plane up to take off speed in reverse, it must have taken a tremendously long time (and tremendously long conveyor belt) But once done, the turbines have no bigger problem pushing the plane up in speed on top of that conveyor belt. When they have done the same amount of work as is normally required to reach take off speed, the plane is standing still relative to the ground. Any further work done by the turbines will accelerate the plane relative to the ground, in the normal manner. The epsilon force from the conveyor belt, through the friction in the bearings, will not stop it.
So getting off the ground will take roughly twice as long (and twice as much fuel), but that doesn't prevent it from happening.
So, simplify it further and take the plane bit out of the equation a moment. Make it a car instead. That makes a completely different situation. The wheels of a car are not free running, but tightly connected to the car engine. The acceleration of the car is caused by the force of the rotating wheels on the ground / belt. Take the belt away, leaving the car floating in free air, and it can neither speed up nor brake down. Suspend a (motorized) model plane in a string, and start the propellers / turbines: It will pull ahead.
So if the original question was talking about a car, then you would be right. But it didn't. That is the essential 'trick question' part: Most of us will think of the way a car accelerates (completely dependent on a solid grip on the road) and overlook that planes are completely different (totally independent of any grip on the runway) with respect to propulsion.
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trønderen wrote: But once done
There is no "once done". It's a conveyor belt. It loops around and keeps happening. It also speeds up such that no, there is no "further work done by turbines" to move the plane relative to ground speed. The definition of conveyor for the thought experiment defines that.
It's no different than a car in neutral with wings slapped onto it and tied to a tree in front so it doesn't go backwards. I doubt you'd say that would lift off just because you spun the wheels fast.
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No.
The reason being Amarnath S wrote: match the speed of the wheels
the result of which is that, no matter how hard the plane pushes forward, the conveyor belt pushes the same wheels, with a nonzero mass, back exactly as fast as the wheels are moving forward relative to the conveyor belt, a process which is not limited by anything (say, the wheels will spin at an incredible rate, and they it were a real-world situation, the wheels would explode from centrifugal forces); so that the total speed of the plane relative to the ground is zero.
So the plane's thrust is used completely to drive the wheels' reaction force.
It cannot be anything different, or else the exactly matching speed in the opposite direction would no longer hold true.
So the wings do not catch any wind and the plane stays on the ground. All the force used by the plane for its attempt to take off is put into the wheels spinning.
And this conveyor belt, having to make those same crazy speeds in the other direction, would be a mighty impressive piece of work.
This is why it is important that you carefully read the question, be precise in what it states and what it does not state, not read carelessly, and not jump to conclusions.
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Martijn Smitshoek wrote: No.
unless
it is a plane designed to take of completely vertically.
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