747 on a treadmill conundrum..

[Angrboda]

"Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?"

The question is misleadingly worded. If the bit about the conveyor belt is read to imply that the conveyor belt is moving as fast as the wheels, then the aircraft is stationary and will not take off. If the plane is moving forward at all, the wheels will be turning faster than the conveyor belt. It's hard to come up with another sensible interpretation of that phrase other than that it will be moving at the same speed. However, it doesn't exactly say that, just that it's designed to do so. If the conveyor moves slower than the wheels so that the plane can move, then the thrust of the engines will propel the plane to take-off speed and it will take off. As stated, however, there's a contrary implication in the question that the plane isn't moving.

[Anomalocaris]

Not if you read fast as describing the velocity of the axle or center of the wheel, as oppose to tangential velocity at the rim or tread of the tire, relative to fixed ground. As the axle of the wheel move forward at X mph relative to fixed point, the treadmill moves backwards at -X mph relative to the same fixed point, and the plane moves with the axle of the wheel.

In the absence of any stated reference point for velocity to be measured against, This is how I interpret treadmill moving backwards at the same velocity as the wheel.

[SteelCurtain]

All of this is moot, though. The plane's wheels have effectively zero effect on the plane's ability to create thrust and move forward on the runway. If the treadmill is always set to match the forward velocity of the aircraft, then the 747's wheels will always be rotating at 2x the indicated airspeed. The plane will take off just fine.

So we could make really short aircraft carriers?

We do make really short aircraft carriers. A normal runway is 1.5 miles long. An aircraft carrier is 1100 feet long. And the takeoff section of the aircraft carrier is only 300 feet long. The plane is hooked to a steam powered catapult and hurled into the air from zero to about 150 knots in less than 2 seconds.

The point still stands. The plane's wheels are not how the aircraft translates thrust into airspeed and therefore lift. The engines push massive amounts of air backwards, which has the equal and opposite effect of pushing the plane forwards. It works the same way as a helicopter. If a helicopter with wheels as landing gear were on a treadmill moving at any speed, the treadmill would have negligible effect on the helicopter's ability to generate thrust, and therefore take off.

It is the propeller/engine/turbofan that makes the thrust by pushing against the air, not by driving the wheels.

[Angrboda]

Not if you read fast as describing the velocity of the axle or center of the wheel, as oppose to tangential velocity at the rim or tread of the tire. As the axle of the wheel move forward at X mph, the treadmill moves backwards at -X mph.

In the absence of any stated reference point for velocity to be measured against, This is how I interpret treadmill moving backwards at the same velocity as the wheel.

I think you're struggling for an interpretation that makes sense of nonsense. You're over-interpreting.

The question is simply badly worded. So badly worded that it yields two contradictory conclusions, both of which are consistent with the wording of the question. One which relies on a literal interpretation of the phrase about the conveyor belt, the other of which relies upon trying to interpret what the author meant, ignoring the literal implication of what was actually said.

[Anomalocaris]

[SteelCurtain]

I think you're struggling for an interpretation that makes sense of nonsense. You're over-interpreting.

The question is simply badly worded. So badly worded that it yields two contradictory conclusions, both of which are consistent with the wording of the question. One which relies on a literal interpretation of the phrase about the conveyor belt, the other of which relies upon trying to interpret what the author meant, ignoring the literal implication of what was actually said.

The problem is that the literal interpretation defines something that is not in line with physics. The treadmill is not going to appreciably effect the airplane's ability to move forward, because the wheels have a nominal effect on this ability. If you interpret the question such that the treadmill somehow makes the plane stationary, then the treadmill itself is not germane at all. Why not ask, "can a 747 take off from a standstill?"

The interpretation that the "ground" is moving at a velocity in precisely the opposite direction with the exact opposite magnitude as the plane's velocity is the only one that makes sense in terms of a thought experiment.

[JuliaL]

From the OP:
"The conveyor belt is designed to exactly match the speed of the wheels"

Case 1: the reference for the speed of the wheel is the tire contact patch. The treadmill top surface is stationary with respect to the airport lounge. I'll call the stationary treadmill "the runway." This is a normal takeoff.
The contact patch of the tire's speed matches the (stationary) treadmill as the plane accelerates away due to thrust from its propeller, fan or reaction motor (jet or rocket) and lifts off normally. The wheel rotates at airframe speed/wheel circumference (in radians/2pi sec).

Case 2: the reference for the speed of the wheel is the axle of the wheel which translates at the same speed as the airframe as the plane accelerates away and lifts off normally. The top of the treadmill also translates at airframe speed. The wheel does not rotate.

Case 3: the reference frame for the speed of the wheel & treadmill is the topmost point on the wheel circumference. For this point to be stationary with respect to the treadmill contact patch, the wheel cannot rotate. This case must be identical to case 2 where the treadmill runs at airframe speed. The aircraft accelerates away and lifts off normally.

Other cases involving rotated reference frames are too silly and will not be considered. I'm also ignoring the difference between velocity and speed for case 3.

In each case the treadmill has to be long enough for the plane to reach liftoff speed, v2min or greater.

[Angrboda]

Why does it make sense to interpret the "speed" as running speed with respect to the conveyor belt, but something else with respect to the wheel?

You're adding unstated assumptions.

[lifesagift]

Can I just say, I didn't think this up or phrase the question, but it has intrigued me for a while on other forums. But I have to say that you guys have gone ten times further in exploring this than other forums I've been on.

My conclusion (which is not necessarily correct) is that the way the question is phrased is crucial in this... firstly, the conveyor and the wheels are used to suggest a plane tethered by an infinitely strong rope therefore the plane will never move, and not get lift, and not take off.

(Credit to Jenny A for this thought process)

[JesusHChrist]

All that matters in this conundrum is the speed of the plane's wing relative to the air.

That's it.

As long as the speed of the air over the wing is below the level to achieve the necessary lift, plane won't become airborne, no matter how fast those wheels are a-spinnin'.

This is kinda basic.