There’s an aspect to this problem that I haven’t seen mentioned before, and this thought experiment illustrates it as well as any. Assuming a strong enough rope, ideal wheel bearings and wheels, there’s still a questions of stability/controlability. Say the treadmill is turning at 300 miles per hour. The rope is holding you in place, the wheels are spinning furiously, no problem. But if anything happens to change he direction of your feet, you’re in trouble. At that rolling speed, get just a hair offline, and your feet will be out from under you in no time.
Same with an airplane. The propeller only affects the air, the lift is only dependent on the air, but during the takeoff run, some of the steering control comes from the wheels and is affected by the rolling speed.
I realize that’s not what people are concerned with when this question comes up, but in the real world, I think that would be the limiting factor.
This was a real-world test and not a thought experiment. Without real-world constraits there are ways for a treadmill to prevent forward airplane motion. I’m not wanting to insult people who look at the problem in that way, they are at least thinking.
There were many, many people who truly believed the tarpaulin behind Jamie’s truck would keep the plane stationary. There’s many, many people who still, even after viewing the results, think the tests were flawed (like the claim that the wheels were still getting traction against the runway through the tarp material) and insist a real conveyor belt could prevent forward motion in a real plane. These are the nitwits.
I was just thinking that if they went by the exact wording of the question, they would have destroyed the universe (infinite speed = infinite size). Good thing they went by the spirit of the question.
Jamie said he gunned the truck’s engine as soon as he saw the plane moving, but the truck-powered treadmill did not keep up with the plane’s acceleration. There was no record of how fast the “conveyor belt” was going.
It matters not. The rolling resistance of the wheel bearings and tires are simply not enough to restrain the plane’s thrust. The wheel speed is irrelevant.
The treadmill has no effect on the airplane. Aircraft thrust is applied to air whether the wheels are on or off the ground. The distance traveled by the plane before it is airborne depends on the speed of the air over the wings. If a plane normally lifts off at 80 knots then it is possible to lift straight off the ground if the plane is facing into an 80 knot wind. A 747 would lift off with no forward movement if it was facing a 140 knot hurricane.
And the exact wording of the original question said nothing about the speed of the wheels. It said the treadmill would, in the opposite direction, match the forward speed of the plane.
if plane is heading east at 50mph, the treadmill will travel west at 50mph. Said treadmill will have no appreciable effect on speed of the airplane. The wheels will just be spinning as if they were travelling 100mph (in relation to the treadmill)
When I first read the problem I thought the airplane would not take off. This is because I assumed a ‘perfect system’ where the treadmill would prevent the aircraft from moving forward. But after reading the explanations in the first (?) thread I came around.
Anyway, why bother with even skis on a sheet of ice? How about a hovercraft? That would eliminate more friction, but the air cushion would still be analogous to wheels. The forward thrust would be provided by the aft-facing prop. The hovercraft would still move forward. If it had wings it would fly.
With a tailwheel airplane, the tail comes off the ground at a rather low airspeed; from that point on, only the main wheels touch the ground and all steering is done aerodynamically.
Some tricycle-geared airplanes - including most small ones - have castoring (unsteerable) nosewheels and thus also rely on aerodynamic steering. Those with steerable nosewheels would on the proposed treadmill be faced with steering at speeds twice what’s normal - not likely to be much problem.
I’m not sure it’s the steerable (or castering) wheels that are the only problem. (I may have phrased things badly; not an unheard of situation when discussing this problem.) Consider the main gear during a normal, non-treadmill takeoff. It’s not rolling perfectly straight all the time; it gets a little off (a gust of wind, uneven pavement, or any of the left-turning tendencies), there’s a sideways force that moves you off the centerline, you apply some steering control (rudder, nosewheel, or differential braking) and track down the runway until you’re going fast enough to leave it.
Compare that to the treadmill case, especially a treadmill going hundreds of miles per hour. Now, in the idealized, theoretical version of this problem, the wheels are spinning their little hearts out but the plane still takes off. But what happens when those wheels aren’t tracking perfectly down the centerline, or with the plane’s velocity vector? The sideways force will be many times stronger. It may carry you off the treadmill before you can react. It would be worse in a taildragger. The wheels are ahead of the CG, so any sideways force would tend to rotate the aircraft in such a way as to increase the deflection. And your steering control is based on airspeed, so you may not have enough rudder authority to redirect the main gear.
To recap: An ideal plane on an ideal treadmill takes off. A real plane on an ideal treadmill, something will eventually break. I’m suggesting that you may lose controlability before you exceed the physical limits of any component.