What if you then magically brought this near infinitely fast treadmill to an abrupt halt? The rotational energy in the wheels would force the plane forward with some insane amount of energy, I think.
Of course, with magic like this, who needs planes?
Shades of The Roads Must Roll
Not only that, my brother reports that when landing/taking off on such a runway in the Himalayas, both sides of the flight path were valley walls. So if there was another airplane coming the opposite direction, their wouldn’t be anywhere to go. And you’re close to maximum altitude, and at landing/ takeoff speed, so nobody is going to quickly climb out of trouble either.
This used to be moderately common in New Zealand, for the purpose of aerial “topdressing” (application of tons of superphosphate fertilizer). The runways took the form of a ski jump (though not quite that steep), which allowed the heavily loaded plane to take off in a reasonably short distance. This video shows a landing and subsequent takeoff; note the (hard to see) operation at the top, where a large amount of fertilizer is quickly dumped into the plane’s hopper.
Yes, the takeoff abort options are decidedly unattractive.
Sounds like Tenzing-Hilary Airport in Lukla, often mentioned as the world’s most dangerous - 12% runway gradient and very poor prospects of a safe takeoff abort or go-around.
Here’s video of a landing and takeoff.
The answer is (C) You would see the tire moving at the same speed as the treadmill, because the only thing making the wheel turn in your scenario is the force given it by the treadmill.
It doesn’t even matter which direction the treadmill is turning. You could switch the treadmill between forward and reverse every five seconds, and the only thing that would happen is the same instant bursts of smoke as the free spinning tires caught up to the new direction., just as non-spinning tires do when the suddenly encounter the 200 mph ground passing underneath them during a normal landing.
The tires are just trying to push straight down against whatever it is touching until the wings generate enough lift so it doesn’t have to touch it anymore.
That’s accurate, assuming that the camera is focused on the wheels so that, when the plane is moving, the camera is panning across to stay focused on the wheels. When watching the video, you’d see the wheels’ speed match the treadmill speed relative to the camera. But that’s not the same as the treadmill speed relative to the ground.
If the treadmill is spinning backwards at 10 kts (relative to the ground) and the plane is moving forwards at 30 kts (relative to the ground), then the plane is moving 40 kts relative to the treadmill and the wheels are spinning at 40 kts. If your camera was focused on the wheels, your camera would be panning across at 30 kts (relative to the ground) and when you watched the video you’d see a treadmill moving at 40 kts (relative to the camera, which is moving) and also wheels spinning at 40 kts.
Well… that’s close. The fact is that the wheels have rolling resistance (a constant force which does not depend on speed) and that force is rearward if the wheels are spinning forward and the force is forward if the wheels are spinning backwards. So changing the direction of the treadmill could reverse the direction of the rolling resistance force. That would have a slight effect on how quickly or how slowly the plane would accelerate or decelerate. That could increase decrease distance required for takeoff, by perhaps 10%.
Yes, the wheels merely hold the plane up until it accelerates to take off speed. The question is how long does that take and how much ground is covered during the acceleration. Putting the plane on a treadmill spinning backwards has exactly the same take off distance as putting the plane on an ordinary runway. But putting the plane on a catapult dramatically decreases the distance required. Putting the plane on a treadmill spinning forwards instead of backwards can act as a very weak catapult and shorten the distance for take off by 10% or so. But it would have the opposite effect during landing, lengthening the landing distance.
Bottom line, the treadmill is pretty damned useless because it needs to be the same length as the runway. It’s cheaper to just use the runway.
Quick question, is that an ideal, or is that in practice. Does that take into account deformation of the wheels of even of the bearings in the axle?
Well, I admit we’ve been simplifying it a little. Rolling resistance generally includes deformation of the tires and the friction in the bearings, and also slippage and vibrations. In most situations, deformation is much larger than the other components. AFAIK, rolling resistance due to deformation does not increase with speed, either in the ideal or in practice. Slippage, on the other hand, increases with torque, but I don’t think that applies to the airplane situation. Vibration would increase with speed but I’m pretty sure we can discount that because the wheels would fall off the airplane long before the increase became significant enough to hold back the airplane. So, when I said rolling resistance does not increase with speed, perhaps I should have said that rolling resistance consists of four components, the largest of which does not increase with speed. The main point is that we are accustomed to thinking about wind resistance increasing with the square of the speed, and rolling resistance does not behave that way. At reasonable speeds, the increase in rolling resistance due to increased speed is negligible.
Here’s a made up example talking about rolling resistance and wind resistance:
5 mph, RR= 40 lb, WR= 50 lb
10 mph, RR= 40 lb, WR= 200 lb
15 mph, RR= 40 lb, WR= 450 lb
20 mph, RR= 40 lb, WR= 800 lb
25 mph, RR= 40 lb, WR= 1250 lb
30 mph, RR= 40 lb, WR= 1800 lb
35 mph, RR= 41 lb, WR= 2450 lb
As you can see, the faster you go, the larger the fraction of your drag is caused by wind resistance. At 5 mph, it’s just over half. But at 35 mph, it’s more than 98% of your drag.
Okay, that’s what I was wondering, I kept going back in forth in my head as to whether it should or shouldn’t. I was thinking it shouldn’t, but wasn’t sure.
Hmmm, I was informed by Chronos, I believe, in a thread about 9/11, that wind resistance actually goes up by the cube of the speed. I wasn’t really sure about that, but I believed him in a sort of reverse ad-hominem way, as he seems to be right about stuff like that. It makes sense if I think about it as the air molecules hitting the vehicle have their energy go up as the square of the speed, and as you increase your speed, you are encountering twice as many.
Is this right, or am I missing something here?
The force due to wind resistance is proportional to the square of the speed. The power lost to wind resistance is proportional to the cube of the speed.