If it makes you feel any better, it took me moment to figure this out.
To simplify, let’s assume there is no wind and the plane’s wheels can turn on frictionless bearings. The pilot fires up the engines. Let’s figure out the net force on the plane. The engines are producing a force on the plane in the direction of travel. The plane is not yet moving, so there is no drag. Because there is a net force on the plane due to engine’s thrust, the plane accelerates in the direction of travel relative to the ground (and the air). As the plane starts moving relative to ground (and the air), the treadmill kicks up. However (and this is very important), because I have frictionless bearings, no force is exerted on the plane by the motion of the treadmill. The motion of the treadmill just makes the wheels spin twice as fast as they would normally.
The plane continues to accelerate relative to the ground and the air until it reaches takeoff speed, at which point it takes off. If takeoff speed is 100 mph, then the plane is moving relative to the ground (and the air) at 100 mph. At the same time, the treadmill is rolling backwards at 100 mph, so the relative speed difference between the plane and the treadmill surface at takeoff is 200 mph.
The real-world situation (with friction in the wheel bearings) simply means that the plane has to overcome a bit more friction to accelerate itself to takeoff speed.
All of you folks trying to figure out how this would work in the real world need to relax, IMHO. It’s a thought experiment, for crying out loud.
And all of you folks who think that the treadmill somehow is exerting a backwards force on the plane are apparently thinking that the plane’s brakes are on, or that the plane is bolted to the treadmill surface.
You got that here, right? Short answer: The conveyor belt doesn’t matter. ll it will do is make the tires spin faster. The plane is pulling against the air, not the pavement. Its propulsion comes from the propeller, not the tires.
If it helps, let’s now compare our plane to an automobile.
Planes accelerate due to their engines, which exert a net force on the plane due to their thrust.
Automobiles accelerate because the contact pad on the tire exerts a force on the ground, and the ground therefore exerts a reactive force on the tire. This is why cars do not accelerate well on ice, for example. Planes, however, will take off just fine on ice, because their acceleration is not dependent on ground friction.
OK, let’s put our automobile on the treadmill now. The driver steps on the gas, and car begins to roll forward. As the car’s wheels begin to turn, the treadmill starts up. In this situation, indeed, the car remains stationary with respect to the ground (and the air, for that matter).
The difference, folks, is that airplanes have engines that produce a net force on the plane irrespective of whether it is on the ground, on a treadmill, or in the air!
The plane is moving, once the propeller starts turning and pushing air aft. Equal and opposite reaction, ya know. The plane’s accelerating motion relative to the air generates increasing lift until it equals weight. What do you think happens normally?
The limitation due to tires gives the “won’t take off” faction their best claim to being correct. Airliners’ tires have maximum speeds, which generally are only a bit (say, 10%) faster than their low-wind, maximum-gross-weight, hot-weather, high-altitude takeoff speed. With the treadmill, the wheels will likely be turning faster than the tires can handle. The tires will fail and the plane will thus not take off.
Any analogy that you make, such as a automobile on the treamill, or a runner on a treadmill, is not a valid comparison because in those situations, the auto or the runner accelerate themselves due to action/reaction frictional forces with the surface of the treadmill. Airplanes do not accelerate themselves with their wheels!
A better analogy would be a car with a rocket engine on the back and free-spinning wheels. Such a car will accelerate just fine relative to the ground, because it’s not acclerating via its wheels.
Another analogy would be a person on roller blades with a huge fan. Again they’re not accelerating themselves via their wheels.
The fundamental problem here is that the original questing is sufficiently vause as to not make it clear if the airplane can move or not. Those who say the plane will not take off are reading it to mean that the treadmill will somehow be able to keep the plane from being able to move. I don’t believe that this is a correct interpretation, or for that matter even physically possible.
It’s pretty clear to me that the people who think the airplane will remain in one place relative to the ground (and not take off) are thinking of how an automobile would act in such a situation, whether they realize it or not.
People may also be thinking of a runner on a treadmill, to wit: runners don’t move relative to the ground, so neither would the plane, so therefore the plane won’t take off.
The mistake, of course, is thinking that the runner on a treadmill and the plane on a treadmill are valid comparisons. They are not.
Well, if the belt is matching the forward speed of the plane, (which is probably what was intended) the wheels are forced to spin at precisely twice that. So, ignoring wind, what is the ratio of the maximum-gross-weight, hot-weather, high-altitude takeoff speed to the unloaded, cold dry air, sea-level takeoff speed? If it’s higher than 2:1, the plane should be able to take off before the wheels melt if you do it in Boston, in December.
My first thought was that the plane will obviously take off because there’s no way for the treadmill to exert a force to match that supplied by the engines. Engine force => acceleration => airspeed => flight.
My second thought was that there’s more than one way for the treadmill to interfere with a takeoff - such as by causing tires to fail.
In my sick twisted mind I found it easiest to compare the airplane to a water skier on a river.
Assume that the skier can swim (ski off course) @ 3mph.
Current is 3 mph
What happens if he tries to swim upstream? He goes no where because 3-3 = 0
Now he puts on his ski and grabs the the rope. The boat takes off up stream. Does the skier stay in the same place? No because he is being towed by the rope. Change the rope for a jet engine and you have your answer.
It’s realists like yourself who make life hard for physics teachers.
Back when I taught college physics, I took pains to always state any simplifying assumptions on tests and quizzes to head off such questions, such as assuming free-fall, or a frictionless surface, or frictionless pulleys, etc. Not doing so, IMHO, was just sloppy test-writing.
I get the jet being towed analogy. It will take off. What I’m having trouble picturing is the concept that if the wheels move forward then the belt isn’t matching the speed of the wheels.
If aircraft tires get hot enough, there’s a plug of rubber, made to melt at a lower temperature than the carcass, which will let the air deflate gradually and, in theory, prevent an explosive blowout. That is usually only a consideration on heavy braking, though.
Let’s say takeoff speed is 300 kph. The circumference of the tires is 5 m.
300 kph = 300,000 m/hr = 5,000 m/min.
The wheels would therefore be turning at 1,000 rpm across normal, level ground at takeoff speeds.
If you were to take a giant invisible hand (or a jet engine) and force the plane forward, the wheels would begin turning at a higher RPM, but the plane would still move.
When the wheels turn at a higher RPM, the conveyor belt moves faster to compensate. . . but the plane still moves. When the plane reaches takeoff velocity, the wheels would be moving at 2,000 rpm which, over level ground, would ordinarily equate to 600 kph. . . but the plane would still move forward at takeoff speed as specified.
We might see feedback between conveyor and wheel RPMs, depending on how the conveyor matches speeds: as one accelerates, the other accelerates farther to compensate. With frictionless bearings this is immaterial, at least until the plane’s wheels leave the treadmill at 2,000 rpm and hit level ground. (With a treadmill long enough for takeoff, even this is immaterial.)
INA aeronautical engineer, but I doubt that a prop driven plane will generate enough airflow over the wings to create lift. I am sure that a jet would not take off. No forward movement, no airflow, no lift, no flight. The wheels have absolutely nothing to do with it. Try this another way. You have a very long cable secured to the tail end of a jet aircraft and secured to an immovable object, if you believe that a plane can take off from a treadmill (w/ no forward movement), then the jet on the tether will also take off? I don’t think so. Again, no airflow, no lift, no flight.
Maybe a small prop plane w/ a very powerful engine might take off, but this would be from the airflow generated by the engine. Think about this, when a jet lands on a carrier the pilot immediately applies full power in case he/she misses the trap and doesn’t reduce power until the aircraft is almost at a standstill.
