Here's my $.02: (Note: Not an expert, just an interested person)

I say the answer is both yes and no, depending on the specifics of the question (i've seen it told with slightly different specifics), how you with to interpret the question, and how realistic you want to be.

The Question as posed here
Okay, first and fore-most, planes/jets/rockets develop their forward movement (not lift) through thrust. Unlike I.C.E.s, wheels are not required to generate forward movement. A prop plane (or boat) requires a fluid to pass it's aerofoil/airfoil through in order to generate thrust (bernoulli's principle). Jets generate thrust by combusting air and fuel and exhausting that rearward (Newton's 2nd & 3rd laws), rockets by combusting their onboard fuel (in solid or liquid form). Again, in all of these cases, wheels are not needed to generate forward movement.



Now, this is another significant bit of info - the treadmill does in fact exert some amount of force on the plane - but it should be (due to bearing design) relatively insignificant. That is, a large amount of force turning the treadmill, will exert a large amount of force on the wheels, which will exert a small amount of force through the wheel bearings to the plane's body. So, as the treadmill gains speed, the wheels gain speed to match, but the plane's body does not gain any significant speed.


So, what does all that mean? Well, let's go back to the question:

This seems to indicates that the plane and conveyor are moving in opposite directions. That right there suggests the plane is not standing still, which should indicate that it should be able to use it's thrust to accellerate on the ground until such time as it reaches it's lift velocity.

This is the key component of the problem and could use clarification (I've seen it worded/represented differently). We know that the treadmill can impart some force on the plane, but most of the force will be imparted on the wheels of the plane and not the body. So let's look at a couple of interpretations/variations of the problem.

1) If the speed of the conveyor is trying to match the airspeed of plane, than the plane will eventually take off.

In this case, one cannot say: "The plane will move at '+x'mph and the conveyor at '-x'mph which cancl each other out (x-x=0)" Again, the amount of force transfered to the wheels of the plane from the treadmill is greater than that transfered to the body of the plane.

That is, the plane will generate 'N' (big N) thrust pushing itself forward at 'x' mph, the treadmill will spin-up to 'x' mph to try to counteract, which will spin the wheels to '-x' mph (same speed, different direction), which will impart 'n' (little n) force on the body of the plane. In this case, the plane's thrust (big N) will always overcome the force of the treadmill (little n) at any given air/ground speed. The plane will accelerate through the air (moving forward on the treadmill), while the treadmill and the plane's wheels continue to increase their speed equally, which will impart a small amount of drag on the plane. Eventually the plane should reach whatever speed it needs to in order to lift-off (having expended a bit more energy than normal to get there).


2) If the conveyor is trying to match whatever speed will keep the plane stationary, the plane will not take off.

In this case, we're assuming the conveyor belt is not trying to match the plane's air/ground speed, but is trying to spin the plane's wheels to whatever [great] speed would be required negate the plane's thrust. That is, while the plane tries to move forward at 'x' mph, the conveyor will need to spin at 'x^y' mph, where 'y' is determined by the efficieny of the wheel bearings. The better the wheel bearings (the lower the friction), the faster the conveyor will have to spin in order to impart enough force to keep the plane stationary. Thus, as the friction of the bearings approaches zero, the speed at which the conveyor must spin needs to approach infinity, in order to keep the plane stationary (or vice-versa).

Now, bearing friction, by design, is approaching zero (we're trying to make them better not worse). Thus it should be easier to get the plane to take off, then to increase the speed of the conveyor to keep the plane stationary. But, for the sake of this argument, we'll assume that the conveyor can spin at any speed, while the plane's thrust is limited to that of a normal plane. Assuming the bearings don't melt, and there are no other complications, the conveyor will eventually spin-up fast enough to overcome the maximum thrust the plane can produce - the result being the plane will remain stationary, or the plane will displace backwards on the conveyor (the plane will move towards the back end of the conveyor).



So, depending on the wording of the riddle, I say the answer could be "yes" or "no". In this case though, I'd say yes.

 

WRONG!
WRONG!
WRONG!

Please read other posts to understand whay this CANNOT happen-

There is no way for the treadmill to "keep the plane stationary"

 

I see what he's saying...if the friction from the turning bearings were to grow to the point that it was equall to the force of thrust (impossible in the real world by my estimation), then the wheels could slow the plane down. I mean...there is a top speed bearings can spin, and if that speed were less than 2X the take off speed, then the plane would be slowed by the wheels and may not take off.

However that situation is more unrealistic than having a treadmill the size of a runway in the first place.


An analogy for this would be skateboarding down hill. In that case, gravity is the propeling force and the wheel bearings are the friction force; when they are equall, you cannot go any faster on a skateboard without being pulled by a motorized vehicle (it happens at ~ 65mph according to the nut ball known as Biker Sherlock).

 

That is my point exactly.

Realistically, the Thrust of the plane "Big N" (Big O for you Big O notation peeps) should always overcome "little n", the amount of force the conveyor can exert on the plane's body (where "little n" is determined by bearing friction).

You have to understand though that, theoretically, unless the wheel bearings are frictionless, the conveyor will exert a lateral force on the plane's body (albeit miniscule), through the plane's wheel bearings.

Yes, this is completely unrealistic, but that is why it is up to the specific wording/ideal of the original question. If the question is trying to be realistic (a real world problem) then it should always be much, much, easier to power the plane to take-off speed, then to power the conveyor to keep the plane from moving.

But, if the question was meant to be a theoretical exercise, and you're assuming the treadmill can spin to any speed at all, at any time, then yes it could be possible to keep the plane from moving. Of course, we have no way of making tires or wheels or wheel bearings that would survive at the speed needed to pull this off - but again, in the case of a theoretical exercise (or trick question) you can assume we do have such things and you should merely look at the forces involved.


As was mentioned earlier, the treadmill does exert a force on the plane, albeit, too miniscule to realistically couter-act the thrust of the plane. But, theoretically, if it can apply any force onto the plane, then there is a rotational speed at which it will cause the plane to not move.


Let's go back to the "skateboard on a treadmill analogy". If you stand on a skateboard that is atop a stopped treadmill, you won't move anywhere (zero relative displacement). Turn on the treadmill and what happens - do you not move at all? If you speed the treadmill up to 100mph, can you just stand on the skateboard (without touching anything) and freewheel in place? No, you can't.

As soon as the treadmill is started, you have to apply some amount of force to remain stationary in relation to the treadmill. Mind you, this force is relatively insignificant (you can just stick your hands on the treadmill's handles), but it is there.

The same thing is true of the plane. The plane will have to exert a force to keep from moving backwards, which is simple for it to do (what's 1 extra pound of thrust when you can produce 20,000). But the faster the conveyor spins, the more thrust the plane will have to produce in order to reach any given speed.


So, in most cases, I'd say the plane will take off. But, theoretically, it is possible for a theoretical "Super Treadmill" to stop a plane from taking off.


P.S. 65mph on a skateboard.... dat's just crazy. :-o

 

Actually wind resistance- (OK so it would be combined with wheel bearing friction) is probably Biker Sherlock's bigger obstacle to going over 65, that and finding pavement steep enough to fully exploit gravity's potential.

Speed skiers have acheived speed of over 150 mph, and their ground friction is very close to nil, since they only have about 10% of their skis in contact with the snow at that speed, but the air keeps them from going any faster.

That's why Harry Eggar is taking his next efforts to the Himalayans, thinner air over 20,000 feet he thinks may allow him to get up to 170mph.

Biker Sherlock is actually a pretty mellow dude, Harry is a genuine nut-case

 

And as far as this "super treadmill" the one in the problem statement is limited to the speed of the plane, you guys are hanging it WAY out there to try and defend your flawed reasoning-

give it up.

 

Geez, seriously. I've said multiple times that it is FAR more realistic for the plane to take off - I agree with you and EQ Tuning on that point.

But as even EQ Tuning pointed out, the treadmill does exert a force on the plane, albeit miniscule. The treadmill will exert a force on the plane proportional to friction of the wheel bearings. With that in mind, it is theoretically possible (though realistically improbable) for the treadmill to counter-act the thrust of a plane, as long as the plane is sitting on the treadmill, and their is friction/traction between the plane's wheels/tires and the treadmill.


Edit: Sorry didn't understand what you were saying at first.
Ok, so if you limit the speed of the treadmill, to the airspeed of the plane (that was situation 1 in my original post), the plane will have no problem in taking off. The treadmill would have to move much faster than the airspeed of the plane (unrealistically faster, I'd assume) to keep the plane from gaining airspeed.

That is why I said it depends on the specific wording of the question. If you look around, sometimes the question (as posed by others) suggests a treadmill that can move infinetly fast, not one that is limited.

 

I don't see why not. If the jets are keeping the airplane at a constant 100mph (not accelerating) and the treadmill is moving 100mph the other direction, the plane will be stationary, have no air-speed which means no lift, and no take off

 

I would have thought that too but he got towed behind a motorbike up to 90+ and didn't slow down very quickly when he let go. You're right though...air friction pretty much takes over at those speeds. But I've never looked at a chart of resistance to rolling vs rotational speed for any type of bearing...the resistance could ramp up really hard at a certain rotational speed.

 

Oh dude...you are way too late...go reread the thread....you're wrong.

 

So this hurts my brain....

what will it take to keep the airplane stationary?

 

Straps to the ground. Or putting on the brakes. Or using a wind tunnel (an 'air treadmill').


Since the engines of a plane push against air and not the ground, a treadmill does nothing to keep it from moving.

 

Now, say you put an airplane (with the jets OFF) on a treadmill moving the airplane backwards going 100mph.

When you turn on the jets, it will move the airplane forward on the treadmill's surface some... and with enough jet-force wouldn't it be able to match the speed of the treadmill?

 

If you put a plane on a treadmill and turned on the treadmill (and left the engines of the plane off), the wheels would start turning under the plane and the planes inertia would keep it standing almost still. It might move a little...but not much.