Hmm... read a couple of explanations here, doesn't explain the fundamentals.

I’ll try to explain here why an aeroplane can take off from a coveyor belt and do a comparison with cars on a coveyor belt. Here are the steps we'll take towards understanding the concept:

Step 1 = How a car moves forward ("gains forward momentum" in TS's words) on a normal road
Step 2 = How an aeroplane moves forward ("gains forward momentum" in TS's words) on a normal runway
Step 3 = How a car reacts on a conveyor belt
Step 4 = How an aeroplane reacts on a conveyor belt

But before we get into all that, let me first describe how a car works and how an aeroplane works in a normal scenario... meaning like normal lah, without coveyor belt and all that.

The Car
First let’s take a look at the car. A car is able to move forward because it has an engine which is connected to a gear box which in turn is connected to the wheels. When you pay money to Petronas, they’ll give you petrol which when poured into your engine will cause the engine to rotate and spin the gears in the gearbox which will then cause your wheels to spin.



The rotation from the wheels then cause the tires to push backwards against the road surface and move the car forwards, thus making you spend more money and making Petronas richer. But also making your car go forwards at the same time.

So at this point, I'd like to introduce a new term just to make the explanations easier. The term is "Medium-of-motion". For a car, the petrol gives power to the engine which gives power to the gearbox which then gives power to the wheels and then finally the wheels push against the "medium-of-motion" which is the road surface.

One important thing to note here in this scenario, is that the "medium-of-motion" (ie. the road surface) is not moving and is fixed.



The Aeroplane
Okay now let's look at how the aeroplane moves forward. Forget about the whole thing about "aerofoil" and "lift" and all that stuff, we don't really care about that, let's just look at how an aeroplane moves forward on the ground at an airport runway.

Click to see how aeroplane moves on ground:


Lol... but seriously though, an aeroplane moves forward both on ground and on air by using massive engines that suck in air from the front and blows it out the back. If you're wondering how something can move forwards by blowing air out the back, try inflating a balloon and then letting it go with the end untied. It flies forwards doesn't it? Same thing with the aeroplane, just that instead of keeping a tank full of air to blow backwards, it simply sucks in the air from the front and blows it out backwards.



At this point, let's stop for a while and look at the "medium-of-motion" for aeroplanes. Again, Petronas's petrol gives power to the engines which give power to the propeller which then sucks in the air from the front and blows it out the back really fast causing the high speed air to push against the stationary air behind the propeller causing the plane to move forward. So the "medium-of-motion" here in this case is the air.

So now we begin to see that the wheels of the airplane serves no purpose at all, except to hold the aeroplane up. That's why aeroplanes can also use skis or floaters instead of wheels.


Introducing the conveyor belt!
Okay, now that we understand the fundamental difference between how a car moves forward and an air-plane moves forward, let's bring in the problem... the conveyor belt!

Before looking further into this problem, let's make a few assumptions:
1. The conveyor belt is very long, longer than the distance required for an air-plane to take off.
2. The conveyor belt is going 'backward' (i.e. opposing the direction we want to go).
3. The conveyor belt, aeroplane and car speeds can be controlled *very* accurately.
4. The car and the aeroplane has a GPS receiver that reads out the coordinates of it's current location.
5. The aeroplane has an additional speedometer that is connected to the aeroplane's wheels

So now, we put the car on the conveyor belt. Let's see what happens.

So we put the car on the belt, turn on the conveyor belt at 35KM/h. At the same time, we on the engine and start driving forwards at car-speedometer's 35KM/h in the opposite direction. Guess what? The car doesn't move forward or backward! It stays in the same spot although the wheels are spinning furiously. The GPS reading too doesn't change and it shows the same coordinates and a speed of 0KM/h!

But why?

Well, when you burn your petrol to make the wheels spin, normally the wheels will push against the road surface to go forwards. But this time, since the conveyor belt too is going backwards, the pushing action of the wheels is negated. It's like driving on a normal road, but instead of you going forwards, the whole world is going backwards!



Because of this, all the force that the wheels is trying to apply to the ground to make the car go forwards is instead being used to stop the car from going backwards. In other words the forward momentum that the car is generating is being cancelled by the backward momentum of the conveyor belt. What does this mean? Well let's jam on the breaks!!

Suddenly when you jam on the breaks, you realise that you're hurtling backwards at 35KM/h. Your GPS reader shows your previously stationary coordinates now moving backwards and your speed as 35KM/h!! So you press the fuel again all the way until your speedometer shows 35KM/h. Look now at your GPS reader and you see that your coordinates are stationary again and you're going at 0KM/h.

Therefore, when your car says you're moving at 35KM/h, this is the speed of your car relative to the "medium-of-motion (ie. conveyor belt), it's not the true speed of the car. The true speed - relative to the Earth - of the car is given by the GPS receiver, which is 0KM/h.

This happens to the car, because the "medium-of-motion" for the car is the road surface and this scenario, the conveyor belt simulates the road surface moving in the opposite direction.

Now let's do it again, this time with an air-plane! Since air-planes move faster, we're going to set the conveyor to 100km/h! Also remember that we have connected an extra speedometer to the aeroplane's wheels, we'll be monitoring this as we do the experiment.

Okay, first lets turn on the conveyor belt at 100KM/h. Immediately we see that the aeroplane is hurtling back at 100KM/h. Let's check the meters:

GPS: 100KM/h -- going backwards
Aeroplane's speed indicator: 0KM/h
Speedometer connected to the wheels: 0KM/h

Okay now we turn on the engines, and start pressing the fuel just a little. Look at the meters!

GPS: 1KM/h -- going forwards
Aeroplane's speed indicator: 1KM/h
Speedometer connected to the wheels: 101KM/h

Notice how fast the speedometer readings jumped? Why?

To understand why we notice such a huge difference in the speedometer reading, we need to look again at the "medium-of-motion" idea. Remember that the speedometer is connected to the tires of the aeroplane, so it measures the velocity of the aeroplane relative to the conveyor  belt.

When the propellers of the aeroplane starts spinning, they begin to suck the air from the front and blow it backwards, this creates a forward momentum for the air-plane.

Now this is where the magic happens:

Because the "medium-of-motion" that the aeroplane uses is air and not the road surface, the conveyor belt cannot cancel off the forward momentum of the aeroplane! So almost immediately when you press the fuel, the air-plane will start moving forwards relative to the Earth! That's why your GPS will start showing you that you're moving forwards at a certain velocity.

Now let's go full throttle!!

GPS: 100KM/h
Aeroplane's speed indicator: 100KM/h
Speedometer connected to the wheels: 200KM/h

So now we see that as we really start burning the petrol and making Petronas richer, we continue moving forwards relative to the Earth really fast!

And since we are now moving forwards relative to the Earth really fast, it's exactly like taking off from a normal runway!

But wait! Let's try something extra here. Let's maintain the aeroplane throttle but now pump up the conveyor belt speed to 300KM/h!!! Look at the meters!! 

GPS: 100KM/h
Aeroplane's speed indicator: 100KM/h
Speedometer connected to the wheels: 400KM/h

Now the speed of the aeroplane - relative to the conveyor belt - increases to 400KM/h but the aeroplane's true speed relative to the Earth is unaffected!

This is because the conveyor belt - which simulates the road as "medium-of-motion" - does not affect the aeroplane which uses the air as "medium-of-motion".

Conclusions
1. Yes, an aeroplane can gain forward momentum while on a conveyor belt regardless of the speed of the conveyor belt.
2. Yes, an aeroplane can take off from a conveyor belt, because of #1.
3. No, taking off on a conveyor belt will not result in the aeroplane needing a shorter runway, it will still need the same length of runway to take off.
4. It's almost impossible to make an aeroplane stay stationary with it's engines on on a conveyor belt, unless you put the whole experiment inside a wind-tunnel. A wind-tunnel is to an air-plane what a conveyor belt is to a car.

Alternative way to look at it
Another way to look at this is through seeing humans running on a treadmill.

In the first scenario depicted below, the man has to run faster than the treadmill if he wants to reach the wall. His legs are representative of the engines, gearbox and wheels of the car.



In the second scenario depicted below, the man needs just to pull himself with the string to reach the wall. The string is representative of the air as the medium-of-motion while the hands are representative of the engines of an air-plane and it's propellers.



Hope this helps you understand the concepts.

Don't focus on wheel part or thrust from engine. Whether the plane can fly or not, all depended on the wing speed on its both wings only.

Thrust of engine main objective is to let the plane having enough wing speed which created the differentiate pressure between upper and lower part of the wing eventually lifting force.

Don't focus on wheel part or thrust from engine. Whether the plane can fly or not, all depended on the wing speed on its both wings only.

Thrust of engine main objective is to let the plane having enough wing speed which created the differentiate pressure between upper and lower part of the wing eventually lifting force.

It does matter in this because in this case it is an additional constraint that may affect the airspeed over the wings.

Whether the plane can fly or not, the airspeed between the wing dictate all.

If the conveyor belt is moving backwards, resulted net zero movement of the plane, i.e. no airspeed between the wing on the plane the the airplane won't fly, as simply as that.

The forwards movement of the plane by engine thrust is to create enough airspeed between the wing. Engine thrust is not the direct reason plane is flying. That's why F1 car doesn't fly even though they are running at 300km/h as almost similar to take-off speed of airplanes because they don't have wing. 
Engine thrust -> plane move doesn't move forwards due to reverse underneath conveyou belt -> no enought airspeed, plane won't fly

Engine thrust -> plane move forwards -> enough airspeed between wing -> plane can fly

No engine thrust -> plane doesn't move forwards -> no airspeed between wing -> no fly

No engine thrust -> plane doesn't move forward -> but if there is natural airspeed blowing in front on the plane (air speed more than 300km/h which doesn't happen naturally -> plane can fly. Just like you see some bird just spread their wing, then can fly/float in the air already, which need to flapping.

So all are about the airspeed blowing/flowing on the wing.

The question is that can a plane take off on a conveyor belt moving at the same speed in opposite direction?

Actually you're agreeing with me, except that you're missing out on a minor point

But if the conveyor belt moves backwards and still the plane moves, will the plane fly, or no? And that's why many of us have been stressing on the fact that it's the wheels also matter.

We are all agreed that the critical condition is that the wing achieves sufficient airspeed, but whether it achieves sufficient airspeed depends on the wheels, thrust, and the curveball in this question, the conveyor belt, which is why they all also count.

What I mean airspeed, is not the speed of the plane moving forwards, what I meant or want to point outed previously is the air flowing speed between the wing.

Again, just measure the airflow between wing, if the airflow speed is enough that produce the lifting force, then it can fly, as simple as that. Don't need to care much about wheel spinning or converyor belt.

If the plane speed/velocity/accelaration is as same as conveyor belt moving backwards, which resulted zero movement or not enough airspeed between the wing i.e. no enough lifting force, the plane won't fly.

If the conveyor belt moving backwards not fast enough than the plane moving forwards, i.e plane is still moving forwards despite backwards moving conveyor, when it reach enough lifting force produce by air flow then it can fly. But in this case it mean more thrust needed i.e. engine is working 2x harder ro propel the plane forwards.

Concentrate on lifting force that enable plane to fly by then you have cleared mind. Lifting force come from differentiate pressure between upper and lower part of the wing which created from air flowing.

The plane still can fly if the plane has net forwards moving motion which enable it to produce the lifting force let say in ordinary situation300km/h airspeed needed, while in a backward moving conveyor belt let say at 100km/h, it means that plane need to move 400km/h to have a net moving motion of 300km/h to enable it to fly, you still see a moving plane at 300km/h net forward in a backward moving conveyor in this case.

Don't focus on wheel part or thrust from engine.

This is highly controversial debate that's still raging I think. And I've edited some details to make it clear of the scenario. It is reflected in bold text

The question is that can a plane take off on a conveyor belt moving at the same speed in opposite direction?

In simple form - [Can a plane take off while being stationary? Having it's forward rolling momentum elliminated?[/b]

Yes, except that how would you know if there is sufficient lifting force for a plane on a conveyor belt, with the air being still relative to say a tree on the ground, and the conveyor belt moving backwards [relative to the ground] and the plane moving forwards [relative to the conveyor belt - we do not know whether it moves forward/backward/being stationary wrt that tree on the ground, which is the problem which is being asked in the first post]

It isn't cleared up at all if you concentrate on lifting force alone, because without considering the wheels and the conveyor belt you would not be able to conclude whether the wings do move relative to that tree on the ground (and hence the air around the wings itself)

The only way your solution would work is if an experiment is done, which I don't think any of us can.

[Note: before we all get confused again, here's my issue with cherroy's initial statement:

I'm simply saying that it is very important in this case, because it ultimately decides what the airspeed over the wings are, and thus whether the critical condition for lift (which is what cherroy has been stressing on) is achiveable]

Which is why in an earlier post, I've stated that the initial question itself is flawed - it's just that Seagates never changed it

Also, we're only playing hypothetical positions here, so we just assume that the plane will fly once it reaches some critical velocity required for flight.

In that case, there's no point arguing/debating since we DON'T really know the exact answer.
That is because the question itself is flawed.