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 CarFirst 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 AeroplaneOkay 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:
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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".
Conclusions1. 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 itAnother 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.
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Lol... took me a few hours just to write this one post! Need to rest
This post has been edited by DeniseLau: Jul 27 2009, 05:15 PM