The case of the plane and conveyor belt

Following on from the post at kottke.org.

“A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?”

Let’s try and clear this up once and for all.

The plane will take off. And here’s why:

Let’s first make some assumptions.

1. The plane is on a runway (albeit one that can move), therefore the landing gear is extended.
2. The pilot will follow normal takeoff procedure, therefore the brakes on the wheels will be released, allowing the wheels to spin freely.
3. Takeoff speed for this plane is 300mph.
4. There is no wind.

“This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction).”

From this we can assertain that if the plane is stationary, so is the runway conveyer. The runway will only move if the plane does. This means that at no point can the runway go fast enough to keep the plane still, as the wheels will prevent it having any affect on the plane. Also, if the plane slows down, so does the runway. This also means that the speed of the plane relative to the runway will always be twice the speed of the plane relative to the surrounding air and nearby buildings.

The other thing to remember is that the planes propulsion acts on the surrounding air in order to propel the plane forward.

As the plane speeds up, so does the runway until the plane hits takeoff speed (300mph). At the same time the runway also hits takeoff speed (not that it takes off!) and at this point the planes speed relative to the runway is 2 times the takeoff speed, which would be 600mph and the wheels of the plane are spinning accordingly. The planes speed relative to the surrounding air (and the stationary control tower) is still 300mph.

At this point the plane takes off.

PS. Comments are off for a reason. If you want to discuss this, use the comments at kottke.org.

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