The Mythbusters on the Discovery Channel did an episode in 2008 regarding an airplane on a treadmill/conveyor belt. The myth as stated on the official discussion board of the show was:
"A plane is standing on a runway that can move (some sort of band conveyor). The plane moves in one direction, while the conveyor moves in the opposite direction. This conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction). Can the plane take off?"
I enjoy watching the show, but sometimes I wish they’d add an engineer Mythbuster to draw some free body diagrams or inform them that there is better testing equipment available then making their own contraptions for testing strengths of materials. This episode sure could have used an engineer or physics professional.
I was thinking about the episode on my way down to Southern California this weekend and figured they didn’t test the myth, but they created their own myth. In the episode they took a long length of fabric, put an ultra light airplane on it. A truck was attached to the fabric and pulled it at the takeoff speed of the plane.
You’ll notice their goal was to have the fabric match the takeoff speed of the airplane, which they did. To satisfy the myth the fabric would have to match the speed of the plane, which it didn’t. When the Mythbusters tested the myth the plane accelerated to match the speed of the fabric and then kept accelerating until it reached the takeoff speed.
In exploring the myth they took a RC car and put it on at treadmill to show matching the speed of the car and treadmill would have the car be stationary on the treadmill.
If you take a look at the forces involved we’ll notice that the force from the engine is transferred to the tires. The main concept to look at is the what friction provides in the relationship. Friction is what allows the tire to rotate on the treadmill and match the rotation of the treadmill. If the tire and treadmill didn’t have any friction the tire would spin and not go anywhere, regardless of the force the engine of the car applied.
In this case I’m looking at the front tires, since my car is front wheel drive. The force of the tire matches the force of treadmill keeping the car stationary. Since the front tire is the driving force we need to look at the rear tire to relate it to the tire of a plane. The both the rear tire of the car and the tires on the planes see only external forces applied to it. The speed of the tire will match the speed of the treadmill because of the force of friction. If there was no friction the tire would stay stationary. If I increased the coefficient of friction of the rear tire it would make the tire go slower, causing the engine to have to apply more force to the front tires in order to stay stationary on the treadmill.
If we take a look at the free body diagram of the plane’s tire we’ll see that the tire sees the force of the treadmill and the propulsion forces of the plane. The force of the treadmill is transferred to the tire through friction. If the friction force is greater than the force of propulsion the plane will move backwards. If the friction force is less than the force of propulsion the plane will move forward.
The tire will go as fast as necessary to reflect the forces applied to it. As the mythbusters provided in their video if the speed of the treadmill matches the takeoff speed of the plane it will take off. That is because the propulsion of the plane is greater than the friction force applied to the plane’s tires.
In order to get the plane to stay on the treadmill we have to either make the plane weigh more to make the force of friction match the plane’s propulsion (which wouldn’t be preferred since the takeoff speed of the plane would be increased or the area on the wings would need to be increased to provide additional lift to keep the takeoff speed the same) or increase the friction coefficient by changing the treadmill/tire surfaces. The easiest way to test it would be to increase the coefficient of friction of the tire/treadmill interaction. Unfortunately I don’t have a plane or treadmill to test out the theory. Maybe the Mythbusters will try it out in a follow up episode.