The Airplane on a Conveyor Belt Conundrum

Today I will address a question/meme that has been around for decades, and resurfaces every now and then to create controversy.

​So the question/meme is: if an airplane is on top of a conveyor belt as wide and long as a runway, with such belt being designed to match the speed of the wheels of the airplane moving in the opposite direction, will it be able to take off?
 
This question has generated multiple debates in the communities of both science and laypeople for many years. The question even inspired an attempt to test it in the television program series Mythbusters where its hosts, Jamie Hyneman and Adam Savage, found that the airplane was able to take off from a tarp that was being pulled by a vehicle in the opposite direction.
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However, many people claimed that this was a bogus test because the vehicle pulling the tarp did not precisely match the speed of the wheels of the airplane moving in the opposite direction.
 
As to the original question, initially I thought the airplane would not be able to take off. I reasoned that if the conveyor belt can exactly match the speed of the wheels, the airplane would not be able to move forward to generate the wind speed necessary for the air flowing around the wings to generate lift. However, I realized that I was thinking about the wheels of an airplane much in the same way I think about the wheels of a car, and this is a mistake.
 
Let’s compare a car vs. an airplane on a conveyor belt that moves in the direction opposite to the movement of the car/airplane matching the speed of the wheels. In order to advance, the car needs friction between the tires and the ground (traction). A car on a very slippery surface will just spin its wheels hopelessly. So by moving in the opposite direction, you can imagine that the conveyor belt essentially nullifies traction for the car. In the case of the airplane, friction between the wheels and the ground is not needed for the airplane to advance. This is why airplanes on skis can take off on runaways made out of ice and snow. The motor of the car pushes against the ground via the wheels. Its forward movement depends on the wheels and their friction with the ground. On the other hand, the engines of the airplane push against the surrounding air. Its forward movement does not depend on the wheels. The airplane’s wheels are passive – they spin freely. The conveyor belt will not slow down the plane.
 
Although this made sense to me, I still had problems visualizing the situation, because in my mind every turn of the wheel is matched by the conveyor belt moving in the opposite direction, so how can the airplane possibly move forward?
 
Then I realized that the conveyor belt will also make the wheels of the airplane spin due to the friction between the wheels and the belt. In which direction does the conveyor belt make the airplane’s wheels spin? If you look at the airplane from the side, say with the nose facing your left and the tail your right (wheels down, of course), the airplane will be trying to move from right to left, the conveyor belt surface will be moving from left to right, and the wheels will be spinning…counterclockwise. In other words, in the direction of the airplane’s movement! But here is the problem: when the airplane moves forward, it will also make the wheels spin in that direction. Therefore, the speed of the wheels will be a result of that caused by the airplane plus that caused by the conveyor belt. This creates a problem in the basic premise of the problem.
 
For example, if the airplane accelerates, making the wheels spin forward at say 100 revolutions per minute (rpm), and the conveyor belt accelerates in the opposite direction to match the spin of the wheels at 100 rpm, then the combined (total) forward spin of the wheels will be 200 rpm (100 + 100). And if the conveyor belt accelerates to match that total speed (200 rpm), then the final speed of the wheels will be 300 rpm (200 + 100). So we have an impossibility. Even if the conveyor belt accelerates to infinity, it will never be able to match the total speed of the wheels because it will always add an extra amount of spin to them! Of course, if the conveyor belt moves instead in the direction of the airplane’s movement, which will make the wheels spin backwards (against the movement of the plane), it will be able to counter the spin of the wheels, which will remain stationary. But then the conveyor belt will just keep accelerating and dragging the airplane along until it takes off.
 
Finally, if you are curious about applying my reasoning above to the case of the car, consider that the car’s wheels are not freely moving. Only the motor of the car can move the wheels. The conveyor belt moving in the opposite direction will not make the car’s wheels spin forward. It will not add an additional spin to the wheels. So in the case of the car, the conveyor belt can match the speed of the wheels. Of course, if you put the car in neutral (disengaging the wheels from the motor) and attach a jet engine to its roof, you will end up with a situation similar to that of the airplane.
 
Although thinking this way made sense to me, I went online and found that communities of scientists had tackled this question before and conducted computer simulations. The long and short of it is that, yes, the airplane indeed will take off. Nevertheless, there would be insurmountable problems in trying to test the question in a real-world situation where a conveyor belt tries to exactly match the speed of the wheels. The conveyor belt and the wheels of the airplane would accelerate to speeds so great that it would destroy them, and the conveyor belt would move so fast that it would begin to generate a wind current against the airplane that could provide a certain amount of lift.
 
This concludes my foray into the Airplane on a Conveyor Belt conundrum. What do you think?

The drawing of the plane on the conveyor belt belongs to the author and can only be used with permission.