What would happen if a car were put on a treadmill duplicating the scenario from the plane on a treadmill problem:
“A car is standing on a runway that can move (some sort of band conveyer). The car moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the car’s speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the car move?”
Remember, the conveyor matches the speed of the car, not its wheels.
Since the speed of the conveyor belt directly effects the car, it could easily be made to keep the car motionless in relation to the rest of the world.
The key point with a plane is that the wheels of the plane don’t provide any forward movement. A car’s wheels do.
Such a system already exists. You see them often in emissions testing stations. It’s called a “set of rollers” and the magic control mechanism is called “friction”. And no, the car never moves, not when the system is operating correctly.
If the car doesn’t move, the rollers can’t either. The premise is “The car moves in one direction, while the conveyor moves in the opposite direction”.
Just like the plane version, you aren’t clearly stating the problem. Does the car move in one direction relative to the treadmill surface, or relative to the ground?
A car on a treadmill is basically the same problem as a person on a treadmill; both get motive force by pushing back against the conveyor belt. And we all know what happens when you walk on a treadmill.
I’m assuming that’s what the plane on a treadmill problem was describing also. At least that’s the way it’s worded.
“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.”
If the plane doesn’t move (relative to the ground), the conveyor doesn’t move either.
So, let’s consider such a conveyor belt with a car. It’s specially made to move in the opposite direction of the car (relative to the bystanders on the ground) with perfect accuracy and instantly. You step hard on the gas. What happens?
In that case, the car’s wheels are spinning twice as fast as normal. So the car’s speedometer reads 120, the car’s going 60 one way (relative to the ground), and the belt’s going 60 the other way.
By the definition you just gave, the wheels spin, the belt compensates, and the car remains stationary with respect to the rest of the world. What is hard about this?
And it is exactly the same as putting the car on a roller-type dynometer. Just think of the roller as a very small conveyor belt.
That’s not at all equivalent to the situation posited in the OP: the front wheels are completely constrained. The car’s not going anywhere even if the roller seize up at full power.
I’m not sure why you’re having trouble. Here is a video of a motorcycle on a dynamometer. Clearly, the motorcycle does not move forward. The roller, which as has been said is sort of a ‘very short treadmill’ does indeed move – as it would have to.
