After a good night’s sleep, I do get it – but not in the way you think.
What I get is that the way this question / problem is phrased, you can’t really answer it. You can’t answer it without making some basic assumptions about speed. This is the reason so many people can disagree so vehemently about the answer and still all be right. Cecil sez:
"Your version straightforwardly states that the conveyor moves backward at the same rate that the plane moves forward. If the plane’s forward speed is 100 miles per hour, the conveyor rolls 100 MPH backward, and the wheels rotate at 200 MPH. "
But it’s actually not that straightforward. Speed is always relative to something else. The question involves two different frames of reference – the conveyor belt and the world outside the conveyor belt. What you use to determine the measurement of speed determines how you answer this question in your own mind.
If you decide that all speed will be measured relative to the world outside the treadmill, the plane very clearly takes off, and the question becomes kind of silly. Once you measure the speed of the plane relative to the outside world, the treadmill is completely irrelevant – it can be going 10,000 times as fast as the plane, but the plane will still take off. The plane need only go faster than take-off speed relative to the world (and air) outside the treadmill. Cecil is dead-on in his assessment of the question when interpreted this way.
Cecil notes an alternative interpretation:
“However, some versions put matters this way: ‘The conveyer belt is designed to exactly match the speed of the wheels at any given time, moving in the opposite direction of rotation.’”
He says this is a paradox, but he’s not quite accurate in his assessment of the situation. This language does not automatically imply a paradox. He says:
"This language leads to a paradox: If the plane moves forward at 5 MPH, then its wheels will do likewise, and the treadmill will go 5 MPH backward. But if the treadmill is going 5 MPH backward, then the wheels are really turning 10 MPH forward. But if the wheels are going 10 MPH forward . . . "
The key to the “paradox” here is a constantly shifting frame of reference. Look at how speed is measured here:
“If the plane moves forward at 5 MPH” --> This is relative to the outside
“the treadmill will go 5 MPH backward” --> This is also relative to the outside
“then the wheels are really turning 10 MPH forward” --> This is relative to the treadmill
Cecil sees a paradox because he jumps between two very distinct frames of reference – the world on the treadmill and the world off – while treating them as if they’re both part of the same frame. There need not be a paradox here – the speed of the plane and treadmill relative to the outside world will always be different from the speed of the plane and treadmill relative to each other, as long as the treadmill is moving relative to the outside world. To stay consistent, it should be worded as follows:
If the plane moves forward at 5 mph relative to the treadmill, and the treadmill moves to 5 mph backward relative to the outside, then the wheels are really turning at 5 mph relative to the treadmill, and the plane is moving 0 mph relative to the outside. If the plane accelerates to 6 mph relative to the treadmill, then the treadmill will accelerate to 6 mph relative to the outside world, and the plane remains moving 0 mph relative to the outside world.
The reason, then, that I argued that the plane does not take off is becuase in my mind, I constructed the situation as such: the conveyor belt’s control system tracks the plane’s speed relative to the conveyor belt and increases the the conveyor belt’s speed relative to the outside world accordingly. Until the plane takes flight, it’s resting on the conveyor belt, and its forward thrust is pushing it and its tires along that conveyor belt. If, as the plane moves forward on the conveyor belt, you increase the speed with which the conveyor belt feeds beneath it, the plane doesn’t move relative to the world outside the conveyor belt, and it doesn’t move relative to the air around it. It never takes off.
You might counter this by saying, “Yes, but because the engine is pushing against the air, it doesn’t matter how fast the plane is going relative to the conveyor belt. The plane moves forwards, and its wheels spin really fast.” Ah, but here’s the rub: as the plane’s wheels spin faster, its speed relative to the conveyor belt increases. The tangle here is that people correlate thrust and speed as if it were absolute: x amount of thrust = x amount of speed. Speed vis a vis the conveyor belt probably requires less thrust than speed vis-a-vis the outside world. If 10% thrust pushed us to 100 mph on a stationary runway, 10% thrust might push us to 1000 mph on our moving conveyor belt.
If this doesn’t make any sense, imagine this: you are a man standing on that runway conveyor belt. When the belt is turned off, the plane’s wheels are still. The plane turns on its engines, and the belt begins to move. If you don’t move, you see the plane rolling down the conveyor belt, moving away from you at a given speed (say, 5 mph). You then jump off the conveyor belt, but before you do, you drop a penny where you were standing. Now, you’re standing next the conveyor belt. Your worldview has changed. The plane, which appeared to be moving away from you when you were on the belt, is now standing still. The penny, which was motionless at your feet, is now moving away from you at 5 mph. You have changed your frame of reference.
If, at any point, the speed of the plane relative to the conveyor is faster than the speed of the conveyor belt relative to the outside world, then the plane moves forward relative to the outside world – but at the cost of violating the terms of the problem. The problem states that the conveyor belt’s speed matches the speed of the plane at all times. If you interpret the conveyor belt’s speed to mean “the speed of the conveyor belt relative to an observor standing next to the belt” and the plane’s speed as “the speed of the plane relative to the conveyor belt”, then ouila: the plane never moves relative to the outside observor. If it never moves, it never attains an airspeed greater than zero. The plane never takes off.
So really, at the end of the day, the thought experiment is interesting, but not for the reasons we might think. It’s interesting because it’s a great demonstration of the principle of relativity and its impact on interpreting the world. I, for one, am done trying to argue it – there is no way to ever win the argument one way or another.
