<p>The case that you are stuck on is of a perfect world where an airplane would stay stationary relative to the ground, with its wheels spinning freely, even as the treadmill moves underneath it. Now the jet engines turn on, applying forward force to the airplane. The plane moves forward, and the only effect of the treadmill is to make the wheels spin faster. Right? Wrong. Just kidding, I’ll buy it in this case. The distinction, since there is some friction in the wheels, is between the conveyor belt moving at the same rate but in the opposite direction as the plane, VS the conveyor belt moving at such a rate that the plane is stationary. Neither interpretation is an unreasonable brainteaser. That’s what I meant about the question being vague - Ben read it as being the latter. You should lay off him, because Ben is a decent guy, whereas you don’t seem that nice. </p>
<p>Anyway, if the plane’s wheels rotate at twice takeoff speed, its tires will decompose and it will explode in a burning fit of agony.</p>
<p>“The situation is NOTHING like what you describe.”</p>
<p>There is no ambiguity in the problem statement:</p>
<p>An airplane is sitting on an enormous treadmill. As the plane starts its engines, the treadmill runs in the opposite direction at the same speed the plane is moving. Can the plane take off?</p>
<p>The treadmill runs in the opposite direction at the same speed as the plane is moving. Your scenario where the treadmill would have to move at a rate fast enough to keep the plane stationary is not relevant, but more importantly, not even feasible. Bearing friction can be assumed to be independent of speed. Do some research, and you will discover that this is a pretty good assumption. It doesn’t matter if the bearing speed is 100rpm or 10,000 rpm, the frictional force isn’t going to change all that much… certainly not enough to affect the plane’s ability to take off. The plane will always be able to overcome this force and take off.</p>
<p>The problem statement does not say that the treadmill moves as fast as necessary to keep the plane stationary. It says it is moving at a speed as fast as the plane is moving. If the plane is moving 50 mph forward, the treadmill is moving 50 mph backwards, and the tires are spinning at an equivalent rate of 100mph. Simple as that… and as I said to Ben… “READ THE PROBLEM STATEMENT CAREFULLY!” </p>
<p>I am not assuming any perfect world… I am assuming a normal world where tire and bearing frictions are assumed to be reasonable. </p>
<p>Remember that the wheels can spin in any direction necessary due to the relative motion of the airplane and runway, but that the force they exert on the airplane (in this case backwards) is small compared to the thrust of the engine.</p>
<p>And by the way… I’m a real nice guy… I just get a bit sick of reading these nonsense answers … I expected much better from the posters on the MIT and Engineering boards…</p>
<p>The problem says only that the conveyor belt moves at the same speed as the plane. The distinction is between the conveyor belt moving at the same ground speed as the plane relative to the ground, versus the conveyor belt moving at the same speed as the plane relative to the conveyor. In the second case, as soon as the plane starts accelerating, the conveyor will accelerate to whatever speed is necessary (no upper bound on the conveyor’s speed is posted), such that the conveyor’s speed relative to the ground is not less than the plane’s speed relative to the conveyor. This means that the plane will not move forward relative to the ground. The qualification of “perfect world” is related to the fact that in a perfect world with no moment of inertia in the axles, the plane could take off even under the second interpretation.</p>
<p>EDIT: PS - Regarding your displeasure at the responses, even Caltech & MIT students can get tricked by brainteasers, this of course being the purpose of brainteasers. However, compared to other schools, you will find that the MIT students are basing their dissenting opinions on internally consistent arguments, rather than probabilistically equating an analogy, intuition, or single piece of evidence, with the truth - this being the attitude of most of the general public. The MIT student, while possibly stubborn and heavy-handed, will eventually recognize deductively-derived facts when they see them. This is a strong point of our school, rather than a failure.</p>
<p>Do you disagree that another, technically not wrong, reading of “same speed as the plane”, means “same speed as the plane, relative to the treadmill”? This is equivalent to “the treadmill moves at whatever speed necessary to keep the plane steady relative to the ground”. If you disagree with that, I can prove it. While this requires a treadmill of unreasonable power, there is some finite power that would suffice. And the question is whimsical anyway.</p>
<p>Don’t get me wrong - I personally believe that your view of the problem statement is the intended one. But don’t be so hasty - the other version isn’t uninteresting, and it’s the first interpretation that occurs to the majority of people. Again, the reason the question draws so many posts is that this other version seems to qualify as a brainteaser, since its relatively simple solution requires one small insight: that it is motion relative to the air (not motion of the wheels) that creates lift.</p>
<p>I’m just trying to bring this topic to a close.</p>
<p>“Do you disagree that another, technically not wrong, reading of “same speed as the plane”, means “same speed as the plane, relative to the treadmill”? This is equivalent to “the treadmill moves at whatever speed necessary to keep the plane steady relative to the ground”.”</p>
<p>The second reading is not equivalent to your last statement. Let’s say the airplane accelerates to 1 unit (an infinitessimal value), the runway would instantenously accelerate to 1 unit in the opposite direction. At this instant, the acceleration relative to the original reference point is now 2 units. Let’s ignore the absurdity of establishing this accelerating reference frame and all of the nonsense it causes, the airplane is still free to accelerate forward because the treadmill’s action on the airplane is independent of the speed at which it moves. These are free spinning wheels with constant, reasonable values of friction. At the next instant, the plane accelerates one more unit, and … now the relative motion of the airplane with the original point on the treadmill is now 4 units … but the forces on the plane haven’t changed. Spinning wheels will exert a constant force on the airplane. The airplane will still accelerate to a speed relative to the ground. </p>
<p>All of these scenarios of wind tunnels and infinite speeds and all of the other absurd statements are absolutely ridiculous and, and even if considered (considered: interpreated as the progression towards a limit, not at infinity itself), do not affect the answer.</p>
<p>By the way, I tried to, and thought the topic was closed last night with Ben’s last statement… but I guess that was not to be…</p>
<p>Edited to add: by the way… I was in no way implying that the above analysis dealing with the relative accelerations and non-inertial reference frames is accurate. I was just showing that the forces on the airplane itself (with viewed from an inertial reference frame) do not change regardless of the nonsense that your second interpretation has on the problem. Once again… let’s all go back to this problem and solve it the right way… Newtonian physics within an inertial reference frame… it’s beautiful… All of the nonsense; however “internally consistent” it is, is just garbage. Start from the basics… then innovate…</p>
<p>“Do you disagree that another, technically not wrong, reading of “same speed as the plane”, means “same speed as the plane, relative to the treadmill”? This is equivalent to “the treadmill moves at whatever speed necessary to keep the plane steady relative to the ground”. If you disagree with that, I can prove it.”</p>
<p>Those statements are not equivalent. The plane and the treadmill surface will move at the same speed in opposite directions. But even so, the plane still moves forward because it has wheels.</p>
<p>Yeah, I know I sound like a jerk. I don’t mind people being fooled by brainteasers or even being wrong - I’m frequently fooled by straightforward problems. I have a problem with people who think they’re right when they’re not. I like people like Lowofo who keep an open mind. Here’s his explanation:</p>
<p>"Yes, you’re right. So work done by static friction from treadmill is converted into KE rotational and KE translational, on the wheel. Goddammit. So, I have been assuming this would be completely transformed into KEtranslational. If that were so, the plane would not take off.</p>
<p>You assume, and correctly so, that it will transform into some mixture of KE rotational and KE translational (like you said, dependent upon bearing efficiency). As such, the plane will eventually take off, but much only after much greater time. Under these circumstances, the treadmill would have to out-accelerate the potential acceleration of the thrust of the aircraft in order to keep it from moving forward (relative to something stationary)."</p>
<p>“Anyway, if the plane’s wheels rotate at twice takeoff speed, its tires will decompose and it will explode in a burning fit of agony.”</p>
<p>Fo shizzle?</p>
<p>I imagine the wheels can spin twice as fast and still maintain an efficiency of over 0%. Planes are pretty safe, and I doubt the wheels would explode at twice the speed required for a takeoff with perfect conditions (windless, perfect track, perfect wheels, etc).</p>
<p>No, that is your reading of the problem (the speed of the treadmill relative to the ground is the same as the speed of the plane relative to the ground). The reading I suggested is that the speed of the treadmill relative to the ground is the same as the speed of the plane… relative to the treadmill. Now, as the plane accelerates, the treadmill accelerates as neccessary so that the speed of the plane relative to the treadmill is the same as the speed of the treadmill relative to the ground, which means that the plane isn’t moving.</p>
<p>As I said, the first interpretation is what was intended, but the second is also interesting - at least to me. It is good to keep an open mind, and not be dogmatic at all.</p>
<p>“I imagine the wheels can spin twice as fast and still maintain an efficiency of over 0%. Planes are pretty safe, and I doubt the wheels would explode at twice the speed required for a takeoff with perfect conditions (windless, perfect track, perfect wheels, etc).”</p>
<p>The wheels burning up is the reason that the Concorde crashed. Because it was moving so quickly, the tire punctured on takeoff when it taxi-ed across a small scrap of metal that was on runway. (However, it is not a certainty that wheels explode just because of the speed)</p>
<p>You are stupid. Not for being wrong, but because you’re trying to twist the problem so your answer is right.</p>
<p>You can’t define the treadmill’s speed based on its speed. You have a circular defintion that still wouldn’t affect the outcome (takeoff).</p>
<p>Let’s say the plane is moving at 1 mph relative to the air/ground/observer. The difference is velocity is 1 mph so the treadmill moves a 1 mph in the opposite direction. Now the difference is 2 mph, so it instantly accelerates to 2 mph. Now it’s 3 so it accelerates to 3. And so on until it’s moving at an infinite speed and the plane is still moving at 1 mph. The plane can continue to accelerate and take off.</p>
<p>Anyway, the harmless riddle states they move in opposite directions. That implies a uniform frame of reference.</p>
<p>Dogmatic? How was he dogmatic? If anything, you were. So not only is your knowledge of physics and mathematics lacking… so is your understanding of vocabulary. </p>
<p>I am relatively new to this board, and know nothing of you. But my view of the quality of students at MIT has been severely damaged by reading the nonsense in this thread. While I realize not all of the posters are current students at MIT, it nevertheless damaging to your and your classmates’ credibility.</p>
<p>I disagree. Looking at a problem in a factually incorrect way is one of the fundamental principles of lateral thinking.</p>
<p>The thread closed with Ben’s last post. It opened again with this:
“God, some of you guys are idiots. Ben Golub, I used to respect you. I don’t anymore (not just because of this).”</p>
<p>I cannot disregard this kind of a statement.</p>
<p>That statement was 100% factually correct. Some people posting on this thread, such as you and me, are idiots. I once respected Ben Golub. Then I got to know him better. I lost respect for him, but not because of this thread.</p>
<p>Looking at a problem in a factually correct way is one of the fundamental principles of thinking, in my opinion.</p>
<p>@Sky, he was dogmatic because he refuted me despite me not saying anything that was technically wrong. Furthermore, you are not in a position to categorically decide that my knowledge of mathematics and physics are lacking. I never found any of the correct solutions distasteful, what I objected to was the use of the descriptions “stupid” and “idiot”, applied to some very intelligent people. There is no cause for this “failure of vocabulary”.</p>
<p>Here’s an example of you being technically wrong:</p>
<p>“The case that you are stuck on is of a perfect world where an airplane would stay stationary relative to the ground, with its wheels spinning freely, even as the treadmill moves underneath it. Now the jet engines turn on, applying forward force to the airplane. The plane moves forward, and the only effect of the treadmill is to make the wheels spin faster. Right? Wrong.”</p>
<p>"Do you disagree that another, technically not wrong, reading of “same speed as the plane”, means “same speed as the plane, relative to the treadmill”? This is equivalent to “the treadmill moves at whatever speed necessary to keep the plane steady relative to the ground”</p>
<p>Slorg, the next sentence in the first passage is “Just kidding…”. And that wasn’t an edit, either. The second passage is not wrong, although I admit to intentionally twisting the wording so that you would be likely to disagree.</p>
<p>You might find this passage from the wikipedia interesting:
*Critical thinking is primarily concerned with judging the truth value of statements and seeking error. Lateral Thinking is more concerned with the movement value of statements and ideas, how to move from them to other statements and ideas.</p>
<p>For example the statement “cars should have square wheels” when considered with critical thinking would be evaluated as a poor suggestion, as there are many engineering problems with square wheels. The Lateral Thinking treatment of the same statement would be to see where it leads. Square wheels would produce predictable bumps. If bumps can be predicted then suspension can be designed to compensate. Another way to predict bumps would be a laser or sonar on the front of the car examining the road surface ahead. This leads to the idea of active suspension with a sensor on the car that has normal wheels. The initial statement has been left behind.*</p>