陶哲轩的数学题

wayne

Suppose you are trying to get from one end A of a terminal to the other end B. (For simplicity, assume the terminal is a one-dimensional line segment.) Some portions of the terminal have moving walkways (in both directions); other portions do not. Your walking speed is a constant v, but while on a walkway, it is boosted by the speed u of the walkway for a net speed of v+u. (Obviously, given a choice, one would only take those walkways that are going in the direction one wishes to travel in.) Your objective is to get from A to B in the shortest time possible. 在机场中，你想从A点前往B点。（为了将问题简化，假设机场是一条线性通道。）一些区域有电动扶梯（双向的），另一些区域没有。你的步行速度恒定为v，电动扶梯的运行速度为u，因此在扶梯上，你的实际速度为v+u。（显然，你不会搭乘与你前进方向不一致的扶梯。）你的目标是尽可能快地从A点到达B点。 1. Suppose you need to pause for some period of time, say to tie your shoe. Is it more efficient to do so while on a walkway, or off the walkway? Assume the period of time required is the same in both cases. 1. 假定你需要暂停片刻，比如系鞋带。请问你应该在电动扶梯上系，还是在没有上电动扶梯时系？假定两种情况下，系鞋带的时间相同。 2. Suppose you have a limited amount of energy available to run and increase your speed to a higher quantity v' (or v'+u, if you are on a walkway). Is it more efficient to run while on a walkway, or off the walkway? Assume that the energy expenditure is the same in both cases. 2. 假定你有有限数量的多余能量，用来奔跑。在跑动时，你的速度提高到v'（如果在电动扶梯上，就相应为v'+u）。请问你应该在电动扶梯上跑，还是在没有上电动扶梯时跑？假定两种情况下，你可供奔跑的能量相同。 3. Do the answers to the above questions change if one takes into account the various effects of special relativity? (This is of course an academic question rather than a practical one. But presumably it should be the time in the airport frame that one wants to minimise, not time in one’s personal frame.) 3. 在狭义相对论的情况下，上述答案是否发生改变？

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