逻辑推理十日谈（8）陶哲轩的一维航空站

在机场的一个航站楼，你想从 A 点前往 B 点，为简化问题，假设该航站楼是一条直线。

该航站楼的一些区域有双向的电动扶梯，另一些区域则没有，显然你不会搭乘与你前进方向相反的扶梯。

你的步行速度恒定为 v，电动扶梯的运行速度恒定为 u，因此在扶梯上，你的实际速度为 v+u。

你的目标是尽可能快地从 A 点到达 B 点。

问题 1: 假定你需要暂停片刻来系鞋带，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。

问题 2: 假定你用有限的多余能量用来奔跑。在跑动时，你的速度提高到 v'；相应的，在电动扶梯上跑，速度为 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。

 

这是陶哲轩在他博客里出的另外一道题。原题是英文，有三个问题。前面的中文只翻译了第一、二部分，同学们不要想得太复杂了。把英文全题留在下面，有心的同学可以试试。

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.

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.

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.

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.)