一道有趣的數學題

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.

在機場中，你想從A點前往B點。（爲了將問題簡化，假設機場是一條線性通道。）一些區域有電動扶梯（雙向的），另外一些區域沒有。你的步行速度恆定爲v，電動扶梯的運行速度爲u，所以在扶梯上，你的實際速度爲v+u。（顯然，你不會搭乘與你前進方向不一致的扶梯。）你的目標是儘量快地從A點到達B點。

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. 假定你須要暫停片刻，好比繫鞋帶。請問你應該在電動扶梯上系，仍是在沒有上電動扶梯時系？假定兩種狀況下，繫鞋帶的時間相同。

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. 假定你有有限數量的多餘能量，用來奔跑。在跑動時，你的速度提升到v'（若是在電動扶梯上，就相應爲v'+u）。請問你應該在電動扶梯上跑，仍是在沒有上電動扶梯時跑？假定兩種狀況下，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. 在狹義相對論的狀況下，上述答案是否發生改變？