邏輯推理十日談（8）陶哲軒的一維航空站

在機場的一個航站樓，你想從 A 點前往 B 點，為簡化問題，假設該航站樓是一條直線。

該航站樓的一些區域有雙向的電動扶梯，另一些區域則沒有，顯然你不會搭乘與你前進方向相反的扶梯。

你的步行速度恒定為 v，電動扶梯的運行速度恒定為 u，因此在扶梯上，你的實際速度為 v+u。

你的目標是盡可能快地從 A 點到達 B 點。

問題 1: 假定你需要暫停片刻來係鞋帶，你應該在電動扶梯上係，還是在沒有上電動扶梯時係？假定兩種情況下，係鞋帶的時間相同。

問題 2: 假定你用有限的多餘能量用來奔跑。在跑動時，你的速度提高到 v'；相應的，在電動扶梯上跑，速度為 v'+u。請問你應該在電動扶梯上跑，還是在沒有上電動扶梯時跑？假定兩種情況下，你可供奔跑的能量相同。

 

這是陶哲軒在他博客裏出的另外一道題。原題是英文，有三個問題。前麵的中文隻翻譯了第一、二部分，同學們不要想得太複雜了。把英文全題留在下麵，有心的同學可以試試。

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.)