A Response to Terence Tao’s “An airport-inspired puzzle”
In Terence Tao’s latest post he poses three questions. Here are my solutions.
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, one would only take those walkway 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.
- 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.
- 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.
- Do the answers to the above questions change if one takes into account the effects of special relativity? (This is of course an academic question rather than a practical one.)
Source: Terence Tao, http://terrytao.wordpress.com/2008/12/09/an-airport-inspired-puzzle/
After just thinking about it without any mathematics I was not to sure so I used a mathematical approach. The first thing I did was to draw a diagram,
Admittedly, I did simplify the problem in my diagram, however I am confident that this will not affect the final answer. (How do I prove this? I don’t know.) Along with this diagram I also had to define some things in terms of variables.
As shown in the diagram, A is the starting point, B is the ending point, C is an arbitrary point in between which separates the escalator section from the non-escalator sections.
t = time it takes to tie shoe lace
v = walking speed
u = escalator speed
= time it takes to get from A to C
= time it takes to get from C to B
= time it takes to get from A to B
We also know, .
Now lets consider two scenarios. Scenario A, the person ties their shoe lace in the non-escalator section. Scenario B, the person ties their shoe lace in the escalator section.
I shall now make some reasonable assumptions (also formalising things a bit more),
All variables are real, and we shall assume that the person has time to tie their shoe lace while on the escalator. I.e.
I shall denote to be from scenario A and to be from scenario B. Now to see which is larger or we can examine the sign of . If it is positive then , if it is negative then .
By some algebra and as , . Hence . Therefore it would be more efficient pause for a moment while on an escalator walkway.\
I will take a similar approach for Q2, examining the two cases and then comparing the resultant time.
(I’ll re-edit the post when I get around to working out the solution)