q1:let the total time taken be T if you dont tie your shoeson your optimal route.let t be the time you take to tie your shoes.

1:if you tie your shoes on ground you take T+t total time.

2:if you tie your shoes on walkway you take T+tv/(u+v).since v/(u+v)<1,therefore T+tv/(u+v)<T+t.therefore tie your shoes on walkway ]]>

so it’s an issue of maximising te spent on the platforms therefore your downtime tying your shoelace should be wgile on the platform while spending up your energy to run should happen off the platform since running on the platform would get you less time od the speed boost.

I did happen to have the same thoughts on my last trip to england as well and did in fact postpone tying my shoelace till i got on a platform :) I ran without regard for optimising my route though as they were shouting my name for i was late..

regarding Q3 i’m not familiar with relativity so i cant give an opinion but i’d like to consider how the problem would change given a stamina regeneration factor(although i guess the answer there would be run off platforms, rest on platforms unless youre full rested while on thr platform. also run with a platform within sight to regen stamina on it :p)

]]>Let G be the length of the ground, W the length of the walkways in total.

Let T be the amount of time you can run at the higher speed, v’. Let T_G be the portion of that time running on the ground.

Assume that you spend T time running at speed v’ and don’t complete the journey, otherwise obviously you just run at full speed the whole time.

Then the time to complete the journey is simply:

T + (G-v’ * T_G)/v + w – (v’+u)(T-T_G)/(v+u). This is a linear function of T_G and the coefficient of T_G is:

(v’+u)/(v+u) – v’/v which can easily be verified to be negative because v’ > v. Therefore to minimize the time, T_G should be as large as possible (i.e you should run on the ground as long as you can).

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