

An airport-inspired puzzle from Terence Tao - nsrivast
http://terrytao.wordpress.com/2008/12/09/an-airport-inspired-puzzle/

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AlexeyMK
1\. its better to rest on the walkway - not seeing any disagreement about this
one.

2\. its better to run off the walkway. Why? Walking normally is more expensive
(in terms of time required to cover N distance) than walking on the walkway.
The question can be rephrased as, all things considered, "would you rather
spend less time on the walkway or the ground?" You would rather spend less
time on the ground, because it takes you more time to cross it.

Naive proof: your normal speed is 1m/s, run speed is 2m/s (can be done for
1s), walkway moves at 1m/s. Track is: 2m of ground, 6m of walkway.

Running on walkway: 2s to cover ground, 1s to cover first 3 meters of walkway
running, 1.5s to cover last three meters = 4.5s. Running on ground: 1s to
cover ground, 3s to cover walkway = 4s.

Look at it this way: the normal (no-run) time is 5s. Running on the ground
chops off an entire second; running off the walkway chops off only half a
second.

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Darmani
For #2, running off the walkway gives a greater boost to speed percentage-
wise; since travel time is inversely proportional to speed, it is thus reduced
greater.

(For positive u and v`>v>1, v`/v<(v`+u)/(v+u))

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cousin_it
The comment by Harald Hanche-Olsen has the best explanation.

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ars
As far as the non-relativistic version of the question, I'm pretty sure it
makes no different if you pause while on or off. Both for question #1 and #2 -
it makes not difference.

Relativistically I'm not sure, but I think it's best to pause while on the
walkway, since while running on the walkway most of your energy goes to
increasing your mass rather than your speed.

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fizx
> As far as the non-relativistic version of the question, I'm pretty sure it
> makes no different if you pause while on or off.

Nope.

~~~
ars
Is this going to become like that thing about airplanes on treadmills that
move backward at the same speed as the plane, where everyone is 100% sure they
have it right?

I see the reasoning that if you walk on the treadmill you enjoy it's speed for
less time - but that's not what you care about. You care about how long it
takes you to get from the start of the treadmill to the end, and if you walk
you get to the end faster.

So if you can walk 100 meters in 1 minute, and that's what you do on the
treadmill, you will have covered an extra 100 meters in 1 minute.

Or you could cover that 100 meters on the ground, and it will also take you 1
minute.

The speed of the conveyor makes no difference - that is totally fixed, the
only difference is do you cover that 100 meters on the conveyor or off of it.

I typed and rejected many examples to give you, but none were intuitive. So if
it's this hard to explain then I'm 100% sure this is going to become like that
airplane on a conveyor belt.

People are going to argue about this forever.

What is it about conveyor belts that makes it hard for people?

~~~
ars
OK, the post below made it easier to me to explain: You can NOT add miles per
hour to miles per hour. So you can not add the speed to walking to the speed
of the belt.

You need to do hours per mile + hours per mile.

So the conveyor takes a certain number of hours for each mile, and you walking
takes a certain number of hours for each mile.

The length of the conveyor is fixed. So the number of hours the conveyor
supplies is fixed - nothing you can do will change that. You can add in your
hours per mile on it or off of it, it doesn't matter - it's exactly the same
in total.

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greendestiny
Well like the airplane on the conveyor belt thing, there really is a right
answer to this and you don't have it. This is a similar explanation to one of
the comments on the blog: imagine one section of ground followed by one
section of conveyor belt. Two people going exactly the same speed walk along
the ground section and one stops just before the conveyor belt and one stops
on the conveyor belt. They both stop for the same time. When they resume
walking the one who stopped on the conveyor belt is ahead. No matter how many
section of belt and ground follow this, the one behind will never catch up.

~~~
ars
No, that's not correct.

You forgot that they both have to get off the conveyor belt at some point.

The guy who got on first will then walk slowly, and the guy who got on second
meanwhile will be both walking and being pulled forward by the conveyor belt,
while the first guy is walking slowly. This will catch him up to the first
guy.

And if you tell me that the first guy is already walking so of course he's
ahead, then you forgot that, for both of them to stop at the same time, and
yet one be on the belt and one on the ground, the first guy must have had a
head start.

Better have two belts, one placed one meter ahead of the other. So, one guy
will be walking for that meter, while the other will be riding the belt and
walking, and both will reach the end at exactly the same time.

~~~
greendestiny
It's confusing alright. When the guy in the lead gets off the conveyor belt
the distance between them will close. But the second guy won't catch up. The
distance between them will vary according to speed but the time between them
will stay the same.

~~~
ars
I concede. You are right, it's better to pause on the belt. I thought about it
all night, using edge cases, like a really slow walker, fast belt, etc.

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markessien
In the first case, you go faster when you pause on the treadmill, and the time
required does not change if you run on or off the treadmill.

Those seem like the intuitive solutions to me.

