4 ms·

An airport-inspired puzzle from Terence Tao

28 points by nsrivast 18y ago

deleted 18y ago
[deleted]
ars 18y ago
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.
- fizx 18y ago
  > 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.
  - Sam_Odio 18y ago
    I have no idea if this is right or not, but I just thought I'd mention how I conceptualized the problem: I thought of the distance from point A to B as Y. I thought of walkways as simply devices that reduced Y by n times t. n is the velocity of the walkway and t was the time spent on it. Therefore one would want to maximize t while maintaing the highest possible average walking/running speed. E.g. it might not make sense for me to stand still on the walkway but I'd tie my shoes on it and try to reserve sprinting for regular surfaces.
    - fizx 18y ago
      You know that classic problem: You are going to a city 50 miles away. Would it be faster to travel there and back at 30mph, or at 40mph leaving and 20mph returning. (hint: http://en.wikipedia.org/wiki/Harmonic_mean http://en.wikipedia.org/wiki/Harmonic_mean)
      - ars 18y ago
        If you ever need to solve such a thing just swap miles per hour to hours per mile: So .0333 + .0333 = .0666 (i.e. it would take you .0666 hours for each mile you need to cover). Or .025 + .05 = .075 Clearly the second is better.
        andrew1 18y ago
        You need to invert your answers, the first is then better...
        ars 18y ago
        Oops! Yes, .0666 hours is less time than .075 hours. Thanks.
  - ars 18y ago
    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 18y ago
      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.
      - greendestiny 18y ago
        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.
- Tichy 18y ago
  Suppose there is one conveyor belt that goes all the way from A to B. You could pause the whole time on the belt and arrive at the destination, whereas if you pause off the belt all the time you'll never get there. So pausing has to make a difference ;-)
- ojbyrne 18y ago
  I thought that too until I made up a simple example and worked through the numbers. Lets say you have a 100 ft non-walkway 400 ft walkway you walk at 10 ft/s and the walkway moves at 10ft/s it takes 10 s to tie your shoes. Case 1 tie your shoes 10s walk 100ft 10s walkway 400 ft (400/(10+10)) 20s Total 40s Case 2 Walk 100 ft 10s Tie shoes on walkway 10s (in which time you travel 100ft) Rest of Walkway 300/(10+10) 15s Total 35s
markessien 18y ago
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.
AlexeyMK 18y ago
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.
- deleted 18y ago
  [deleted]
- Darmani 18y ago
  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))
cousin_it 18y ago
The comment by Harald Hanche-Olsen has the best explanation.

28 points by nsrivast 18y ago

deleted 18y ago

[deleted]

ars 18y ago

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.

fizx 18y ago
> 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.
- Sam_Odio 18y ago
  I have no idea if this is right or not, but I just thought I'd mention how I conceptualized the problem: I thought of the distance from point A to B as Y. I thought of walkways as simply devices that reduced Y by n times t. n is the velocity of the walkway and t was the time spent on it. Therefore one would want to maximize t while maintaing the highest possible average walking/running speed. E.g. it might not make sense for me to stand still on the walkway but I'd tie my shoes on it and try to reserve sprinting for regular surfaces.
  - fizx 18y ago
    You know that classic problem: You are going to a city 50 miles away. Would it be faster to travel there and back at 30mph, or at 40mph leaving and 20mph returning. (hint: http://en.wikipedia.org/wiki/Harmonic_mean http://en.wikipedia.org/wiki/Harmonic_mean)
    - ars 18y ago
      If you ever need to solve such a thing just swap miles per hour to hours per mile: So .0333 + .0333 = .0666 (i.e. it would take you .0666 hours for each mile you need to cover). Or .025 + .05 = .075 Clearly the second is better.
      - andrew1 18y ago
        You need to invert your answers, the first is then better...
        ars 18y ago
        Oops! Yes, .0666 hours is less time than .075 hours. Thanks.
- ars 18y ago
  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 18y ago
    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.
    - greendestiny 18y ago
      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.
Tichy 18y ago
Suppose there is one conveyor belt that goes all the way from A to B. You could pause the whole time on the belt and arrive at the destination, whereas if you pause off the belt all the time you'll never get there. So pausing has to make a difference ;-)
ojbyrne 18y ago
I thought that too until I made up a simple example and worked through the numbers. Lets say you have a 100 ft non-walkway 400 ft walkway you walk at 10 ft/s and the walkway moves at 10ft/s it takes 10 s to tie your shoes. Case 1 tie your shoes 10s walk 100ft 10s walkway 400 ft (400/(10+10)) 20s Total 40s Case 2 Walk 100 ft 10s Tie shoes on walkway 10s (in which time you travel 100ft) Rest of Walkway 300/(10+10) 15s Total 35s

markessien 18y ago

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.

AlexeyMK 18y ago

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.

deleted 18y ago
[deleted]
Darmani 18y ago
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))

cousin_it 18y ago

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