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        riddles >> medium >> Night Journey
        
(Message started by: Barukh on Sep 19th, 2007, 7:37am)
Title: Night Journey
Post by Barukh on Sep 19th, 2007, 7:37am
Two cities A and B are at the same longitude on Earth.

On a specific day in city A, the Sun sets at 10pm and rises next day at 4am. On the same day, the night duration at B is 1 hour longer.

Train leaves B at the sunset, moves to A on the shortest route and constant speed and arrives at A at the sunrise.

Is this information sufficient to estimate approximately the darkest time train’s passengers will experience on their journey?
Title: Re: Night Journey
Post by towr on Sep 19th, 2007, 8:37am
Why wouldn't it just be 1 am?
The middle of the night is the same everywhere along a meridian. The only remaining thing that could influence it (aside from the moon, stars, clouds, cities, etc) is how dark it gets at night; but I wouldn't know where to begin to guess how that depends on the length of the night.
Sure when the nights so short the sun hardly sets, the middle of the night could be bright as day. But after a point, I'd think it's pretty much all the same with the sun well on the other side of the earth.
Title: Re: Night Journey
Post by Barukh on Sep 19th, 2007, 10:17am
1am is not a bad approximation, but can we improve on it?

Afterall, there are additional data. And your only remaining thing does have some influence...
Title: Re: Night Journey
Post by hiyathere on Sep 19th, 2007, 10:46am
It all depends on when the sun set at city B.

The darkest time the train experiences is most likely the mean between the darkest time at city A and city B. So I would guess somewhere between 12:30 - 1:30, depending on when the sun set and rose at city B.
Title: Re: Night Journey
Post by hiyathere on Sep 19th, 2007, 10:48am

on 09/19/07 at 08:37:09, towr wrote:
Why wouldn't it just be 1 am?
The middle of the night is the same everywhere along a meridian. The only remaining thing that could influence it (aside from the moon, stars, clouds, cities, etc) is how dark it gets at night; but I wouldn't know where to begin to guess how that depends on the length of the night.
Sure when the nights so short the sun hardly sets, the middle of the night could be bright as day. But after a point, I'd think it's pretty much all the same with the sun well on the other side of the earth.


Towr, the two cities needs not to be on the same longitude for them to have different day and night lengths.
Title: Re: Night Journey
Post by towr on Sep 19th, 2007, 10:55am

on 09/19/07 at 10:48:03, hiyathere wrote:
Towr, the two cities needs not to be on the same longitude for them to have different day and night lengths.
Indeed not, they would have to be on different latitudes. That they are on the same longitude is given in the premise of the puzzle.
Title: Re: Night Journey
Post by hiyathere on Sep 19th, 2007, 10:58am
Gotcha.
Title: Re: Night Journey
Post by Grimbal on Sep 20th, 2007, 4:52am
Can we assume there is no daylight saving time change?  If not I agree with towr.  The darkest time is between sunset and sunrise.
Or do you take into accout the time the sun crosses the horizon?
Title: Re: Night Journey
Post by Barukh on Sep 20th, 2007, 7:24am

on 09/20/07 at 04:52:07, Grimbal wrote:
Can we assume there is no daylight saving time change?

Yes.

Quote:
Or do you take into accout the time the sun crosses the horizon?

No.
Title: Re: Night Journey
Post by Joe Fendel on Sep 20th, 2007, 8:00pm

on 09/19/07 at 10:46:31, hiyathere wrote:
The darkest time the train experiences is most likely the mean between the darkest time at city A and city B. So I would guess somewhere between 12:30 - 1:30, depending on when the sun set and rose at city B.


In fact it should be a bit before 1am, since after 1am the train is both moving to a "lighter" latitude and the sun is "rising".

My instinct would be to create parametric equations for both the darkest spot on the earth, D (presumably the exact opposite side from the sun), as it moves through the night, and also equations for our friend, and use these to map the distance between them as a function of time, which can be minimized using calculus.  

But if you think I'm gonna actually try it...   ;)
Title: Re: Night Journey
Post by Grimbal on Sep 22nd, 2007, 6:34am
I see my mistake.  1 am is the darkest time for anybody who doesn't move along the meridian.  But the train moves.
Title: Re: Night Journey
Post by GhengisKhan on Sep 22nd, 2007, 8:25am
Yes, the night at B should be from 9:30 until 4:30
The train will depart at 9:30pm and arrive at 4:00am.  I am assuming that the darkest time is a single moment in time rather than a sustained maximum darkness.  (My guess is that in reality maximum darkness may be achieved about 2 hours after sunset and end about 2 hours before sunrise.)  In any case, I was thinking the answer would be 12:45am  But now I've decided that it is darkest at 1am regardless of where on the longitudinal line the train is located.  I think the answer is 1am.
Title: Re: Night Journey
Post by Barukh on Sep 22nd, 2007, 9:44am

on 09/22/07 at 08:25:56, GhengisKhan wrote:
In any case, I was thinking the answer would be 12:45am

I agree with you!  :)
Title: Re: Night Journey
Post by GhengisKhan on Sep 22nd, 2007, 12:49pm
Wait Barukh, you quoted where I said 12:45, but my final conclusion is 1am.  Are you saying you think it's 12:45?  
Title: Re: Night Journey
Post by Barukh on Sep 24th, 2007, 4:53am

on 09/22/07 at 12:49:18, GhengisKhan wrote:
Wait Barukh, you quoted where I said 12:45, but my final conclusion is 1am.  Are you saying you think it's 12:45?  

Yes. If we plot the "darkness" at different locations as functions of time, we get something similar to the attached picture. The curves at A and B are symmetric about time 1am. Assuming (I admit this assumption is not based on solid arguments, but afterall we want something approximate)  that the train's curve is also symmetric, its axis should be at 12:45am.
Title: Re: Night Journey
Post by towr on Sep 24th, 2007, 7:43am
That assumes the maximum darkness at B is quite a bit darker than the maximum darkness at A. The smaller the difference in maximum darkness, the closer it'll be to 1 am; and I'm not seeing a reason to assume a linear connection between minimum darkness at one pole, and maximum darkness somewhere between the equator and the opposite pole.
Title: Re: Night Journey
Post by Barukh on Sep 24th, 2007, 9:16am

on 09/24/07 at 07:43:50, towr wrote:
That assumes the maximum darkness at B is quite a bit darker than the maximum darkness at A. The smaller the difference in maximum darkness, the closer it'll be to 1 am;

I don’t think so. If the difference in maximum darkness at A and B gets smaller, so will be the difference at the train and both A and B, but the 15 min shift will persist.

This 15 min difference follows from the 30 min difference in sunset (and sunrise) times at A and B.


Quote:
I'm not seeing a reason to assume a linear connection between minimum darkness at one pole, and maximum darkness somewhere between the equator and the opposite pole.

Neither see I. Still, I believe it’s a good approximation for the conditions given.
Title: Re: Night Journey
Post by towr on Sep 24th, 2007, 9:23am

on 09/24/07 at 09:16:08, Barukh wrote:
I don’t think so. If the difference in maximum darkness at A and B gets smaller, so will be the difference at the train and both A and B, but the 15 min shift will persist.
If the curves touch (i.e. the curve for B is only wider than A's), then the maximum for the train must be at 1 am, I can't see where else it could be.
And it would be odd if it's discontinuous, i.e always 15 minutes shifted unless the maximum darkness becomes the same when it's suddenly 0.
Title: Re: Night Journey
Post by Barukh on Sep 24th, 2007, 9:31am

on 09/24/07 at 09:23:06, towr wrote:
If the curves touch (i.e. the curve for B is only wider than A's),

The curves can't touch: more night hours at B means the sum is deeper under the horizon, therefore the night is darker.
Title: Re: Night Journey
Post by Eigenray on Sep 24th, 2007, 9:57am
I think it should be just 1am.  Let us consider the Earth as rotating about the Z-axis with its center at the origin, with the Sun fixed (in the XZ plane, making an angle http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif with the Z-axis, and off in the negative X direction).  Then a point which is fixed wrt Earth, at a latitude of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, will be at some point P=(coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif, coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif, sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif) in space, where http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif/(2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif)*(24 hours) is the time since 1am.  Thus P is in sunrise/sunset when

coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif = tanhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif/tanhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif,

assuming my diagram is correct.  As the train moves, we have http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif goes from -7http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/24 to 6http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/24, and http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif goes from http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifB to http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifA, both linear wrt time.

Now, how do we measure the darkness at some point?  If we measure it as time to nearest sunset/sunrise, then

dark(P) ~ arccos(tanhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif/tanhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif) - |http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif|,

but this is maximized when http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif=0, which is at 9.5 + (7/13)(16-9.5) = 1 am.

On the other hand, if we measure darkness as

dark(P) ~ coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif - tanhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif/tanhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif,

then numerically I find darkness maximized somewhere between 46.776 (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif=23.5o) and 46.803 (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gif0) minutes past midnight, i.e., approximately 12:45.

What is the correct way to measure darkness?  Distance to the terminator?
Title: Re: Night Journey
Post by towr on Sep 24th, 2007, 10:15am

on 09/24/07 at 09:31:24, Barukh wrote:
The curves can't touch: more night hours at B means the sum is deeper under the horizon, therefore the night is darker.
That doesn't change the argument. How much darker it is is at issue; if the difference in maximum darkness goes to zero, the timeshift goes to zero; or it is discontinuous.

Also, if the latitude with the darkest night is between A and B, I think you could have a time shift in the other direction.
Title: Re: Night Journey
Post by denis on Sep 24th, 2007, 11:06am
Take the case when the sun is highest in the sky (at zenith). Does this not occur at the same time of day for all points along the same longitude of the same hemisphere? I believe it does.

By the same token, as Towr mentioned, the darkest part of the night, when the sun is furthest below the horizon, would occur as well at the same time along the points of the same longitude in the same hemisphere.

If this assumption is correct, and if the train goes along the longitude (the shortest surface route), then it should get its darkest part of the night also at 1am. Note the fact that A,B and the train all get the darkest at the same time does not mean that A, B and the train reach the same "absolute" darkness index.

Did I miss something here?

Perhaps the shortest geodesic route is not the same as the shortest surface route and perhaps therein lies the difference? But that is beyond my scope.
Title: Re: Night Journey
Post by Joe Fendel on Sep 24th, 2007, 11:45am

on 09/24/07 at 11:06:39, denis wrote:
By the same token, as Towr mentioned, the darkest part of the night, when the sun is furthest below the horizon, would occur as well at the same time along the points of the same longitude in the same hemisphere.

If this assumption is correct, and if the train goes along the longitude (the shortest surface route), then it should get its darkest part of the night also at 1am.

Did I miss something here?


This puzzle is clever for precisely this reason.  At all points along the journey, the darkest moment occurs at 1:00am.  Yet the darkest moment for the train occurs at 12:45am.  How is this apparent contradiction possible?

Barukh's graph illustrates the reasoning very nicely (which isn't surprising, given that this is his puzzle!), but here's another, more extreme variant which may make things easier to understand:

Suppose instead our train were moving along a longitude line from a spot near the equator to a spot near the pole, leaving at sunrise and arriving at mid-day.  At let's imagine that this journey takes place on the equinox, to simplify things.  Then when is the sun at its highest point in the sky, from the perspective of the train?

Certainly at any single point along the route, the answer is mid-day.  Yet at mid-day, when the train reaches its destination, the sun is basically just at the horizon, exactly where it was when the train left.  But during the journey, the sun was in the sky the whole time, and clearly got well above the horizon.  So from the perspective of the train, the answer is sometime in the middle of the journey, at a point marked X.  The "contradiction" is resolved when we realize that after the train passes X, although the sun will continue to get higher at point X throughout the day, it will start to "set" from the perspective of the pole-bound train.
Title: Re: Night Journey
Post by denis on Sep 24th, 2007, 12:07pm
Yes I see how the paradox is resolved now... Good explanation Joe. The train is traveling towards a point where maximum darkness is less even though all along the longitude maximum darkness is reached at the same time. So it reaches its own absolute darkness a little earlier than at each stationary point along the route

Good puzzle.
Title: Re: Night Journey
Post by Barukh on Sep 25th, 2007, 6:47am
Excellent demonstration, Joe!  :D

I am glad you liked the puzzle.
Title: Re: Night Journey
Post by Eigenray on Sep 28th, 2007, 8:47am
To be precise, the angle http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/delta.gif between the point P=(coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif, coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifsinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif, sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif) and the terminator (-sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif, 0, coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/perp.gif is

sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/delta.gif= sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gifcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif- coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gifsinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif,

and this is maximized somewhere between 12:46:14 (solstice) and 12:46:45 (near equinox), rounding to the nearest second.
Title: Re: Night Journey
Post by GhengisKhan on Sep 29th, 2007, 9:46pm
Thank you Eigenray.  After watching all that debate, I became convinced that Joe's example proved it wasn't 1am, but that Joe's example didn't prove 12:45am was correct.  I was thinking that it had to be somewhere in between, but I don't know how to do that kind of math.  In my mind, I guessed it would be closer to 12:59:58

Also, It seems like there might be other possibilities.  Like; depending on time of year, the journey could be closer to the pole and that would change the correct answer.

I imagine it as a sort of doppler effect.
If the time is 12:45, whatever point the train is located will be continuing to darken for 15 more minutes.  The train's movement could cause a slight shift, but anything more than seconds seems like the train's speed would have to increase greatly, or that it would need to be closer to the pole.

Of course it is only a theoretical problem not accounting for moonlight, starlight, and the true behavior of light in our atmosphere.  I've enjoyed participating for the first time here, and am humbled by some of you.  I hope I'm not too anoying.


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