Text 28754, 106 rader
Skriven 2009-03-08 22:31:42 av Michiel van der Vlist (2:280/5555)
  Kommentar till text 28715 av Dale Shipp (1:261/1466.0)
Ärende: Question
================
Hello Dale,

On Saturday March 07 2009 23:11, you wrote to me:

 DS> Isn't it custom when solving these kind of problems to assume
 DS> otherwise  ideal conditions by default? Like it is over here?

 DS> There have been many times in my career when I was asked to solve a
 DS> problem which had been abstracted by someone from a real situation.
 DS> More often than not, although the abstracted problem might have a
 DS> solution, it ended up not fitting reality as much as desired.  That is
 DS> why I always preferred to work with the real problem and raw data.

There is no way one can deal with the "real problem" and the "raw data". Out of
necessity one always has to settle for an approximation. The "real problem" is
infinitely complex and the "raw data" are alwasy incomplete. In practice this
is seldom a probem as most of the time an approximation is good enough.

In the case of the problem I forwarded, one can argue that the the earth is not
a perfcet sphere, a better approximation is an ellisoid. But that is just an
approximation as well. So how far fo you want to go? Foothills? Mole hills?
Sand grains? Individual molucules?  Or would the assumption that he earth is a
perfect sphere be good enough an approximation?

Note that you are already using an approximation when not using the full
decimal expansion of pi....

 DS>> I suspect that if the bird actually follows the true ground, the
 DS>> extra distance might also be close to that 1.57 meters -- but I
 DS>> can easily see that there might be some pathological ground
 DS>> conditions that would change that.

 MVDV>> My first anwser is that it would be a tiny little bit less
 MVDV>> as the 100 km as measured with the rope on the ground no
 MVDV>> longer corresponds to 1/400 of the circumference of the
 MVDV>> earth, but a little bit less.

 MVDV>> But this is a problem of a totally different level, it can
 MVDV>> not be solved with simple geometry.

 DS> Consider the ground looking something like:
 DS> ___-------\            /--------------------
 DS>            \          /
 DS>             \        /
 DS>              \      /
 DS>               \    /
 DS>                \  /
 DS>                 \/

 DS> Assume that the bird is flying left to right.  The bird would follow
 DS> the "level" parts, and begin to dip down into the gully, but would
 DS> curve back up without following at 100 meter parallel to the left wall
 DS> of the gully all the way down.  He would turn back when he intersected
 DS> the 100 meter distance from the right wall.  Hence his path would be
 DS> somewhat shorter than the ground path.

You are making a lot of assumptions.. It would be very stupid bird. The
sensible thing would be not to dip down at all and simply keep the same
altitude when croosing the "gorge".

If the bird has to maitain a constant altitude over ground, it would follow a
line like this:

 ___-------\            /--------------------  Bird
            \          /
             \        /
              \      /
               \    /
                \  /
                 \/
 ___-------\            /--------------------  Ground
            \          /
             \        /
              \      /
               \    /
                \  /
                 \/



 DS> You can imagine that if the ground path gully were very sharp, say 100
 DS> meter deep and 10 meter across, then the bird might hardly dip at all.
 DS> He would fly a bit more than 10 meter over a ground path of slightly
 DS> more than 200 meter.


Yes, but than teh bird would violate the given that it has to maintain a
constant altitude over ground.

 DS> If you can express the ground curve as a mathematical function, then
 DS> it should be possible to define an relationship that gives the
 DS> air function.  Then you could do a line integral along that air
 DS> function. That is conceptually how the general problem might be
 DS> solved, but I would have to dig a lot to come up with the exact way of
 DS> getting the solution.

But it would still be an approximation. If you want to solve the "real problem"
you would have to make the bird follow *every* fluctuation of the ground no
matter how small.



Cheers, Michiel

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