Big brachistochrone
in [Math]
|
Prev: A Local Property of Convex Sets
Next: HISTORIC 12 POINT CHARGE AGAINST THE INCOMPETENCE OF CURRENT MATHEMATICS THEORY
From: gudi on 7 Aug 2010 05:51 From an altitude H = 5000 km above equator a rollercoaster is built in the equatorial plane touching the equator. What should be its shape so that the time of descent is a minimum? (we neglect wind resistance etc., and consider only gravity). For altitudes << H we should have the brachistochrone cycloid as a special case. Regards Narasimham
From: William Elliot on 7 Aug 2010 07:49 On Sat, 7 Aug 2010, gudi wrote: > From an altitude H = 5000 km above equator a rollercoaster is built in > the equatorial plane touching the equator. What should be its shape so > that the time of descent is a minimum? (we neglect wind resistance > etc., and consider only gravity). For altitudes << H we should have > the brachistochrone cycloid as a special case. No need to shape it; just drop it.
From: Thomas Nordhaus on 7 Aug 2010 08:32 William Elliot schrieb: > On Sat, 7 Aug 2010, gudi wrote: > >> From an altitude H = 5000 km above equator a rollercoaster is built in >> the equatorial plane touching the equator. What should be its shape so >> that the time of descent is a minimum? (we neglect wind resistance >> etc., and consider only gravity). For altitudes << H we should have >> the brachistochrone cycloid as a special case. > > No need to shape it; just drop it. Well, well that's just a trivial case ;-) How about a half-pipe brachistochrone going from point A 5000km above the ground to ground-zero at the antipodal point and ending at point A again going once around the equator? -- Thomas Nordhaus
From: spudnik on 7 Aug 2010 15:07 ah, yes; resistanceless!... so, for realism, what'd be the minimum "boost," as the bobsledder approacheth the antipode at sealevel, to get back to the start? I didn't think, though, that the brachistochrone/tautochrone was cycloidal, but that roundtrip makes me wonder. > > just drop it. > > Well, well that's just a trivial case ;-) How about a half-pipe > brachistochrone going from point A 5000km above the ground to > ground-zero at the antipodal point and ending at point A again going > once around the equator? --les ducs d'oil! http://tarpley.net --Light, A History! http://wlym.com
From: James Waldby on 7 Aug 2010 22:37
On Sat, 07 Aug 2010 14:32:31 +0200, Thomas Nordhaus wrote: > William Elliot schrieb: >> On Sat, 7 Aug 2010, gudi wrote: >> >>> From an altitude H = 5000 km above equator a rollercoaster is built in >>> the equatorial plane touching the equator. What should be its shape so >>> that the time of descent is a minimum? (we neglect wind resistance >>> etc., and consider only gravity). For altitudes << H we should have >>> the brachistochrone cycloid as a special case. >> >> No need to shape it; just drop it. > > Well, well that's just a trivial case ;-) How about a half-pipe > brachistochrone going from point A 5000km above the ground to > ground-zero at the antipodal point and ending at point A again going > once around the equator? If earth's rotation is considered too (rather than only gravity), would you shape a half-pipe differently for east-bound travel vs west-bound? <www.paperblog.fr/1888349/espece-de-brachistochrone/> in French seems to be considering half-pipe cases as far as I can tell from google's translation, while problem 7.6 on page 234 of Classical Mechanics by Rana & Joag as shown at long link below asks one to consider the effect of rotational speed on the shape of an underground tunnel for tautochronous motion. It appears that neither of these references contemplate returning to the original point, or limiting the dept of fall (and rise) to 5000km, or direction of travel. <http://books.google.com/books?id=dptKVr-5LJAC&lpg=PA234&ots=AILtY1JJ_D&dq=underground%20tunnel%20%20brachistochrone&pg=PA234#v=onepage&q=underground%20tunnel%20%20brachistochrone&f=false> -- jiw |