20160320, 23:31  #1 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·641 Posts 
The Fastest Path
What is the fastest path from point A to point B?
* Hint 1: It's rarely the shortest path. Don't Google unless you give up. If you do, please do not spoil the puzzle. * Hint 2: Galileo mistakenly thought it was an arc. Last fiddled with by a1call on 20160320 at 23:42 
20160320, 23:52  #2 
If I May
"Chris Halsall"
Sep 2002
Barbados
2·4,643 Posts 

20160320, 23:58  #3  
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·641 Posts 
Quote:
Think GravityAssist (but not of space exploration kind). Last fiddled with by a1call on 20160321 at 00:01 

20160321, 00:27  #4 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3×641 Posts 
Hints:
* It is indeed a curved path but it's not an arc * The path is related to a rolling wheel 
20160321, 00:34  #5 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·641 Posts 

20160321, 00:54  #6 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·641 Posts 
Spoiler alert, I guess it took long enough.
Here are the answers: It's a Cycloyd: [URL]https://en.wikipedia.org/wiki/Cycloid[/URL] [URL]https://en.wikipedia.org/wiki/Brachistochrone_curve[/URL] 
20160321, 03:29  #7 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{2}·1,447 Posts 
Fastest for whom?
If person T travels at near c, and person S remains stationary and observes T, then is the fastest path the same for both T and S? Or would T decide upon a different path than S? Would it depend upon the position of S relative to the travel vector of T, or the positions of A and B relative to T and/or S, or some other combination? 
20160321, 08:27  #8 
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
3×3,371 Posts 
Underspecified problem. You don't put any conditions on the traveller, A, B, any fields existing in their neighbourhood nor the geometry of the latter.
The fastest path for a photon between two points in flat spacetime, one in air and the other in water and where the line AB is not perpendicular to the surface is two straight line segments. That's an extremely good approximation indeed. 
20160322, 02:04  #9 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·641 Posts 
Here is are follow up problems:
* Points AB have an aligned distance of 4176.3 m * Straight line AB is inclined 1Β° relative to horizontal ** Disregarding friction and drag How long would it take a ball to roll from A to B along a straight path with an initial speed of 0 (I know this answer) ** Disregarding friction and drag How long would it take a ball to roll from A to B along a downward cycloid path with an initial speed of 0 (I don't know this answer) This might help: https://en.wikipedia.org/wiki/Tautochrone_curve Last fiddled with by a1call on 20160322 at 02:05 
20160322, 02:37  #10  
"Forget I exist"
Jul 2009
Dumbassville
20B1_{16} Posts 
Quote:
Last fiddled with by science_man_88 on 20160322 at 02:37 

20160322, 03:12  #11  
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·641 Posts 
Quote:
Along a curved rightleft symmetrical path (destination tip trimmed, not withstanding) the ball would accelerate nonlinearly all the way down and decelerate all the way up with the time required to reach the bottom being equal to the time required to reach the top. Speed of light is irrelevant, this is a classical mechanical problem. Last fiddled with by a1call on 20160322 at 03:12 

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