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.

[SPOILER]* Hint 2: Galileo mistakenly thought it was an arc.[/SPOILER]

[QUOTE=a1call;429702]What is the fastest path from point A to point B?[/QUOTE]

It will be a function of what transportation infrastructure exists between point A and point B.

"Beam me up, Scotty."

[QUOTE=chalsall;429703]It will be a function of what transportation infrastructure exists between point A and point B.

"Beam me up, Scotty."[/QUOTE]
Well, whatever The transportation infrastructure is, it would be faster if it followed this type of a path rather than any other.:smile:
Think Gravity-Assist (but not of space exploration kind).

Hints:
* It is indeed a curved path but it's not an arc
* The path is related to a rolling wheel

Hint:
It's not any of these:

[url]https://en.wikipedia.org/wiki/Epicycloid#/media/File:EpitrochoidOn3-generation.gif[/url]

[url]https://en.wikipedia.org/wiki/Hypocycloid#/media/File:Deltoid2.gif[/url]

[url]https://en.wikipedia.org/wiki/Involute#/media/File:Animated_involute_of_circle.gif[/url]

Spoiler alert, I guess it took long enough.

Here are the answers:

[SPOILER]It's a Cycloyd:
[URL]https://en.wikipedia.org/wiki/Cycloid[/URL]

[URL]https://en.wikipedia.org/wiki/Brachistochrone_curve[/URL][/SPOILER]

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?

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.

Here is are follow up problems:
* Points AB have an aligned distance of 4176.3 m
* Straight line AB is inclined 1° 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:
[URL]https://en.wikipedia.org/wiki/Tautochrone_curve[/URL]

[QUOTE=a1call;429757]Here is are follow up problems:
* Points AB have an aligned distance of 4176.3 m
* Straight line AB is inclined 1° 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:
[URL]https://en.wikipedia.org/wiki/Tautochrone_curve[/URL][/QUOTE]

would they not matter if it's accelerating the full time or if it has a top speed lower than the speed of light or not etc.

[QUOTE=science_man_88;429758]would they not matter if it's accelerating the full time or if it has a top speed lower than the speed of light or not etc.[/QUOTE]
Along a straight downhill path the ball will accelerate constantly (linearly) as a function of g and the angle of decline.
Along a curved right-left symmetrical path ([B]destination tip trimmed, not withstanding[/B]) 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.