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Discussion Starter #1 (Edited)
If there are 2 points A (0,0) and B (0,1) horizontally separated by 1 meter, assuming a sliding bead has an initial velocity of 1 meter per second, what is the shortest duration path between the 2 points and what is the travel time, assuming frictionless sliding motion, and the only force acting on the bead is uniform gravity near the Earth's surface? A straight line path is the shortest distance and takes exactly 1 second to transit, but is not the shortest duration path.

I've been trying to solve this problem for a few days but have been having a bit of trouble finding anything better than a partial, approximate solution, which is this (this isn't for homework):

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For reference:
pg 116
http://classicalmechanics.lib.rochester ... f/vpcm.pdf

If there was no initial velocity, the quickest path would be a "cycloid," but when there is initial velocity the path is a partial cycloid, with the upper portion that gives the needed initial velocity "chopped off the top" so to speak. Despite knowing this I'm having a lot of trouble finding a formula that gives the exact solution and travel time. The above solution was found through trial and error and should be within about a percent or so of the exact answer.

A descent of 0.5102m gives a velocity of 1m/s on the Earth's surface...

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It depends very much on wether the bead/ball is a full BEV or just a Plug In, and wether the charging curve is efficient from a medium SOC. o_O
 

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Discussion Starter #5
This shows a different version of my approximate solution, including the portion of the curve above the X axis that accelerates the bead to 1m/s at point (0,0):

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parametric graphing

^The whole curve is a perfect cycloid, but the size and positioning on the axes isn't exact, and we are only interested in the portion below the X axis with initial 1m/s.
 

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I think we need a map of the gravitational field before we can answer that. I also need to dust off my Einstein Field Equation slide rule...
 

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Discussion Starter #12
If you want to one-up Isaac Newton, find the curve where the initial velocity vector is horizontal... If that’s the case, the bead can’t change it’s velocity vector instantly to follow the cycloid, so the 1st part of the curve should be downwardly concave.
 

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Yes i get that, or rather i don't, its way beyond me i am afraid. Still can't fathom our why the acceleration under gravity isn't all lost on the ascent making it nett slower. Sadly my physics education ended about the time we studied friction compensation runways using ticker tape machines!
 

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Probably the testosterone curve, as in; "my 2nd order partial differential equation is bigger than yours!".

Unfortunately, you have put insufficient information into your question, so it doesn't make sense. You mention 'initial velocity' but velocity is a vector and you've only defined a speed. In other words, two projectiles are not in the same state, if one is on the flat it is not experiencing any gravitational acceleration while the one on the curve is. If you are then comparing different curves, then are you ALSO defining the starting gradient, because that will change the fastest curve?

In other words, 'faster than 'what' '? If you start a projectile off at an angle then 'the fastest curve' from that point will be the cycloid that has 'that' gradient at that point, and passes through your end point of choice, symmetrically (assuming no losses i.e. same height). There will only be one.

Or do you want it to come to a halt at the final point? Then that is a different 'set' of curves again, whilst one on the flat would still have energy, so again 'compared with what'? A curve with a ball moving at the end of it at the same initial speed?
 

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Discussion Starter #18
Johann Bernoulli wrote the problem as:

“Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.”
 

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Johann Bernoulli wrote the problem as:

“Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.”
You're asking a different question.
 
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