What is the answer to this physics riddle?

[fresh_42]

Then how come brachistochrone exists, if time on all curves is identical?

But it isn't one. The more on kinetic energy gained downwards is lost again upwards. (But I haven't done the math, maybe it doesn't cancel out, however I think it does. I desperately try to remember the analogous experiment in a math museum I once visited, I thought they were equally fast at the finish ...)

 

[PeroK]

But it isn't one. The more on kinetic energy gained downwards is lost again upwards. (But I haven't done the math, maybe it doesn't cancel out, however I think it does.)

The final speed must be the same (without friction) but the ball will take different times for different paths. For example, you can have a path with a time to the lowest point as large as you wish!