Soupspoon wrote:Makes me wonder about the problem I considered when I was maybe five or six and playing Crazy Golf... Given an initial impetus of a ball rolling over an (arbitrarily profiled, even multi-humped but smooth) hill, with zero rolling/air resistance or other dampening forms of friction, up to a certain initial velocity there's a point up the rise (or maybe one of several rises) subsequent to which the ball will reach. This will never be a/the local maxima (because if it rolls onto it, it will continue to roll over it) or horizontal inflection. It will also never be a downwards-inclined slope beyond a maxima, or indeed any subsequent slope not higher than the prior maximal point. (There's also a velocity beyond which the ball departs the undulating surface, which we either decide is an upper limit of its own or else treat the last concurrently rolled-over point as the solution to "where the ball gets to".)
However simple and pure the hill curve is, the derivative y'=g(x') - distance for attained for each velocity - derived from the y=f(x) - the height of the slope at any given distance (horizontal or slope-hugging, to taste) is discontinuous and not even like a tangent's vertical asymptote to infinity but a distinct up-to-but-not-including limit.
I'm unconvinced there's no nice hill where the ball ends up (in a lim t→∞) stationary. And I mean rolling towards a point where dy/dx=0, so y=-e^-x is disqualified.
If you've already disproven its existence with the second paragraph, can you please elaborate? My geometry/calculus understanding is too rudimentary to even imagine how to properly map time to ball position on a constant slope. (that is, by a derivation that works for general curves, instead of just using y=v₀t-½at²) And if not, can you prove/disprove its existence?
Oh right, it's pretty trivial that a ball with exactly enough speed to reach the top of any continuous hill (at height h) will take forever to get there. It can't move horizontally at y=h (any other direction is fine, however the hill is perfectly flat at h, so it can't be at the top) and it can never stop moving because y<h except at the top means it has kinetic energy (and it can't turn around because that requires a non-zero slope at h).
I'm still very much interested in how you construct the motion equations or parametrization for the ball.