I am writing a program where you chose where a particle starts and the particle is moving in a given graph under uniform gravity for a given amount of time. For my setup, this program needs a range of values $x$ to load the graph and calculate its derivatives for particle movement. However I am having problems finding the upper bounds for values $x$ which load the graph.
I assume the answer is some kind of Brachistochrone curve, as it says in Wikipedia:
Brachistochrone curve or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time.
Since in needs the least amount of time to travel to another point, the curve which will travel the furthest in the $x$ direction given some time $t$ is also a Brachistochrone? I found in another math stack answer that the time to travel the whole Brachistochrone is
$T = \sqrt{\frac{A}{g}}\theta_o$, where $A$ is the radius of the rolling circle, $g$ is the gravitational acceleration and $\theta_o$ is the parameter angle for the rolling circle to make one roll with constant speed, which is equal to $2 \pi$. For example, in the horizontal case I can express $A = \frac{T^2g}{\theta_o^2}$ and $x = 2\pi A$, thus $x = \frac{T^2g}{\theta_o}$, but what if the end point is at a lower height? How can this time equation be written if the line slant of the rolling circle with constant speed is other than horizontal? I think in the slanted case the parameter $A$ increases, but then you need to project it back to the x-plane. How can this be done?