The image above shows two circular planar portals: a blue portal, with its axis oriented in the vertical direction, and an orange portal, with its axis oriented in the horizontal direction. The blue portal is situated at , and its -coordinate is irrelevant. The orange portal is situated at , and its height (-position) varies as described later.
The rules of portal physics are the following:
At time , two things happen simultaneously:
A massive ball drops from its initial resting position at height and falls under the influence of a uniform downward gravitational acceleration . It falls toward the center of the blue portal.
The orange portal (position defined by its center) begins to descend from its initial vertical position at a constant speed .
Consider the -coordinate of the ball at the instant at which it intersects the -axis after emerging from the orange portal. The value of which maximizes this quantity can expressed as:
If and are coprime positive integers, determine the value of .