Portal Physics

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 \(y=0\), and its \(x\)-coordinate is irrelevant. The orange portal is situated at \(x=0\), and its height (\(y\)-position) varies as described later.

The rules of portal physics are the following:

  • An object enters one portal and emerges from the other
  • Portals do not conserve vector momentum or total energy (kinetic + potential)
  • Portals do conserve the scalar speed of objects that enter / exit them
  • An object entering one portal at its center on a trajectory perpendicular to its axis, will emerge from the other portal in like manner
  • Portals do not impart their own velocity to objects which pass through them
  • Aside from their own strange properties, portals do not otherwise alter the physics of nearby objects and environments
  • It takes zero time to go through a pair of portals

At time \(t = 0\), two things happen simultaneously:

  • A massive ball drops from its initial resting position at height \(h_B\) and falls under the influence of a uniform downward gravitational acceleration \(g\). 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 \(h_P\) at a constant speed \(v_P\).

Consider the \(x\)-coordinate of the ball at the instant at which it intersects the \(x\)-axis after emerging from the orange portal. The value of \(h_B\) which maximizes this quantity can expressed as:

\[{h_{B_\text{max}} = \frac{a}{b} g \left(\frac{h_P}{v_P}\right)^2}.\]

If \(a\) and \(b\) are coprime positive integers, determine the value of \(a+b\).


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