# 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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