Crossing Paths - Revisited

Classical Mechanics Level pending

Two astronauts, Alfred and Bob, are at the midpoints of adjacent walls in a rectangular room in a space station. Alfred's wall has length \(3L\), and Bob's wall has length \(2L\). At the same instant, they both push off their walls and begin floating across the room. Alfred has speed \(6v\), and Bob has speed \(13v\). As they float, they both get really tired and fall asleep, so that when they reach their opposite walls, they bounce back at the same angle.

They keep bouncing forth until they finally collide. The time it takes for them to collide can be expressed as \(\frac{aL}{bv}\), where \(a\) and \(b\) are positive, coprime integers.

Find \(a+b\).

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