Crossing Paths in a Space Station

Two astronauts, Alfred and Bob, are positioned 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 \(2L\).

They both push off of their walls at the same instant. Alfred has a speed of \(3v\), and Bob has a speed of \(5v\). They both stop when they reach their opposing walls.

The closest distance they ever get to each other can be written as \(\frac{L}{a\sqrt{b}}\), where \(a\) and \(b\) are positive integers and \(b\) is square free.

Find \(a+b\).

Ignore gravity and air resistance.


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