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 , and Bob's wall has length .
They both push off of their walls at the same instant. Alfred has a speed of , and Bob has a speed of . They both stop when they reach their opposing walls.
The closest distance they ever get to each other can be written as , where and are positive integers and is square free.
Find .
Ignore gravity and air resistance.