I don't think this was linear programming. Basically, all you need to know is that with all the velocities constant, the center of mass will move at the average of the velocities, a straight line. It's not going to curve or anything. Then, you need to know how to find the closest point on this ray (since we're only interested in t>=0) to the origin and that's about all. You only need a simple calculation and it's also really fast.
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