Conundrum - is this scenario deterministic or not?

There is a puzzle that occurred to me several years ago that I have never been able to solve to my satisfaction, and I would like to know your thoughts on it. It is a paradox, in the sense that it seems to have strong arguments supporting completely contradictary results. It is about whether a particular scenario is deterministic or not.

Begin with the old sixth-grade math problem of the train that leaves Chicago headed for New York at a particular time and at a specific speed at the same time another train leaves NY for Chicago on the same track at such-and-such speed, and a bug which can fly extremely fast that zips back and forth between the two, and is able to turn around in zero time, and you are supposed to figure out how far it flies before it gets crushed between the two trains, and of course this is easy given all the right data.

And of course this is a mathematics puzzle so it flies in a perfectly straight line and all speeds are exactly constant and the bug is a mathmatical point and a the end, both trains and bugs are all at a single point. It is not a trick question.

But now consider the opposite question, the one that intrigues me.

If we take this scenario and play it backwards, with the two trains and the bug all beginning from a point, and the trains fly apart at that exact time and specified speed, with the bug bouncing back and forth, is the location of the bug something that is deterministic, in the sense that, for example, at the end of ten minutes we could calculate its exact location?

Answer one: Of course. It is a well-defined algorithm beginning from a mathematical point and following specific rules, it can only be deterministic. There is only one place where the bug could be.

Answer two: Of course not, playing the scenario forward (with the trains converging) the bug could begin anyplace between them and they would all still end up at the same point.