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If a planet is directly between two stars y distance apart (so it's y/2 from each planet), with velocity x such that x is sufficient to continue orbiting at that distance (y/2), which does it orbit, and how does it continue?

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Google "Lagrange points". This is one of the unstable Lagrange points, meaning that a planet exactly in the middle would stay there would stay, but if it's a little bit off, it will tend to drift away. –  Peter Shor Jul 4 '11 at 0:19
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up vote 4 down vote accepted

Given that the mass is at the CoM of the system, there is a solution in which it has no velocity (i.e. it just sits there while the stars orbit around their common center which happens to include a planet). I believe the planet is positionally unstable in that case (that is, it won't stay there in the real world).

Aside from that there is no circular or elliptical orbit at all. I don't know if there are any stable orbits that pass through that point (in fact I tend to doubt it, there will be a sensitive dependence), but if there are they will certainly be strange.

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