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If the two people walked backwards, all the while holding the rope taut, until they were standing back to back on the other side of the Earth, would the rope touch the Earth at any point, or would it be taut and above the surface of the Earth at every point?

A friend and I have been debating this for about a week now, and neither of us, nor any of our teachers, can come up with a foolproof way of giving an answer.

I've been looking for a proper answer for a while now, and Google brought me here. Is there an actual answer, or will it remain a hypothetical?

If the problem isn't clear, I'll be more than happy to try to clarify any misunderstandings. Thanks for your time!