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But if I use the same figures and shoot the watermelon around the Earth in a grazing orbit, with eccentricity of zero, the orbital period is 6 minutes longer.

Why?

BTW: I suspect any minimum energy subway tunnel of hypocycloidal shape has 1/2 the transit time of the watermelon, or 38 minutes, or 76 minutes round trip. A hypocycloid where the generating circle diameter is the same as the base circle radius generates a straight line through the center.

Comments?