Turn this into an experiment that tells the difference between a flat and round earth. The only experiment I see this turning into is the following: start out heading East, and then go in a straight line. If you eventually hit Antarctica, the earth is flat, (or you started out very close to the north pole) and if you eventually come back to where you started, it's round.
The problem with this experiment, despite it being extremely difficult to carry out (much harder, for example, than watching a Foucoult pendulum in Sydney and another in New York for a few hours each, which only requires a couple of plane tickets), is that you need some way of objectively determining whether you are going straight. A compass is completely useless for this purpose, since starting out going due East, a straight line will not continue to go due East (unless the Earth is round and you start out on the Equator), so you need some other way of making sure you are going straight, and it has to guarantee that you go straight for thousands of miles, without reference to visible objects on the Earth or sky, since you don't know the shape of the Earth, and by extension the sky's shape either.