Strictly speaking we should make a distinction between the centre of mass and the centre of gravity of an object, but as long as we are in a uniform gravitational field we can ignore such fine details.
I would like you to do an experiment. Take something flexible — atie, a jumper, a set of masses linked by springs or string… whatever — and throw it across the room. The object could turn and change shape in a really complex way, but the centre of gravity will move predictably: it will move in a parabola under the action of the Earth’s uniform gravitational field. Of course the path it takes will have been affected by air resistance, so it won’t be a perfect parabola, but that’s a factor that could be eliminated by performing the experiment in a vacuum.
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