You don’t rotate with the earth, instead you continue on a straight path, and accelerate downward due to gravity.
It’s just that, for short distances, earth surface can be approximated as flat, and rotation along this surface is not that different from a straight line.
But for long distance, this approximation does not hold. If you fire an artillery shell, the shell follows a parabolic path, but the target does not move on a straight line during the flight time. This will end up having an effect over long distances, so you actually need to account for earth rotation.
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