Theoretically, and if you were going slow enough, I'm sure you could keep your velocity and speed constant. Or am I missing something fundamental?
Constant velocity? No.
Constant speed? Yes.
You have to remember velocity is speed and direction, and going in a circle, your direction changes at every point, therefore your velocity can never be constant even though your speed could be.
On a race track you could have constant velocities at certain points since the track is not a perfect circle, but overall you will not have a constant velocity if you make one lap around the track.
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