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I believe you meant that you had an object on a string that revolved around an axis at a constant velocity. If that is the case, under friction free conditions, when the string snaps, it's velocity would not change, only its direction of movement. The centripetal force that was keeping the mass revolving around the axis would converted into centrifugal force that would make it travel in a straight line away from the axis.

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Q: If I had a mass on a string and rotated it at constant velocity how would its velocity change if the string snapped?
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