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It goes horizontally. The string is the only force that is keeping the rock from going in a straight line (the rock will go in a straight line in the absence of a force because of its "inertia") Thus, if the force is removed when the rock's instantaneous velocity is horizontal, it will continue to travel horizontally thereafter (in the absence of gravity, another force)..

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Q: Suppose you tie a rock to a string and spin it in a vertical plane In what direction will the rock move if the string breaks exactly when the rock has passed the lowest point and is in a horizontal?
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