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So I know the conventional way we determine the maximum distance a given constant force can stretch a spring, namely dividing the applied force by the spring constant, k. My issue is that I feel that would only tell us when the object attached to the spring would stop accelerating, not when it actually stops (v = 0).

Can someone help me reason through this dilemna?

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If you hand a weight on a spring, and let it go, the system oscillates for a while, finally stopping somewhere near the midpoint of the excursions. This is when the stretch is measured: when the system has stopped moving.

To take gravity out of the equation the standard spring test is performed by a machine which can measure the tension, and a system for slowly stretching the spring, as with a worm gear drive.

From this information one can find the spring constant for that spring design, and the limits to which the spring may be repeatedly stretch prior to failure.

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