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Height to Strike Billiard Ball For No Slippage

This problem still haunts me from my freshman physics class at Stanford. It was the last problem on the exam, and nobody got it right, including me! But now, applying everything we've learned in the videos up until now, it turns out to not be so bad: just set up your free body diagram, set up your net force and net torque equations, and look for stuff to cancel out.