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To get a billiard ball to roll without sliding from the start, the cue must hit the ball not at the center (that is, a height above the table equal to the ball's radius R) but exactly at a height 2R/5 above the center. Prove this result. [See Arnold Sommerfeld, Mechanics, volume 2 of Lectures on Theoretical Physics, Academic Press, Orlando (1964 paperback edition), pp. 158-161, for a supplement on the mechanics of billiards.]
Therefore, the billiard ball must be hit at a height 2R/5 from its center.
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