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Although competition rules prohibit significant difference in size, typical coinoperated pool tables may present players with a significant difference in diameter between the typical object ball (i.e., a colored ball) and the cue ball (i.e., the white ball). In fact, once an object ball goes into a pocket, it is captured by the table whereas a cue ball must always be returned to the player; and it is not uncommon for the return mechanism to use the difference in ball diameter to separate the cue ball from the rest. Given this, suppose we want to hit a ball resting against the bumper in such a way that, after the collision, it moves along the bumper. Modeling the contact between balls as frictionless, establish whether or not it is possible to execute the shot in question with (a) an undersized cue ball and (b) an oversized cue ball.

Step-by-step

It is possible to execute the shot in case (a) (undersized cue ball) but not in case (b) (oversized cue ball). The reason is that to execute the shot in question, the LOI of the impact must be parallel to the bumper. Since the object ball is assumed to be at rest and touching the bumper, we have that the LOI can be parallel to the bumper if the cue ball is either of the same size as the object ball or smaller than the object ball. If the cue ball is larger than the object ball then the LOI will not be directed “into the bumper.” Consequently, right after impact, the object ball will tend to rebound off the bumber.

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