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I don't understand why the center of mass' acceleration is equal to 0 when the center of mass should be moving as an external force is exerted on the cue ?

Billiard Balls. Now we can go back and examine the behavior of the billiard balls.Once the cue ball has begun to roll,no net external force acts on the (twoball) system.Thus, because 0, Eq. 9-14 tells us that 0 also. Because acceleration is the rate of change of velocity,we conclude that the velocity of the center of mass of the system of two balls does not change.When the two balls collide,the forces that come into play are internal forces,on one ball from the other. Such forces do not contribute to the net force ,which remains zero.Thus,the center of mass of the system, which was moving forward before the collision, must continue to move forward after the collision,with the same speed and in the same direction.

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    $\begingroup$ 'once the cue ball has begun to roll' means 'after the cue has struck it' $\endgroup$
    – user107153
    Apr 28, 2016 at 12:50

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The external force acts only for the small time when the cue has been struck. Once it moves, there is no force. This means that the ball is moving with zero external force, which means according to Newton's second law, the velocity of the ball is the same. here the act of friction is of less importance as it requires in a billiard play. So the center of mass of the "two-ball" system moves with constant velocity means the acceleration of the center of mass is zero.

On collision, the internal forces acting are equal and opposite. So they will not contribute anything to the net force.

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