# Rolling without slipping effect on velocity of Centre of Mass

Ok, so I was solving a problem about a billiard stick hitting a billiard ball and it revolves around angular impulse. When the stick hits the ball, it gains an initial linear velocity as well as an initial angular velocity. The problem had the following statement about after the impulse: "Because of initial rotation, the velocity of centre of mass increases to maximum value of X". Initially in the problem, you are told that there is kinetic friction, so you would assume that the motion is rotation with slipping and that totally fits in for the initial rotation part. But, in the solution manual, it says that for the final angular velocity, it is rotation without slipping which I don't understand why. Is it related to the statement I wrote above?

• The cue ball must have "top spin", based on your description. At the moment the cue stick strikes the ball, it is spinning forward, and it has an initial linear velocity. The kinetic friction of the cue ball on the felt acts to speed up the linear velocity of the ball until it is no longer slipping on the felt. At that point, the cue ball is moving faster than its initial velocity, but rotating slower than its initial rotation rate. Commented Nov 25, 2016 at 21:21

In general, the motion of the ball after being struck is a mixture of linear velocity $v$ of the CM and rotational velocity $\omega$ about the CM (which is the same as about any point). Another way of describing this is rolling with slipping. When these two velocities are related by $v=R\omega$, where $R$ is the radius of the ball, then the ball is rolling without slipping. This means that the point of contact with the table is stationary for an instant.
If there is relative motion between the cue ball and the table at the point of contact, then friction will increase/decrease the linear velocity of the ball while also decreasing/increasing the angular velocity. When $v=R\omega$ then there is no relative motion at the point of contact and no friction. Initially we could have $v>R\omega$ or $v<R\omega$, but when friction has finished doing work on the ball we have $v=R\omega$.