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A billard ball is struck with a cue. The line of action of the applied impulse is horizontal and passes through the center of the ball. The initial velocity $v_0$ of the ball, its radius $R$, its mass $M$ and coefficient of friction $\mu_k$ between the ball and the table are all known. How far will the ball move before it ceases to slip on the table?

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This looks very like a homework question, and the site rules forbid us from answering homework questions. If you want to edit your question to make it more general e.g. "what controls whether a ball will skid or roll?" we can answer that, but to be honest you could easily Google for that answer. – John Rennie Jul 18 '12 at 12:02
I could not find an answer for this question. I'm editing the question now. – Zeyneb Jul 18 '12 at 12:06
See under the heading "A Closer Look At Relative Slipping" – John Rennie Jul 18 '12 at 12:17

Start by finding the force that is trying to spin the ball. Actually it is a torque. Then try to find out for how long this torque needs to be applied such that the rotational speed matches the transnational speed and there is no slip at the contact.

Depending on your convention (what is positive or negative) you need an expression for the slip amount based on the ball linear speed $v(t)$ and rotational speed $\omega(t)$. What you are after is the instance these two give zero slip.

You use Newton's laws to find these speeds at any instant.

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