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domephilis

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**[Mentor Note: Two threads merged below and the OP has fixed up their LaTeX in later posts]**

**TL;DR Summary:**A billiard ball with zero angular velocity and linear velocity v_0 is released on a horizontal surface with coefficient of friction mu_k. It begins to roll without slipping after travelling a distance d. Show that mu_k = \frac{12v_0^2}{49gd}.

Question: A billiard ball with zero angular velocity and linear velocity v_0 is released on a horizontal surface with coefficient of friction mu_k. It begins to roll without slipping after travelling a distance d. Show that $mu_k = \frac{12v_0^2}{49gd}$.

Using energy principles, I was able to get to this point:

$\mu_k = \frac{1}{gd}(\frac{v_0^2}{2}-\frac{7}{10}v^2)$

What I am currently struggling to do is relating the velocity at distance d and the velocity at the start. I have tried using the torque to find the angular velocity at distance d and use ##v = r\omega## (we can do so since we are rolling without slipping at that point). That didn't lead anywhere because I have no idea how long it takes to reach distance d nor do I know what angle theta has been rotated after travelling d. I am really stuck here and would greatly appreciate any hint or help. Thank you.

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