One more hint

11.13

On 11.13, we managed to find alpha, but we can't find how long the ball slides because we don't know omega at that point. We know we can solve for the rest of the problem, but we're stumped right here.

Ah, but you do know omega at the point that slipping stops - at that point, it is pure rolling motion, and v = r * (omega). That's what you found in the first part.

You know that the linear velocity starts out at v_i, and ends up at v at the moment t that the sliding stops. That means

v(t) = v_i + at

where a is the acceleration you already found. For the angular part, you know that the ball starts out *without* rotation, so (omega)_i = 0. At the moment rolling without slipping starts, you know (omega) = (omega)_i + (alpha)t = (alpha)t. You also know what when rolling without slipping starts, v = r(omega). Put that together ...

v(t) = v_i + at = r(omega) = r*(alpha)*t

Now you know everything but t in the equation above ...