A bowling ball is thrown horizontally with initial velocity $v$ and no initial rotation over a surface with static coefficient of friction $\mu$. Determine the distance the ball travels ($d$) until it stops sliding and starts rotating.
I can't even start this problem (how do I know, mathematically, that the ball will stop sliding and will start rotating?).
Answer: $d=\frac{12v^2}{49\mu g}$
The coefficient of friction, $\mu$, will determine the torque on the ball as it slides. This torque is constant and can be defined as $\tau=\mu\,m\,g\,r$, where $m$ is the ball's mass, $r$ its radius and $g$ the acceleration due gravity.
So the ball's angular speed will increase according to $I\,\dot{\omega} =\tau$, where $I$ is the ball's mass moments of inertia. If you can work out $I$ and you can find the linear rate of increase of angular speed.
At the same time, the ball's speed $v$ will decrease following Newton's second law.
So now you can work out the speed $v(t)$ and angular speed $\omega(t)$ as a function of time $t$.
When the angular speed reaches the point where
$$\omega(t_0)\,r = v(t_0)\tag{1}$$
there is no relative slip between the ball and the surface. At this point, the friction force theoretically stops and the ball rolls onwards at a constant speed.
Your mission, should you choose to accept, is to solve (1) for $t_0$ and then you can work out the distance travelled.
