### Rigid Body

The ball starts to move with sliding so that the friction force $F=\mu mg$ accelerates the ball but the torque $N=Fr=\mu mgr$ decelerates the rotation:
$m\dot{v}=\mu m g \Rightarrow v=\mu gt\\I\dot{\omega}=-\mu mgr \Rightarrow \omega=\omega_0-(\mu mgr/I)t$
where $I=(2/5)mr^2$. When$v=\omega r$ the sliding stops, the friction force vanishes and the ball continues to roll without sliding so that $v$ and $\omega$ no longer change. This happens when
$\mu g t=\omega_0r-(\mu mgr^2/I) t \Rightarrow t=\omega_0r/\mu g(1+mr^2/I)$
and the velocity is
$v=\omega_0r / (1+mr^2/I)$