Let it roll

F.B.D of sphere for no vertical acceleration $N=mg$

$\Rightarrow f= \mu mg$ (as $f= \mu N$ )

$\Rightarrow a=\mu g$

$v=\mu gt$ , Also $\alpha=\frac{\mu mgr}{I} \Rightarrow \omega =\frac{\mu mgrt}{I}$ where $I$ is moment of inertia.

The velocity of the lower most point of the sphere at time $t$ is

$v_{s}=v+ \omega r$

$\large{v_{s}=\mu gt + \frac{\mu mg r^2}{I}t}$

From F.B.D of plank

$\large{\Rightarrow a_{p}=- \left(\frac{F+f}{M} \right)=- \left(\frac{F+\mu mg}{M}\right)}$

$\large{v_{p}=v_{o}- \left(\frac{F+ \mu mg}{M} \right)t}$

For no slipping $v_{s}=v_{p}$

$\large{\Rightarrow \mu gt+ \frac{\mu mg r^2}{I}t=v_{o}- \left(\frac{F+ \mu mg}{M} \right)t}$

as $I=\frac{2}{5} mr^2$

$\Large{\Rightarrow t=\frac{v_{o}}{\frac{7}{2}\mu g+\left(\frac{F+\mu mg}{M} \right)}}$