Minimum Length of Cart + Pure Rolling!

From the conservation of energy velocity of $m$ when it lands on $M$ is $5m/s$

Now from the figure

$Ma=f_{k}$ ............(1)

$f_{k}=0.4(12)g=48N$

So from (1) $a=8$

So the acceleration of $m$ is $\frac{ma+f_{k}}{m} =12$

Now

$f_{k}R=I \alpha$

So $8=R \alpha$

Now using $v=u+at$ we get

$v=5-12t$

and $\omega=8t/R$

Now for pure rolling $v=R \omega$

So $8t=5-12t$ So $t=0.25$

So the pure rolling motion after $0.25 sec$ .

As $S=ut+0.5t^{2}$

So $S=7/8$

So the length of the plank should be greater than of equal to $7/8$ m.