Figure shows a small block A of mass m kept at the left end of a plank B  of mass M=2m and length L. The system can slide on a horizontal road the system is started towards right with the initial velocity  v .The friction coefficient between the road and the plank is 1/2 and that between the plank and the block is 1/4. Find the time elapsed before the smaller blocks separates from the bigger block.

 

There will be relative motion between block and plank and road. So at each surface limiting friction will act. The direction of friction forces at different surface are as shown in figure

Here, $f_1=\left(\frac{1}{4}\right)\left(mg\right)$ and $f_2=\left(\frac{1}{2}\right)(m+2m)g=\left(\frac{3}{2}\right)mg$

Retardation of A  is $a_1=\frac{f_1}{m}=\frac{g}{4}$ and retardation of B is $a_2=\frac{f_2-f_1}{2m}=\frac{5}{8}g$

since, $a_2>a_1$ 

Relative acceleration of A with respect to B is $a_r=a_2-a_1=\frac{3}{8}g$

Initial velocity of both A and B is v. so there is no relative initial velocity. Hence, 

Applying $s=\frac{1}{2}a{t}^2$ Or $L=\frac{1}{2}{a}_r{t}^2=\frac{3}{16}g{t}^2$

$\therefore t=4\sqrt{\frac{L}{3g}}$