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 side 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

(a) the time elapsed before the block separates from the plank
(b) displacement of block and plank relative to ground till that moment.

Explanation

Let `a_(1)` and `a_(2) `be the retardation of `m and M` respectively

` rArr a_(rel) = la_(1) - a_(2)l `.
where `a_(1) = (mu //2 mg)/(m)= (mu)/(2) g`
and `a_(2) = (mu (M + m)g - (mu //2) mg)/(M)`
The relative acceleration between the block
`a_(rel) = |a_(2) - a_(1)| = (mu(M + mg))/(2M)`
Now , `l = (1)/(2) a_(rel) t^(2) `, where `t =` time of sliding of `m` over `M`
`rArr t = sqrt((21)/(a_(rel))) = sqrt((4ml)/((M + m) mu g))`