A uniform board of length L is sliding along a smooth (frictionless) horizontal plane as in figure. The board then slides across the boundary with a rough horizontal surface. The coefficient of kinetic friction between the board and the second surface surface is `mu_k`

a. Find the acceleration of the board at the moment its front end has traveled a distance x beyond the boundary.
b. The board stops at the moment its back end reaches the boundary, as in figure. Find the initail speed `v` of the board.

Explanation

a. Let `m` be the mass of the whole board. The portion on the rough surface has mass `mx//L`. The normal force supporting it is `mxg//L` and the frictional force is `mu_kmgx//L=ma`.
Then `a=mu_kgx//L` opposite to the motion.
b. In an incremental bit of forward motion `dx`, the kinetic energy converted into internal energy is `f_xdx=(mu_kmgx)/(L)dx`. The whole energy converted is
`1/2mv^2=underset0oversetLint(mu_kmgx)/(L)dx=(mu_kmg)/(L)x^2/2 :|_0^L=(mu_kmgL)/(2)`
`v=sqrt(mu_kgL)`