Question

A block of mass m starts at rest at height h on a frictionless inclined plane. The block slides down the plane, travels across a rough horizontal surface with coefficient of kinetic friction u, and compresses a spring with force constant k a distance x before momentarily coming to rest Then the spring extends and the block travels back across the rough surface, sliding up the plane. The block travels a total distance d on rough horizontal surface . the correct expression for the maximum height h' that block reaches on its return

Answer 1

It's an big question, it is really very hard.
I think you should ask your teacher

Answer 2

As the blocks comes down from a friction-less inclined plane and travels a distance d on a friction surface the force with which it will strike the spring displacing it by x is given by the expression as,

                                    Friction = u m g.

And after the displacement of x, spring comes back, and force the block to gain kinetic energy and reach a height h’, therefore by the work energy theorem,

                                    m g h – 2 u m g d – m g h’ = 0

                                    $\Rightarrow$ h = h’+ 2 u d