Question

Please, can any one help me solve this difficult problem??

Answer 1

The problem's statement gives you a clue on how to approach the solution. It ask for a terminal velocity for the plank , the plank will reach a maximum speed at some point.
If the plank were sliding on a frictionless inclined plane it would accelerate indefinitely until it reach the bottom. This is because gravity is doing work on the plank and this work is converted to kinetic energy. The longer gravity works the faster the plank will go.

However in the problem the plank is passing over rollers and starting them to rotate. Thus the plank is doing work on the rollers and is thus transferring its energy to the rollers one at a time after it has moved one plank length.

Gravity is doing work at a constant rate, and the plank is doing work at a constant rate after it has moved one plank length. When these two rates become equal the plank is passing all energy that gravity is giving it to the rollers.

What is the rate at which gravity is giving the plank energy? And what is the rate the rollers are acquiring this energy from the plank? both depend on the speed that the plank is moving over the rollers.

Can you take it from here ?