I don't think you're quite getting this.
For the first h that the particle travels it will have a certain force (tension - friction) acting on it. Immediately after, there is no tension, since the 2m particle has hit the floor and is not accelerating downwards anymore creating tension in the string.
So for the last 1/3 h the m particle is travelling under
only the force of friction (this frictional force is
μR i.e the same as before). So the acceleration will be different for the last 1/3 h distance it travels.
Therefore you need to consider the motion for the first h and the motion for the last 1/3 h travelled separately (different accelerations for each part which you need to work out).