A solid cylinder of mass M and radius R rolls without slipping on a rough horizontal surface with a speed v . It then enters a smooth horizontal region and strikes a smooth vertical wall elastically. After rebounding from the wall, the cylinder eventually re-enters the rough horizontal surface. The final steady speed of the cylinder when pure rolling is re-established on the rough surface is:

Step-by-step solution

Let the initial velocity of the cylinder be v towards the right. For pure rolling, its initial angular velocity is = v R (clockwise). When the cylinder enters the smooth region and strikes the smooth vertical wall elastically, its linear velocity reverses to v towards the left. Since both the wall and the floor are smooth, no frictional torque acts on the cylinder. Thus, its angular velocity remains = v R (clockwise). When the cylinder re-enters the rough surface, it is moving to the left with speed v but still rot