A little rolling sphere

A small solid steel sphere of total mass \(0.1~\mbox{kg}\) and radius \(1~\mbox{cm}\) rolls without slipping with an initial speed \(v_i\) on a table. It eventually rolls onto a thin sheet of paper. As the ball is moving on the paper you suddenly move the paper horizontally at a speed of \(1~\mbox{m/s}\) for \(2~\mbox{seconds}\) perpendicular to initial velocity of the sphere. The sphere may slip on the paper during this time. You then stop the paper, the sphere keeps rolling/slipping, but eventually rolls off the paper back onto the table. When the sphere is back to rolling without slipping on the table, it has a final speed \(v_f\). What is \(v_f/v_i\)?

Details and assumptions

• The coefficients of static and kinetic friction of the paper with the sphere are \(\mu_s=0.5\) and \(\mu_k=0.4\).