# 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 $$\mu_k=0.4$$.
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