The story of a mercilessly thrown sphere...

Classical Mechanics Level 3

A spherical ball of mass M and radius R is thrown along a rough horizontal surface so that initially, it slides with a linear speed \(\displaystyle{\upsilon_0}\),but does not rotate. As it slides, it begins to spin and eventually rolls without slipping. The time taken to start rolling can be expressed as \[\dfrac{a}{b} \times \dfrac{\upsilon_0}{\mu_k g}\] Where \(\displaystyle{\mu_k}\) is the coefficient of kinetic friction between the surface and the ball .What is \(\displaystyle{a+b}\)?

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