Impulse-driven slipping-to-rolling transition of a sphere

For a uniform sphere struck horizontally with impulse $J$ at height $d$ above its centre, $u_0=J/M$ and clockwise spin ${\Omega }_0=5Jd/(2M{a}^2)$. friction brings the contact slip $u-a\Omega$ to zero. The final rolling speed is $u_f=5J(1+d/a)/(7M)$, independent of a positive constant friction coefficient; friction can accelerate an initially overspinning centre of mass.

For the ideal uniform solid sphere with negligible table impulse during the cue strike, immediate rolling requires $u_0=a{\Omega }_0$, hence the horizontal cue line lies $2a/5$ above the centre. A different inertia distribution changes this height.