The physics
A blow J on a free body moves its centre and spins it. The centre moves at J/m, wherever the blow lands. The spin comes from the blow's arm about the centre.
v = J / m J d = Icmω
A pulled roller feels friction at the floor. Write two equations: the forces for the centre and the torques for the spin. Rolling ties them together: a = αR.
F ± f = ma torque about the centre = Iα
A ball that slides skids first. Friction works on the slip until v = ωR. Then it rolls on at a steady speed.
Friction acts along the floor. So L about a point on the floor stays the same through the skid.
mv₀R + Icmω₀ = (Icm + mR²) v / R no skid: h = R(1 + β)