Sliding, Then Rolling

The cue ball is still a picture. Give it a speed and it should slow down. It does, but not the way you would guess: for its first half metre it skids, and only then does it roll.

The push goes where the ball touches

The cloth touches the ball at one point, under its centre. Friction acts there, against that point's slip, its speed over the cloth: u = v + ω × r. Here ω is the spin and r runs from the centre down to the cloth.

A ball struck at centre height leaves with no spin, so its bottom skids at the full speed. The push on that bottom point slows the ball and spins it up at once, so the slip shrinks three and a half times as fast as the speed. When it reaches zero the ball rolls, at 5/7 of its speed (the 5/7 result, worked through). Rolling resistance takes over, twenty times weaker:

sliding   μs·g = 0.2 × 9.81  = 1.962 m/s²
rolling   μr·g = 0.01 × 9.81 = 0.098 m/s²
at 2 m/s  skids 2v / (7·μs·g) = 0.29 s over 0.50 m, then rolls at 5/7 × 2 = 1.43 m/s
          and stops 10.9 m out, 14.85 s after the strike