Let M be the mass of the ring and m that of the ball and let V and $$\upsilon $$ be their velocity before collision. The initial momentum of the system (ring and ball) in the horizontal direction is

$${\overrightarrow p _i} = M\overrightarrow V + m\overrightarrow \upsilon $$

$$ = 2 \times 1 + 0.1 \times ( - 20)$$

$$ = 2 - 2 = 0$$

From conservation of momentum, the final momentum of the system $${\overrightarrow p _f} = 0$$ in the horizontal direction. Hence $${V_{cm}} = 0$$ for the ring, i.e. the ring has pure rotation about its centre of mass. So choice (a) is correct.

The total initial angular momentum of the system about the point of collision is

$${L_i} = m\upsilon r - I\omega $$

$$ = m\upsilon r - M{R^2}{V \over R}$$

$$ = m\upsilon r - MRV$$

$$ = 0.1 \times 20 \times 0.75 - 2 \times 0.5 \times 1$$

$$ = 1.5 - 1 = 0.5$$ kg m² s^$$-$$1

From the conservation of angular momentum, the final angular velocity must be anticlockwise. Hence the friction between the ring and the ground is to the left. So the correct choices are (a) and (c).