A spherical ball of radius r initially at rest on a rough horizontal surface is hit horizontally at a point at a distance x above the central line. Due to this sharp impulse the centre of the ball acquires a velocity vc. After some time, the ball will start pure rolling with a velocity equal to

Initially ball will gain both linear and angular velocity. Linear impulse: I0=mvC Angular impulse: I0x=Iω0⇒mvCx=Iω0I is moment of Inertia about C.Apply conservation of angular momentum about lowest point.Iω0+mvCr=Iω+mvr where ω=vr at the time of pure rolling ⇒ mvCx+mvCr=25mr2vr+mvr⇒ vC(x+r)=75rv⇒ v=57vCx+rr