A smooth sphere A is moving on a frictionless horizontal plane with angular velocity and centre of mass velocity $v$. It collides elastically and head on with an identical sphere B at rest. Neglect friction everywhere. After the collision their angular speeds are ${\omega }_{\mathrm{A}}$ and ${\omega }_B$ respectively. Then  

Since, it is head on elastic collision between two identical balls, they will exchange their linear velocities

i.e., A comes to rest and B starts moving with linear velocity v. As there is no friction anywhere, torque on both the spheres about their centre of mass is zero and their angular velocities remain unchanged. Therefore,
${\omega }_A=\omega$ and ${\omega }_{\mathrm{B}}=0$