Question

A smooth solid sphere A of radius R and of mass m moves on a frictionless horizontal plane surface with an angular speed ω and with a velocity v of center of its mass. v and ω need not be related. The sphere may or may not be rolling. It collides perfectly elastically head on with another identical sphere B at rest. Neglect friction every where between any two surfaces.
After the collision, their angular speeds are ωA and ωB, respectively. Then what is true among these following:

(A) ωA < ωB (B) ωA = ωB (C) ωA = ω (D) ωB = ω

What is the linear velocity of sphere A and what is the linear velocity of sphere B.

Answer 1

When identical spheres or objects collide elastically head on, one moving with a velocity v and the other being stationary, on a frictionless surface, then stationary sphere moves with the velocity v and the colliding sphere becomes stationary.

 we apply the conservation of linear momentum during collision.  Here the angular velocity is not considered.
       m v = m v1 + m v2      =>  v = v1 + v2      --- (1)

Let I be the moment of Inertia of the spheres about their center of mass C.
       I = 2/5 * m R² = K m R²,  say, where   K  = 2/5

Conservation of angular momentum :
       I ω = I ω₁ + I ω₂        => ω = ω₁ + ω₂        --- (2)

As there is no friction between the sphere 1, and sphere 2,  the angular momentum is not transfered to sphere 2.  The contact between the spheres allows only transfer of linear momentum.  Only if there is a friction between them, then the second sphere will be forced to rotate.

           ω₂ = 0.    =>    ω₁ = ω  --- (3)

  we cannot say  v = R ω  as the sphere may or may not be rolling.

Conservation of kinetic energy:
    1/2 m v² + 1/2 I ω² = 1/2 m v₁² + 1/2 I ω₁² + 1/2 m v₂² + 1/2 I ω₂²
     v² + K R²  ω²  = v₁² + v₂² + K R² ( ω₁² + ω₂² )
 => v² = v₁²  +  v₂²        --- (4)
=> v₁ v₂ = 0  from 1 and 4.
  So  v₁ = 0  and  v₂ = v    or,    v₁ = v  and  v₂ = 0.

It is physically impossible that v₁ = v  and  v₂ = 0.  As after collision sphere1 moves, it implies that  second sphere moves faster than sphere1.

Hence,    v₁ = 0.  So first sphere remains at one point and starts rotating with angular speed on the surface.   As there is no friction on the floor, it is possible.  Hence, the second sphere moves with a linear speed v  and does not rotate.

So,      v₁ = 0 ,  ω₁ =  ω ,  v₂ = v  and  ω₂  = 0