Peculiar question

At the highest point no vertical velocity exists.Also there is no net force in the horizontal direction.Hence the linear momentum of the system remains conserved along the horizontal direction.As one part retraces the path so it's horizontal velocity just after collision is vcosθ.
Now conserving linear momentum we get,
m*vcosθ= -(m/2)*vcosθ + (m/2)*u
or, u=3vcosθ.

Initially the whole ball moves right.finally one part has horizontal velocity towards left after collision.