^Linear momentum of earth ball system

When a ball is thrown upwards in vacuum an external force equal to its weight acts on it which changes its linear momentum with the time.

However the linear momentum of earth – ball system is conserved & earth moves backward in accordance with LCLM. If we take earth ball as one system, then forces between two are purely internal, consequently the linear momentum of earth ball system remains conserved & we can write.

m_b v_b + m_e v_e = 0

As m_e >> m_b thus recoil speed of the earth is too small to be of significant consideration. Also as the ball moves away from earth, earth moves away from the ball in the opposite direction such that the potential energy of the earth – ball system increase & KE decrease in such a way that mechanical energy of the earth-ball system remains conserved (provided non-conservative force are absent).