A 50-kg pitching machine (excluding the baseballis placed on a frozen pond. The machine fires a 0.40-kg baseball with a speed of 35 m/s in the horizontal direction. What is the recoil speed of the pitching machine? (Assume negligible friction.)"

Newton's second law in terms of momentum:  The rate of change of momentum of an object is equal to the net force applied to it.
The law of conservation of momentum: The total momentum of an isolated system of objects remains constant.

Newton's second law for a system of particles: The sum of all forces acting on the system is equal to the total mass of the system times the acceleration of its center of mass.