A small bomb, of mass 10 kg, is moving toward the North with a velocity of 4.0 m/s. It explodes into three fragments: a 5.0-kg fragment moving west with a speed of 8.0 m/s; a 4.0-kg fragment moving east with a speed of 10 m/s; and a third fragment with a mass of 1.0 kg. What is the velocity of the third fragment? (Neglect air 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.