Principle or law of conservation of linear momentum:
When the resultant external force acting on a system is zero, the total linear momentum of the system is constant; it is conserved.
The total linear momentum of an isolated system, i.e., a system subject only to internal forces, is constant, i.e., it is conserved.
The principal is a consequence of Newton's third law of motion.
Example:(1)As a person jumps out of a stationary boat, the boat moves backward. Initially, the momentum of the system was zero because both of them were at rest. While jumping, the forward momentum of the person must be equal in magnitude to the backward momentum of the boat such that their sum is zero and the total linear momentum of the person and boat is conserved.
(2) When a bomb explodes in mid-air, the fragments fly off in different directions. However, it is found that the vector sum of the momenta of all the fragments at any instant always remains equal to the initial momentum of the bomb in both magnitude and direction. Here, there is an increase in KE and heat, sound, and light production because the chemical PE of its explosive ingredients is partly converted to these forms.
(3) Rockets propulsion depends on the law of conservation of momentum as applied to a system consisting of the rocket and the exhaust gases. Most of a rocket on its launching pad is oxygen and fuel, all of which are eventually burnt to accelerate the small payload. Exhaust gases (combustion products)are ejected at very high speed from the nozzle at the tail end of the rocket engine. Because of the momentum of the ejected gases, the rocket receives a compensating momentum in the opposite direction to conserve the original momentum of the rocket-gas system.
Law of conservation of linear momentum from Newton's laws of motion:
Consider the collision of two particles of masses m1 and m2, moving before the collision with constant velocities u1 and u2 in the laboratory frame of reference.
We shall assume that there are no external forces on the two-particle system. The contact forces, which the particles exert on one another, act only over the short time interval ∆t for which the particles are in contact during a collision. The contact forces during collision act only between the system's particles: they are internal forces that originate in bodies not belonging to the system.
Expression for the recoil velocity of a gun in terms of the muzzle velocity of the bullet:
Consider a gun of mass M which can fire a bullet of mass m. Before firing, the system's momentum consisting of the loaded gun (the gun and the bullet) is zero. When the gun is fired, the bullet acquires a forward momentum. At the same time, the gun recoils with equal and opposite momentum so that the total momentum of the system remains zero.
If u is the muzzle velocity of the bullet and V is the recoil velocity of the gun, by the law of conservation of linear momentum.
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