Conservation of linear momentum is based on newton’s second law of motion which states that in an isolated system the total momentum remains the same.
Remember: In the above experiment we did not consider any loss of energy due to friction, heat, etc. and all the collisions were elastic in nature i.e. there was a total transfer of energy, Actual observations may differ.
Let’s consider a case where a football of mass \(M_2\) is resting on the ground, a bowling ball with a comparatively heavier mass of \(M_1\) is thrown at the football at a velocity of \(U_1\).
When the bowling ball hits the football the energy is transferred and the bowling ball loses some velocity and moves at a new velocity \(V_1\), the football moves at velocity \(V_2\), why did the football move?
To conserve linear momentum, i.e. the bowling ball had an initial momentum of \(M_1\) so as \(M_2\lt M_1\) and the momentum of football should be equal to the momentum lost by the bowling ball according to the law of conservation of momentum, the football had no other option than moving at a velocity \(V_2\).
Mathematically it is given by,
\(m_1u_1~+~m_2u_2\) = \(m_1v_1~+~m_2v_2\)
Note: \(m_1u_1 \neq m_1v_1\)
\(m_1\) is mass of the bowling ball
\(m_2\) is the mass of the football
\(u_1\) and \(u_2\) are the initial velocities and \(v_1\) and \(v_2\) are the final velocities.
Even though the momentum of each particle changes, altogether the momentum of the system remains constant as long as there is no external force acting on it. In a head to head car accident the momentum is transferred from one car to the other, but so much force is applied that the car structure cannot handle it, this is why a car wrecks. If the cars were able to deal with the amount of force, and the collision was elastic they both will move in opposite directions, considering their weights are same.
In short momentum is always conserved in any collision, whether it be an elastic or a non-elastic collision, though kinetic energy is not conserved in a non-elastic collision, the kinetic energy is converted into heat energy or potential energy etc.
Consider this example of a balloon, the particles of gas move rapidly colliding with each other and the walls of the balloon, even though the particles themselves move faster and slower when they loose or gain momentum when they collide, the total momentum of the system remains the same. Hence, the balloon doesn’t change in size, if we add external energy by heating it, the balloon should expand because it increases the velocity of the particles and this increases their momentum in turn increasing the force exerted by them on the walls of the balloon.
Having said so the energy of a system is always conserved, one of the best applications of the law of conservation of momentum would be in space travel, there is no medium in space to exert a force on, then how do rockets travel? Well, they eject matter at very high speed so in an isolated system the momentum should remain constant therefore the rocket will move in the opposite direction with the same momentum as that of the exhaust.