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Question: When two masses collide inelastically, the kinetic energy is not conserved. This is obvious because kinetic energy is the energy associated with a mass in motion. In an inelastic collision, some of the energy of the collision is converted to other types of energy, such as heat, sound, and deformation.
But, if we look at the total momentum in a collision, (i.e., the sum of the momentum of all the masses in the interaction) the total momentum before and after the collision does not change.
Answer: Kinetic Energy and momentum are always conserved in elastic collisions. As a result, it is relatively easy (and pedagogically instructive) to measure the kinetic energy and momentum before and after the collision in an elastic collision to compute the post-collision velocities of two masses. In this special/idealized case, it is possible to solve for the velocity and direction of both masses after a collision by using the simple p=mv and KE=(1/2)mv^2 conservation equations.
But, in an inelastic collision, the number of masses to measure and compute in the before and after-collision state makes the numerical/formulaic computation of the post-collision direction and velocity of the masses impractical. Still, the conservation of total energy and momentum is hidden behind the complexity of the motion and numerous entities involved after the collision.
Since this problem stipulates that the collision is inelastic, we shall attempt to illuminate how both momentum and total energy are conserved amidst the confusion of kinetic energy conversion to numerous forms.
Obviously, in an inelastic collision, some of the kinetic energy of the masses will be lost to various other forms of energy. The total system energy will always be conserved, it is just complex to track the details of every molecules conversion of kinetic energy into another form of energy after the collision.
Listed explicitly, the kinetic energy of the pre-collision masses will be converted into a variety of post-collision energy-species, such as: 1) the residual kinetic energy of the masses, 2) the kinetic energy of the thermal energy of molecular motion, 3) the acoustic energy of molecular air vibration, 4) the photonic energy of IR photons released from the higher than ambient temperature masses, and 5) the energy held in activated atomic orbitals, etc.
But, the fact that kinetic energy converts into many kinds of energy whose total energy is conserved, does not intuitively resolve the apparent discrepancy between kinetic energy and momentum. In other words, what is the nature of momentum that it is apparently conserved in an inelastic collision, while kinetic energy is not?
Regarding momentum, if we calculate the vector sum of the momentum of the two masses pre- and post-inelastic collision, the sum of the momentum vectors will be equal before and after the collision.
To intuitively understand the distinction between momentum and kinetic energy that produces this apparent difference in conservation, we note that the collision of every molecule involves an equal and opposite action and reaction. This principle of interaction is at the heart of the conservation of momentum.
Energy is a different aspect of the properties of various types/configurations of space (e.g., mass is a type of energy, as is the motion of a mass, as is the electromagnetic packet of a photon, as is the attraction of charged bodies, etc.). Energy reflects the magnitude of an (unknown, not fully understood, but real) property of space. Energy is conserved because that property cannot be created or destroyed, only converted.
Example: When two equal masses collide and rebound, in the center of mass frame, their momentum are opposite and equal, and the net momentum of the system is zero, both before and after the collision, whether elastic or inelastic. The total energy associated with the two masses before the collision is equal to the total energy after the collision.
Momentum is conserved on the molecular level in every interaction/collision between molecules. Thus, even though an inelastic collision is complex, the momentum changes of all the constituent molecular collisions all cancel each other out. The result is a net zero change in momentum as measured before and after the macroscopic collision.
The total energy is conserved because the energy associated with the moving bodies is neither created nor destroyed in this, or any interaction. Thus, the magnitude of the total energy is conserved.