We know that in the case of a collision, the force acting between the two objects is irrelevant to momentum conservation, but is very important to determining the amount of energy converted to thermal energy. For example, if two blocks collide with a spring between them, then all the kinetic energy they come in with, they will also go out with, since the internal spring force involved in the collision is conservative, and there is no spring potential energy before or after. On the other hand, if an internal non-conservative force is present between the colliding objects, then some of the incoming kinetic energy is converted into thermal energy. The former sort of collision (where kinetic energy is conserved) we call elastic, and the second type of collision we call inelastic.
From our discussion in Section 4.4, it's clear that what determines the inelasticity of a collision is the deformation of the colliding objects. When a colliding object deforms, it's because the particles directly involved in the contact are accelerated more than other particles in the same object, thus introducing internal energy, and reducing the amount of mechanical energy available to go back into the motions of the objects.
A collision where the objects continue together with the same velocity after the collision (i.e. they remain stuck together), is often referred to as totally or perfectly inelastic. This of course does not mean that all of the kinetic energy is lost (the objects do continue moving at the end in most such collisions), only that they don't bounce off each other. From the perspective of the center of mass frame, we can see that such a collision maximizes the amount of internal energy that the collision can create: In this frame, the objects stop entirely after the collision, so all of the mechanical energy becomes internal. Changing frames doesn't change the amount of internal energy created (it only changes the mechanical energy we see), so having the objects stick together results in the largest possible creation of internal energy.
If we are told that a given collision is elastic (or at least can be approximated as such), then that gives us an additional condition that we can use to solve the problem. Let’s see a couple of examples. in each case, the diagram will show the experimental result, which we will then show mathematically using the combination of momentum and kinetic energy conservation.