Equation for a Perfectly Inelastic Collision:
m1v1i + m2v2i = (m1 + m2)vf
Proving Kinetic Energy Loss
You can prove that when two objects stick together, there will be a loss of kinetic energy. Let's assume that the first mass, m1, is moving at velocity vi and the second mass, m2, is moving at velocity 0.
This may seem like a really contrived example, but keep in mind that you could set up your coordinate system so that it moves, with the origin fixed at m2, so that the motion is measured relative to that position. So really any situation of two objects moving at a constant speed could be described in this way. If they were accelerating, of course, things would get much more complicated ... but this simplified example is a good starting point.
m1vi = (m1 + m2)vf
[m1 / (m1 + m2)] * vi = vf
You can then use these equations to look at the kinetic energy at the beginning and end of the situation.
Ki = 0.5m1Vi2
Kf = 0.5(m1 + m2)Vf2
Now substitute the earlier equation for Vf, to get:
Kf = 0.5(m1 + m2)*[m1 / (m1 + m2)]2*Vi2
Kf = 0.5 [m12 / (m1 + m2)]*Vi2
Now set the kinetic energy up as a ratio, and the 0.5 and Vi2 cancel out, as well as one of the m1 values, leaving you with:
Kf / Ki = m1 / (m1 + m2)
Some basic mathematical analysis will allow you look at the expression m1 / (m1 + m2) and see that for any objects with mass, the denominator will be larger than the numerator. So any objects that collide in this way will reduce the total kinetic energy (and total velocity) by this ratio. We have now proven that any collision where the two objects collide together results in a loss of total kinetic energy.