Sweeney

I'm finding it difficult to understand how momentum is conserved in an inelastic collision, but kinetic energy isn't. I understand some energy leaves the system in the form of sound and heat.

If we look at the momentum of the system before and after the collision, P=mV, they are equal, due to CoM. Assuming a constant mass we can also say that the velocity of the system is constant.

Now looking at kinetic energy, E=(1/2)mv^2 we know that it changes. Again assuming a constant mass we can say that v^2 changes and thus v changes. Obviously these two statements contradict each other, but I don't understand where I went wrong.