Force, Work, and Energy

Simple question. Okay, not so simple question.

Both momentum and energy must be conserved. Momentum is mv, and kinetic energy is 1/2 mv^2. Just by looking at this, it's not possible that both of them can be conserved, except at very specific velocities. If both quantities are conserved, how come in a collision, momentum is considered the "more important" quantity? Why can kinetic energy just be assumed to disappear into heat? What mathematical basis is there for such an assumption?

You take newton's law that states that F = dp/dt (derivative of momentum on time), after you consider your system of particles entirely. You say there are no external forces (say a collision on vaccum) and clearly dp/dt = 0, so momentum doesn't change.

Also, regarding your problem with two equations and 1 variable (v). You have to see that you are considering multiple particles so you have that the TOTAL momentum and energy is conserved. So you have {v1, v2, v3, ...} with their components on all directions of space (3 equations for momentum) plus the conservation of mechanical energy.