Rigid bodies and inelastic collision If two rigid bodies collide - how the collision can be inelastic?  where the energy goes?
If the energy transffered to heat, doesn't it contradict that the bodies are rigid?
 A: Your intuition is mostly correct here. Perfectly rigid bodies would not be able to collide perfectly inelastically (assuming they interact via contact forces)
A perfectly inelastic collision is one where the final total kinetic energy in the center of momentum frame is zero. In the center of momentum frame you would have the two objects move towards each other and then come into contact. Each object needs to lose its kinetic energy. Since $\Delta K = F\Delta x_{CM}$ that means the center of mass each object must move at least a little for work to be done. But if the object is perfectly rigid the center of mass cannot move relative to its boundary, and its boundary can't move because it is in contact with the other object. Without some kind of infinite force the work done would have to be zero and inelastic collision would be impossible. (Actually, this proves that basically any kind of collision is impossible for perfectly rigid bodies!)
If you remove the assumption of contact forces things are a little different. For example, a positron and an electron are both perfectly rigid bodies (because they are point particles), and yet they can collide perfectly inelastically by forming a  positronium particle. The difference is that the electromagnetic force acts over a distance, starting well before they come in contact.
A: If the collision is inelastic, momentum is conserved, but energy is not. Some of the energy is converted to heat (and sound). 
A: Yes,
Two inelastic rigid bodies is an idealization that does not reflect reality. It is just a shortcut for when the details are not important.
