[Physics] Why do objects always stick together in perfectly inelastic collisions

Question

From what I understand, perfectly inelastic collisions are those in which the maximum possible amount of kinetic energy is lost from the system. This means that the kinetic energy after the collision must be minimized. If this is true, then for the case in which an object with a constant velocity crashes into a stationary object, why is the post-collision kinetic energy minimized when the two objects stick together?

Answer 1

In the center of momentum frame:

$$ \vec p_1 = - \vec p_2 \equiv \vec p$$

The total energy is:

$$ T = \frac{p^2}{2m_1} + \frac{p^2}{2m_2} $$

After the collision:

$$ \vec p_1' = -\vec p_2' \equiv \vec p' $$

and the kinetic energy is:

$$ T' = \frac{p'^2}{2m_1} + \frac{p'^2}{2m_2} =p'^2(\frac 1 {2m_1}+\frac 1 {2m_2})$$

which is clearly minimized by:

$$ p' = 0 $$

which implies the 2 masses have no relative motion.