In a perfectly inelastic collision, the colliding particles stick together.

Assume there are two particles with mass and velocity as $m_1, \vec{v_1}$ and $m_2,\vec{v_2}$
If the velocity after the collision is $\vec{u}$
From conservation of momentum, $m_1\vec{v_1}+m_2 \vec{v_2}=(m_1+m_2)\vec{u}$
So $\vec{u}=\frac{m_1\vec{v_1}+m_2 \vec{v_2}}{m_1+m_2}=V_{cm}$

Energy loss due to collision is
$\Delta k=\frac{1}{2}m_1 \vec{v}_1^2 +\frac{1}{2}m_2 \vec{v}_2^2 -\frac{1}{2} (m_1+m_2) \vec{u}^2=\frac{1}{2}\frac{m_1m_2}{m_1+m_2}(\vec{v}_1-\vec{v}_2)^2$