Step 1: Conceptual Clarification:
The problem presents a contradiction. A "perfectly elastic collision" conserves kinetic energy. Conversely, when particles "stick together," it signifies a "perfectly inelastic collision," where momentum is conserved, but kinetic energy is not. Given the condition that particles adhere post-collision, the scenario must be treated as a perfectly inelastic collision, overriding the "perfectly elastic" designation. The fundamental principle governing collisions within an isolated system is the conservation of linear momentum.
Step 2: Governing Principle:
Conservation of linear momentum dictates that the total momentum prior to a collision equals the total momentum subsequent to it.
\[ \vec{P}_{\text{initial}} = \vec{P}_{\text{final}} \]
\[ m_1 \vec{v}_1 + m_2 \vec{v}_2 = (m_1 + m_2) \vec{v}_f \]
Here, \( \vec{v}_f \) represents the shared final velocity of the combined mass.
Step 3: Detailed Derivation:
Let the initial velocities of the two particles be \( v_1 \) and \( v_2 \). The aggregate initial momentum is \( m_1 v_1 + m_2 v_2 \).
Upon collision, the particles coalesce, forming a single entity with mass \( (m_1 + m_2) \) and a uniform final velocity \( v_f \).
The total momentum post-collision is thus \( (m_1 + m_2) v_f \).
Applying momentum conservation:
\[ m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f \]
Rearranging to solve for \( v_f \):
\[ v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \]
Step 4: Conclusion:
The final velocity when particles adhere is determined by the center of mass velocity formula, aligning with option (B).