Question

In a perfectly elastic collision if m₁ and m₂ be the masses and v₁ and v₂ be the velocities of two colliding particles. Then velocity after the collision, if particles stick together will be:

• A. \( \frac{m_1 v_1 + m_2 v_2}{v_1 + v_2} \)
• B. \( \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \)
• C. \( \frac{m_1 v_2}{v_1 + v_2} \)
• D. \( \frac{m_2 v_2}{v_1 - v_2} \)

Hint:

Be aware of contradictory statements in questions. "Sticking together" is the defining characteristic of a perfectly inelastic collision. In such collisions, linear momentum is always conserved, but kinetic energy is lost (converted to heat, sound, etc.). The final velocity is the velocity of the center of mass of the system.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question contains a contradiction. A "perfectly elastic collision" is one where kinetic energy is conserved. A collision where "particles stick together" is a "perfectly inelastic collision", where kinetic energy is not conserved, but momentum is. Since the question asks for the final velocity under the condition that the particles stick together, we must treat it as a perfectly inelastic collision and disregard the "perfectly elastic" part. The governing principle for all collisions in an isolated system is the conservation of linear momentum.
Step 2: Key Formula or Approach:
The principle of conservation of linear momentum states that the total momentum before the collision is equal to the total momentum after the collision.
\[ \vec{P}_{\text{initial}} = \vec{P}_{\text{final}} \] \[ m_1 \vec{v}_1 + m_2 \vec{v}_2 = (m_1 + m_2) \vec{v}_f \] where \( \vec{v}_f \) is the final common velocity of the combined mass.
Step 3: Detailed Explanation:
Let the initial velocities of the two particles be \( v_1 \) and \( v_2 \). The total momentum before the collision is \( m_1 v_1 + m_2 v_2 \).
After the collision, the particles stick together, so they move as a single object with mass \( (m_1 + m_2) \) and a final common velocity \( v_f \).
The total momentum after the collision is \( (m_1 + m_2) v_f \).
By conserving momentum:
\[ m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f \] Solving for the final velocity \( v_f \):
\[ v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \] Step 4: Final Answer:
The velocity after the collision when the particles stick together is given by the formula for the velocity of the center of mass, which matches option (B).