Question

Given below are two statements, one is labelled as Assertion (A) and the other is labelled as Reason (R).

Three identical spheres of the same mass undergo one-dimensional motion as shown in the figure with initial velocities \( v_A = 5 \, \text{m/s}, v_B = 2 \, \text{m/s}, v_C = 4 \, \text{m/s} \). If we wait sufficiently long for elastic collision to happen, then \( v_A = 4 \, \text{m/s}, v_B = 2 \, \text{m/s}, v_C = 5 \, \text{m/s} \) will be the final velocities. Reason (R): In an elastic collision between identical masses, two objects exchange their velocities. In light of the above statements, choose the correct answer from the options given below:

• A. Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
• B. (A) is true but (R) is false
• C. Both (A) and (R) are true and (R) is the correct explanation of (A)
• D. (A) is false but (R) is true

Hint:

In elastic collisions for identical masses, the velocities exchange. This is true when the objects collide directly in a one-dimensional motion.

Answer 1

The Correct Option is D

Step 1: In an elastic collision for the same mass, the velocities of the objects interchange. Before the collision: \[ v_A = 5 \, \text{m/s}, \, v_B = 2 \, \text{m/s}, \, v_C = 4 \, \text{m/s} \] After the collision: \[ v_A' = 2 \, \text{m/s}, \, v_B' = 4 \, \text{m/s}, \, v_C' = 5 \, \text{m/s} \]

Step 2: The assertion states that after a sufficiently long time, the velocities will be \( v_A = 4 \, \text{m/s}, v_B = 2 \, \text{m/s}, v_C = 5 \, \text{m/s} \). This holds true as per the elastic collision property, where velocities exchange.

Step 3: The reason also holds true, as in an elastic collision between identical masses, the velocities of the objects do indeed exchange. However, the assertion describes a situation where the objects have reached final velocities after sufficient time. Therefore, the reason is not the correct explanation for the assertion, which is why option (4) is correct.