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:
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In elastic collisions for identical masses, the velocities exchange. This is true when the objects collide directly in a one-dimensional motion.
Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
(A) is true but (R) is false
Both (A) and (R) are true and (R) is the correct explanation of (A)
(A) is false but (R) is true
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The Correct Option isD
Solution and Explanation
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.
