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

Both (A) and (R) are true but (R) is NOT the correct explanation of (A)

Both (A) and (R) are true and (R) is the correct explanation of (A)

Center of Mass PYQ — JEE Main 2025 · 29 Jan · Shift 2 · Q70 with Solution

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Correct answer: C

Given data and geometry of motion

Three identical spheres move in one dimension with initial velocities: $v_A=5\text{ m/s},\quad v_B=2\text{ m/s},\quad v_C=4\text{ m/s}.$

From the usual figure for such questions, sphere $A$ is behind $B$ , and $B$ is behind $C$ , all moving along the same line to the right.

So initially:

$A$ is fastest,
$C$ is next,
$B$ is slowest.

Hence collisions will occur because faster spheres from behind catch slower ones ahead.

Key fact for 1D elastic collision of identical masses

For a head-on elastic collision between two identical masses, the two bodies exchange velocities.

So the Reason (R) is:

In an elastic collision between identical masses, two objects exchange their velocities.

This statement is true.

Track the sequence of collisions

Since $A$ has speed $5$ m/s and $B$ has speed $2$ m/s, $A$ catches $B$ first.

First collision: $A$ with $B$

Before collision: $v_A=5,\quad v_B=2.$

After elastic collision of identical masses, they exchange velocities: $v_A=2,\quad v_B=5,\quad v_C=4.$

Now check further collisions:

$A=2$ m/s, so it cannot catch anyone.
$B=5$ m/s is behind $C=4$ m/s, so $B$ catches $C$ .

Second collision: $B$ with $C$

Before collision: $v_B=5,\quad v_C=4.$

After exchange of velocities: $v_B=4,\quad v_C=5.$

Thus final velocities become: $v_A=2\text{ m/s},\quad v_B=4\text{ m/s},\quad v_C=5\text{ m/s}.$

Check the Assertion (A)

Assertion says final velocities will be: $v_A=4\text{ m/s},\quad v_B=2\text{ m/s},\quad v_C=5\text{ m/s}.$

But from correct collision analysis, final velocities are: $v_A=2\text{ m/s},\quad v_B=4\text{ m/s},\quad v_C=5\text{ m/s}.$

So Assertion (A) is false.

Check the Reason (R)

The reason states that in an elastic collision between identical masses, velocities are exchanged.

This is a standard correct result for 1D elastic collisions of equal masses.

So Reason (R) is true.

Final conclusion

Assertion (A): False
Reason (R): True

Therefore the correct option is: $\boxed{\text{C}}$

(A) is false but (R) is true

Comparison with stored correct answer

Stored correct answer: C

My derived answer: C

They agree.

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