JEE Advanced · Physics · 7. COM & Collisions

Two balls, having linear momenta $\overrightarrow{\mathbf{p}}_1=p \hat{\mathbf{i}}$ and $\overrightarrow{\mathbf{p}}_2=-p \hat{\mathbf{i}}$, undergo a collision in free space. There is no external force acting on the balls. Let $\overrightarrow{\mathbf{p}}_1^{\prime}$ and $\overrightarrow{\mathbf{p}}_2^{\prime}$ be their final momenta. The following option(s) is/are NOT ALLOWED for any non-zero value of $p, a_1, a_2, b_1, b_2, c_1$ and $c_2$

A $\overrightarrow{\mathbf{p}}_1^{\prime}=a_1 \hat{\mathbf{i}}+b_1 \hat{\mathbf{j}}+c_1 \hat{\mathbf{k}}, \overrightarrow{\mathbf{p}}_2^{\prime}=a_2 \hat{\mathbf{i}}+b_2 \hat{\mathbf{j}}$
B $\overrightarrow{\mathbf{p}}_1^{\prime}=c_1 \hat{\mathbf{k}}, \overrightarrow{\mathbf{p}}_2^{\prime}=c_2 \hat{\mathbf{k}}$
C $\overrightarrow{\mathbf{p}}_1^{\prime}=a_1 \hat{\mathbf{i}}+b_1 \hat{\mathbf{j}}+c_1 \hat{\mathbf{k}}$
D $\overrightarrow{\mathbf{p}}_1^{\prime}=a_1 \hat{\mathbf{i}}+b_1 \hat{\mathbf{j}}$
$\overrightarrow{\mathbf{p}}_2^{\prime}=a_2 \hat{\mathbf{i}}+b_2 \hat{\mathbf{j}}-c_1 \hat{\mathbf{k}}$
$\overrightarrow{\mathbf{p}}_2^{\prime}=a_2 \hat{\mathbf{i}}+b_1 \hat{\mathbf{j}}$

Verified Solution

Answer & Solution

Correct Answer

(D) $\overrightarrow{\mathbf{p}}_1^{\prime}=a_1 \hat{\mathbf{i}}+b_1 \hat{\mathbf{j}}$
$\overrightarrow{\mathbf{p}}_2^{\prime}=a_2 \hat{\mathbf{i}}+b_2 \hat{\mathbf{j}}-c_1 \hat{\mathbf{k}}$
$\overrightarrow{\mathbf{p}}_2^{\prime}=a_2 \hat{\mathbf{i}}+b_1 \hat{\mathbf{j}}$

Step-by-step Solution

Detailed explanation

Initial momentum of the system $\overrightarrow{\mathbf{p}_1}+\overrightarrow{\mathbf{p}_2}=0$
$\therefore$ Final momentum $\overrightarrow{\mathbf{p}_1^{\prime}}+\overrightarrow{\mathbf{p}_2^{\prime}}$ should also be zero.
Option (b) is allowed because if we putc $c_1=-c_2 \neq 0, \overrightarrow{\mathbf{p}_1^{\prime}}+\overrightarrow{\mathbf{p}_2^{\prime}}$ will be zero. Similarly, we can check other options.
$\therefore$ correct options are (a) and (d).

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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