MCQ
The balls, having linear momenta $\vec{p}_1=\hat{p} \hat{i}$ and $\vec{p}_2=-p \hat{i}$, undergo a collision in free space. There is no external force acting on the balls. Let $\vec{p}_1^{\prime}$ and $\vec{p}_2^{\prime}$ be their final momenta. The following option$(s)$ is (are) $NOT ALLOWED$ for any non-zero value of $\mathrm{p}, \mathrm{a}_1, \mathrm{a}_2, \mathrm{~b}_1, \mathrm{~b}_2, \mathrm{c}_1$ and $\mathrm{c}_2$.

$(A)$ $ \overrightarrow{\mathrm{p}}_1^{\prime}=\mathrm{a}_1 \hat{\mathrm{i}}+\mathrm{b}_1 \hat{\mathrm{j}}+\mathrm{c}_1 \hat{\mathrm{k}} $

$ \overrightarrow{\mathrm{p}}_2^{\prime}=\mathrm{a}_2 \hat{\mathrm{i}}+\mathrm{b}_2 \hat{\mathrm{j}}$

$(B)$ $ \overrightarrow{\mathrm{p}}_1^{\prime}=\mathrm{c}_1 \hat{\mathrm{k}} $

$ \overrightarrow{\mathrm{p}}_2^{\prime}=\mathrm{c}_2 \hat{\mathrm{k}}$

$(C)$ $ \overrightarrow{\mathrm{p}}_1^{\prime}=\mathrm{a}_1 \hat{\mathrm{i}}+\mathrm{b}_1 \hat{\mathrm{j}}+\mathrm{c}_1 \hat{\mathrm{k}} $

$ \overrightarrow{\mathrm{p}}_2=\mathrm{a}_2 \hat{\mathrm{i}}+\mathrm{b}_2 \hat{\mathrm{j}}-\mathrm{c}_1 \hat{\mathrm{k}}$

$(D)$ $ \vec{p}_1^{\prime}=a_1 \hat{i}+b_1 \hat{j} $

$ \overrightarrow{\mathrm{p}}_2^{\prime}=a_2 \hat{\mathrm{i}}+b_1 \hat{\mathrm{j}}$

  • A
    $(B)$ and $(D)$
  • $(A)$ and $(D)$
  • C
    $(B)$ and $(C)$
  • D
    $(D)$ and $(A)$

Answer

Correct option: B.
$(A)$ and $(D)$
b
$\overrightarrow{\mathrm{P}}=\overrightarrow{\mathrm{P}}_1+\overrightarrow{\mathrm{P}}_2=\overrightarrow{\mathrm{P}}_1^{\prime}+\overrightarrow{\mathrm{P}}_2^{\prime} $

$ \mathrm{F}_{\text {ext }}=0 $

$ |\overrightarrow{\mathrm{P}}|=0$

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