Two balls, having linear momenta $\overrightarrow{\mathbf{p}}_1=p \hat{\mathbf{i}}$ and…

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 $c_2$
  1. $\overrightarrow{\mathbf{p}}_1^{\prime}=a_1 \hat{\mathbf{i}}+b_1 \hat{\mathbf{j}}+c_1 \hat{\mathbf{k}}, \overrightarrow{\mathbf{p}}_2^{\prime}=a_2 \hat{\mathbf{i}}+b_2 \hat{\mathbf{j}}$
  2. $\overrightarrow{\mathbf{p}}_1^{\prime}=c_1 \hat{\mathbf{k}}, \overrightarrow{\mathbf{p}}_2^{\prime}=c_2 \hat{\mathbf{k}}$
  3. $\overrightarrow{\mathbf{p}}_1^{\prime}=a_1 \hat{\mathbf{i}}+b_1 \hat{\mathbf{j}}+c_1 \hat{\mathbf{k}}$
  4. $\overrightarrow{\mathbf{p}}_1^{\prime}=a_1 \hat{\mathbf{i}}+b_1 \hat{\mathbf{j}}$ $\overrightarrow{\mathbf{p}}_2^{\prime}=a_2 \hat{\mathbf{i}}+b_2 \hat{\mathbf{j}}-c_1 \hat{\mathbf{k}}$ $\overrightarrow{\mathbf{p}}_2^{\prime}=a_2 \hat{\mathbf{i}}+b_1 \hat{\mathbf{j}}$

Solution

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).

Asked in: JEE Advanced 2008 (Paper 1)

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