Question

Two balls, having linear momenta ${\overrightarrow{p}}_1=p\hat{i}$ and ${\overrightarrow{p}}_2=-p\hat{i}$, undergo a collision in free space. There in son external force acting on the ball. Let ${\overrightarrow{p}}_1$ and ${\overrightarrow{p}}_2$be their final momenta. Which of 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. ${\overrightarrow{p}}_1=a_1\hat{i}+b_1\hat{j}+c_1\hat{k},{\overrightarrow{p}}_2=a_2\hat{i}+b_2\hat{j}$
• B. ${\overrightarrow{p}}_1=c_1\hat{k},{\overrightarrow{p}}_2=c_2\hat{k}$
• C. ${\overrightarrow{p}}_1=a_1\hat{i}+b_1\hat{j}+c_1\hat{k},{\overrightarrow{p}}_2=a_2\hat{i}+b_2\hat{j}-c_2\hat{k}$
• D. ${\overrightarrow{p}}_1=a_1\hat{i}+b_1\hat{j},{\overrightarrow{p}}_2=a_2\hat{i}+b_1\hat{j}$

Answer 1

Answer: (A), (D)

${\overrightarrow{p}}_1=a_1\hat{i}+b_1\hat{j},{\overrightarrow{p}}_2=a_2\hat{i}+b_1\hat{j}$
${\overrightarrow{p}}_1=a_1\hat{i}+b_1\hat{j}+c_1\hat{k},{\overrightarrow{p}}_2=a_2\hat{i}+b_2\hat{j}$

as given

$\overrightarrow{P}=P\hat{i}$

$\overrightarrow{P_2}=-P\hat{i}$

$\therefore$ Net Linear Momentum first before collision

$\overrightarrow{Pi}=\overrightarrow{P_1}+\overrightarrow{P_2}=P\hat{i}-P\hat{i}=0$

$\because$ No ecternal force is acting on the balls, momentum will remain conserved.

$P_f^{\prime }=\overrightarrow{P_1}+\overrightarrow{P_2}=0$

$\therefore$ In option $\left(A\right)=\overrightarrow{P_1}+\overrightarrow{P_2}=(a_1+a_2)\hat{i}+(b_1+b_2)\hat{i}+c_1\hat{k}\ne 0$

$Q{c}_1$ in non zero

$\therefore$ In option $\left(D\right)\Rightarrow \overrightarrow{P_1}+\overrightarrow{P_2}=(a_1+a_2)\hat{i}+2b_1\hat{i}\ne 0$

$Q{b}_1$ is non zero