Question

Two balls, having linear momenta $p_1=p\hat{i}$ and $p_2=-p\hat{i}$, undergo a collision in free space. There is no external force acting on the balls. Let $p_1^{\prime }$ and $p_2^{\prime }$ be their final momenta. The following option figure (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. $p_1^{\prime }=a_1\hat{i}+b_1\hat{j}+c_1\hat{k},p_2^{\prime }=a_2\hat{i}+b_2\hat{j}$
• B. $p_1^{\prime }=c_1\hat{k},p_2^{\prime }=c_2\hat{k}$
• C. $p_1^{\prime }=a_1\hat{i}+b_1\hat{j}+c_1\hat{k},p_2^{\prime }=a_2\hat{i}+b_2\hat{j}-c_1\hat{k}$
• D. $p_1^{\prime }=a_1\hat{i}+b_1\hat{j},p_2^{\prime }=a_2\hat{i}+b_1\hat{j}$

Answer 1

Answer: (A), (D)

Initial momentum of the system $p_1+p_2=0$
$\therefore$ Final momentum $p_1^{\prime }+p_2^{\prime }$ should also be zero.
Option (b) is allowed because if we put $c_1=-c_2\ne 0$,
$p_1^{\prime }+p_2^{\prime }$ will be zero. Similary, we can check other options.