question_answer

Two spheres A and B of masses m1 and m2 respectively collide. A is at rest initially and B is moving with-velocity v along x-axis. After collision B has a velocity\[\frac{v}{2}\] in a direction perpendicular to the original direction. The mass A moves after collision in the direction

A) same as that of B

B) opposite to that of B

C) \[\theta ={{\tan }^{-1}}\left( \frac{1}{2} \right)\]to the x-axis

D) \[\theta ={{\tan }^{-1}}\left( \frac{-1}{2} \right)\]

Correct Answer: C

Solution :

Here, \[{{\mathbf{p}}_{i}}={{m}_{2}}v\mathbf{i}+{{m}_{2}}\times 0\]                 \[{{\mathbf{p}}_{f}}={{m}_{2}}\frac{v}{2}\mathbf{j}+{{m}_{1}}\times {{v}_{2}}\] Law of conservation of momentum                 \[{{\mathbf{p}}_{i}}={{\mathbf{p}}_{f}}\] \[{{m}_{2}}v\mathbf{i}={{m}_{2}}\frac{v}{2}\mathbf{j}+{{m}_{2}}\times {{v}_{1}}\] \[{{v}_{1}}=\frac{{{m}_{2}}}{{{m}_{1}}}v\mathbf{i}+\frac{{{m}_{2}}}{{{m}_{1}}}\frac{v}{2}\mathbf{j}\] From this equation we can find\[\tan \theta =\frac{1}{2}\] \[\theta ={{\tan }^{-1}}\left( \frac{1}{2} \right)\]to the x-axis.