question_answer

The ratio of masses of two balls is 2 : 1 and before collision the ratio of their velocities is 1 : 2 in mutually opposite direction. After collision each ball moves in an opposite direction to its initial direction. If e = (5/6), the ratio of speed of each ball before and after collision would be

A) (5/6) times        

B) Equal

C) Not related

D) Double for the first ball and half for the second ball

Correct Answer: A

Solution :

[a] Let masses of the two ball are 2 m and m and their speeds are u and 2u, respectively.

By conservation of momentum,

\[{{m}_{1}}{{\vec{u}}_{1}}+{{m}_{2}}{{\vec{u}}_{2}}={{m}_{1}}{{\vec{v}}_{1}}+{{m}_{2}}{{\vec{v}}_{2}}\]

\[\Rightarrow \,\,\,2mu-2mu=m{{v}_{2}}-2m{{v}_{1}}\Rightarrow {{v}_{2}}=2{{v}_{1}}\]

Coefficient of restitution

\[=-\frac{({{v}_{2}}-{{v}_{1}})}{({{u}_{2}}-{{u}_{1}})}=-\frac{(2{{v}_{1}}+{{v}_{1}})}{(-2u-u)}=\frac{-3{{v}_{1}}}{-3u}=\frac{-3{{v}_{1}}}{-3u}=\frac{{{v}_{1}}}{u}=\frac{5}{6}\] [as  given]

\[\Rightarrow \frac{{{v}_{1}}}{{{u}_{1}}}=\frac{5}{6}=\]ratio of the speed of first ball before and after collision.

Similarly, we can calculate the ratio of second ball before and after collision,

\[\frac{{{v}_{2}}}{{{u}_{2}}}=\frac{2{{v}_{1}}}{2u}=\frac{{{v}_{1}}}{u}=\frac{5}{6}\]