Question

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

Answer 1

Answer: (A)

(5/6) times

Answer 2

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

By conservation of momentum

$m_{1}{\overset{\rightarrow}{u}}_{1} + m_{2}{\overset{\rightarrow}{u}}_{2} = m_{1}{\overset{\rightarrow}{v}}_{1} + m_{2}{\overset{\rightarrow}{v}}_{2}$ ⇒

$2mu - 2mu = mv_{2} - 2mv_{1}$⇒ v₂ = 2v₁

Coefficient of restitution =

$- \frac{({\overset{\rightarrow}{v}}_{2} - {\overset{\rightarrow}{v}}_{1})}{({\overset{\rightarrow}{u}}_{2} - {\overset{\rightarrow}{u}}_{1})} = - \frac{(2v_{1} + v_{1})}{( - 2u - u)} = \frac{- 3v_{1}}{- 3u} = \frac{v_{1}}{u} = \frac{5}{6}$

[As $e = \frac{5}{6}$ given]

⇒ $\frac{v_{1}}{u_{1}} = \frac{5}{6} =$ ratio of the speed of first ball before and after collision.

Similarly we can calculate the ratio of second ball before and after collision, $\frac{v_{2}}{u_{2}} = \frac{2v_{1}}{2u} = \frac{v_{1}}{u} = \frac{5}{6}$.