A ball collides impinges directly on a similar ball at rest. The first ball is brought to rest after the impact. If half of the kinetic energy is lost by impact, the value of coefficient of restitution `( e)` is

Options

A. `(1)/(sqrt3)`
B. `(sqrt3)/(2)`
C. `(1)/(2 sqrt3)`
D. `(1)/(sqrt2)`

Answer

D

Explanation

`:. Initial K.E. K_(i) = (1)/(2) mu_(1)^(2) + (1)/(2) mu_(2)^(2) = (1)/(2) mu_(1)^(2)`
and final `KE, K_(f) = (1)/(2) mv_(1)^(2) + (1)/(2) mv_(2)^(2) = (1)/(2) mv_(1)^(2)`
Loss in KE, `Delta K = K_(i) - K_(f) = (1)/(2) mu_(1)^(2) - (1)/(2) mu_(2)^(2)`
It is given` (1)/(2) ((1)/(2) m u_(1)^(2)) = (1)/(2) m u_(1)^(2) - (1)/(2) mv_(2)^(2)`
(`because` Haff of tis `KE` is lost by impact)
`u_(1)^(2) = 2 v_(2)^(2)` or `v_(1) = (u)/(sqrt2)`
Hence, coefficient of restitution, `e = |(v_(2) - v_(1))/(u_(1)| - u_(2))| = (v_(2))/(u_(1)) = (1)/(sqrt2)`