N smooth spheres are placed in a row and their masses are in the ratio `1 : (1)/(e ): (1)/(e^(2)) …… (1)/(e^(N - 1))` where e = coefficient of restitution. If velocity v is given to the first sphere, find the velocity of the last sphere.

Options

A. v
B. `e^(N_(ɣ))`
C. `e^(N + 1)v`
D. `e^(N - 1)v`

Answer

D

Explanation

Let `v_(2)` be the velocity of 2 after impact. Then by the law of collision `v_(1) - v_(2) = - ev` where `v_(1)` is the velocity of 1 after impact, By the law of conservation of momentum,
`mv = mv_(1) + (mv_(2))/(e) rArr v = v_(1) + (v_(2))/(e)`
`rArr ev = ev_(1) + v_(2)`
`:. - ev_(1) - v_(2) = v_(1) - v_(2)`
`rArr v_(1) = 0 " " :. v_(2) = ev`
Similarly
`v_(3) = ev_(2) = - e^(2) V` and `v_(N) = e^(N - 1)v`