The ball sticks to block `C`, then block `C`collides elastically , head - on with `A`. If the maximum compression is `x_(0)`, the spring constant `k` is

Options

A. `(2)/(3) (mv_(0)^(2))/(x_(0)^(2))`
B. `(mv_(0)^(2))/(2 x_(0)^(2))`
C. `(mv_(0)^(2))/(3 x_(0)^(2))`
D. `(mv_(0)^(2))/(6 x_(0)^(2))`

Answer

D

Explanation

When a ball sticks to block `C` , the common velocity is
`mv_(0) + m xx 0 = ( m + m) v rArr v = (v_(0))/(2)`
Now after the head - on elastic collision , block `C` stops and `A` acquires velocity `v_(0)//2`.
At the maximum compression , the blocks move together.
`2m (v_(0))/(2) + m xx 0 = 2mv' + mv' rArr v' = (v_(0))/(3)`
`(1)/(2) . 2m((v_(0))/(2))^(2) = (1)/(2) ( 2m + m) v'^(2) + (1)/(2) k x_(0)^(2)`
`k = (mv_(0)^(2))/(6 x_(0)^(2))`