Figure shows two blocks A an B having mass 2kg and 4 kg moving with a speed `4 m//sec` and `2 m//sec` respectively. The maximum compression in the spring is when k=10

Options

A. `sqrt((24)/(45)) m`
B. `sqrt((24)/(25)) m`
C. `sqrt((45)/(24))m`
D. `sqrt((25)/(24))m`

Answer

A

Explanation

At the point of max. compression both blocks are moving with same speed. Applying momentum conservation
`2 xx 4 + 4 xx 2 = (2 + 4) v`
`8 + 8 = 6 v`
`v = (16)/(6) rArr v = (8)/(3) m//sec`
Now applying energy conservation
`(1)/(2) xx (2) (4)^(2) + (1)/(2) (4) (2)^(2)`
`= (1)/(2) xx 10 xx (x)^(2) + (1)/(2) xx 2 ((8)/(3))^(2) + (1)/(2) (4)^(2) ((8)/(3))^(2)`
`16 + 8 5x^(2) + (64)/(9) + (128)/(9) rArr 5x^(2) = 24 - (192)/(9)`
`rArr 5x^(2) = (216 - 192)/(9) rArr x^(2) = (24)/(45)`
`rArr x^(2) = (8)/(15) rArr x = sqrt((8)/(15)) m "or" x = sqrt((24)/(45) m`