Block `A` is hanging from a vertical spring and is at rest. Block `B` strikes the block `A` with velocity `v` and sticks to it. Then the value of `v` for which the spring just attains natural length is

Options

A. `sqrt((60 mg^2)/(k))`
B. `sqrt((6 mg^2)/(k))`
C. `sqrt((10 mg^2)/(k))`
D. `sqrt((8 mg^2)/(k))`

Answer

B

Explanation

Linear momentum conservation `mV +0= 2 mV implies U. = V/2`
Final K. E. = `1/(2) .2 m(V/2)^2 = (mV^2)/4`
Energy conservation `(mV^2)/4+ 0 = mgx + 1/2kx^2`
` :. kx = mgx = (mg)/k`
`(mV^2)/4 = mg.(mg)/(k)+1/2k((mg)^2)/(k^2)`
`V^2/4= (mg^2)/k+(mg^2)/(2k)=(3mg^2)/(2k)`
`V^2/2= (3mg^2)/kimpliesV=sqrt((6mg^2)/(k))`