Two masses `m_1 and m_2` re connected by a spring of spring constant k and are placed on as frictionless horizontal surface. Initially the spring is stretched through a distance `x_0` when the system is released from rest. Find the distance moved by the two masses before they again come to rest.

Explanation

Mass of the two blocks asre `m_1,m_2`
Inititally the spring is stretched by `x_0` with spring constant K.
For the block to come to rest again
let the distance travelled by `m_1 and m_2` be `x_1 and x_2` towards right and left respectively. As no exterN/Al force acts in horizontal direction
`m_1x_1=m_2x_2` .........1
Again the energy would be consrved in the spring
`rarr (1/2)kxx x_0^2=(1/2)k(x_1_x_2-x_0)^2`
`rarr x_0=x_1+x_2-x_0`
`rarrx_1+x_2=2x_0`...........2

`x_1=2x_0x_2`
`similarly x_1=((2m_2)/(m_1+m_2)x_0`
`rarr m_1(2x_0-x_0)=m_2x_2`
`rarr 2m_1x_0m-1x_2m_2x_2`
rarr x_2=(2m_2)/(m_1+m_2) x_0`