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Two masses m1 and m2 re connected by a s...

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.

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Blocks again come to rest when spring is compressed by `x_(0)`. Since no external force is acting on the system, so there is no change in the position of cm. of the system i.e. `Deltax_(cm)=0` .
Let mass `m_(1)` displaces by `Deltax_(1)`, and `m_(2)` displaces by `Deltax_(2)`, then
We have `Deltax_(1) + Deltax_(2) = 2x_(0)`.........(i)
and `Deltax_( cm) =(m_(1)Deltax_(1) + m_(2)Deltax_(2))/(m_(1) + m_(2))`
As `Deltax_(cm)=0`
`therefore (m_(1)Deltax_(1) +m_(2)Deltax_(2))/(m_(1) + m_(2)) =0`........(ii)
After solving equation (i) & (ii), we get,
`Deltax_(1) =(2m_(2)x_(0))/(m_(1) + m_(2)), Deltax_(2) = (2m_(1)x_(0))/(m_(1) + m_(2))`
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