Spring transferring half its energy to each mass?

[frog]

My question is:
Does a spring between two differing masses, each with an initial momentum of zero, transfer half of its stored elastic potential energy to each?

My intuition says yes. But it seems to not be the case. If you apply the conservation of energy and momentum, the answer you get is different than if you take half its stored elastic energy, give it to each object and calculate velocity.

If this is not the case, why?

 

[Nugatory]

My question is:
Does a spring between two differing masses, each with an initial momentum of zero, transfer half of its stored elastic potential energy to each?

My intuition says yes. But it seems to not be the case. If you apply the conservation of energy and momentum, the answer you get is different than if you take half its stored elastic energy, give it to each object and calculate velocity.

If this is not the case, why?

Consider the case in which one of the two masses is enormously greater than the other. For example, one end of the spring is connected to a 1kg weight and the other end is fastened to a wall which is attached to the foundation of a building which is attached to the Earth which has a mass of about $5\times{10}^{24)$ kg. How much does the Earth move under the force of the spring's tension? How much does the 1 kg weight move under that same force? What does that tell you about how much the energy of each changes?