U Physics 12E 7.71 Conservation of Energy: An experimental apparatus with mass

NamaeKana

The question is shown below the --- or this question and answerbook is from U Physics 12E #7.71. I uploaded a JPG that can be seen at http://i43.tinypic.com/35j9jja.jpg

I don't understand this problem. I see that to solve this Conservation of Energy and N2L are merged using x, and that h from Ugrav, and a from F=ma are thus related, but why ? I understand the algebra, but how does this merger model the problem ?

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An experimental apparatus with mass m is placed on a vertical spring of negligible mass and pushed down until the spring is compressed a distance x. The apparatus is then released and reaches its maximum height at a distance h above the point where it is released. The apparatus is not attached to the spring, and at its maximum height it is no longer in contact with the spring. The maximum magnitude of acceleration the apparatus can have without being damaged is a, where a > g.
(a) What should the force constant of the spring be?
(b) What distance x must the spring be compressed initially?
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NamaeKana

o.k. i get it now. we are propagating a to limit the forces so the block/animal launched doesn't break/die. so that's why we pull out x from kx^2=ma then plug x into Uel=Ug. the objective again it to limit. then after we get the final equation, we can pick a k and an x-compression that will work.