I don't know what you mean by the direction "away from the mass." There are an infinite number of directions away from a point.
If there is an acceleration of the mass at t=0, there must be a force then, and I don't see how that can happen if the spring is unstretched at that time.
It is not necessary to use Laplace Transforms for this kind of problem. In my opinion, it just confuses matters. In any case, it has been too unclearly stated to be answerable.
A mass 'm'is atached to one of an unstreched spring,spring constant k,at time t=0 the free end of the spring experiences a constant acceleration'a',away from the mass. Using the Laplace transformations,
a)Find the position 'x' of 'm' as a function of time.
b)Determine the limiting form of x(t) for small 't'.
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