1. A particle \(P\), of mass \(m\), is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity kmg.

The other end of the spring is attached to a fixed point \(O\) on a ceiling.
The point \(A\) is vertically below \(O\) such that \(O A = 3 a\)
The point \(B\) is vertically below \(O\) such that \(O B = \frac { 1 } { 2 } a\)
The particle is held at rest at \(A\), then released and first comes to instantaneous rest at the point \(B\).

1. Show that \(k = \frac { 4 } { 3 }\)
2. Find, in terms of \(g\), the acceleration of \(P\) immediately after it is released from rest at \(A\).
3. Find, in terms of \(g\) and \(a\), the maximum speed attained by \(P\) as it moves from \(A\) to \(B\).