7. A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 3 } { 2 } m g\). A particle \(P\) of mass \(m\) is attached to one end of the string. The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and falls vertically. When \(P\) has fallen a distance \(a + x\), where \(x > 0\), the speed of \(P\) is \(v\).

1. Show that \(v ^ { 2 } = 2 g ( a + x ) - \frac { 3 g x ^ { 2 } } { 2 a }\).
2. Find the greatest speed attained by \(P\) as it falls. After release, \(P\) next comes to instantaneous rest at a point \(D\).
3. Find the magnitude of the acceleration of \(P\) at \(D\).