6 A light elastic string has natural length 4 m and modulus of elasticity 2 N . One end of the string is attached to a fixed point \(O\) of a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass \(0.1 \mathrm {~kg} . P\) is held at rest at \(O\) and then released. The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(x \mathrm {~m}\).

1. Show that \(v ^ { 2 } = 45 - 5 ( x - 1 ) ^ { 2 }\). Hence find
2. the distance of \(P\) from \(O\) when \(P\) is at its lowest point,
3. the maximum speed of \(P\).