5. One end of a light elastic string, of natural length 2.5 m and modulus of elasticity \(30 g \mathrm {~N}\), is fixed to a point \(O\). A particle \(P\), of mass 2 kg , is attached to the other end of the string. Initially, \(P\) is held at rest at the point \(O\). It is then released and allowed to fall under gravity.

1. Show that, while the string is taut, $$v ^ { 2 } = g \left( 5 + 2 x - 6 x ^ { 2 } \right)$$ where \(v \mathrm {~ms} ^ { - 1 }\) denotes the velocity of the particle when the extension in the string is \(x \mathrm {~m}\).
2. Calculate the maximum extension of the string.
3. 1. Find the extension of the string when \(P\) attains its maximum speed.
   2. Hence determine the maximum speed of \(P\).