7 One end of a light elastic string of natural length 3 m and modulus of elasticity 24 N is attached to a fixed point \(O\). A particle \(P\) of mass 0.4 kg is attached to the other end of the string. \(P\) is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the extension of the string is \(x \mathrm {~m}\) the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 64 + 20 x - 20 x ^ { 2 }\).
2. Find the greatest speed of the particle.
3. Calculate the greatest tension in the string.