Bungee!

A man of height \(h_{0}\) \(=\) \(2\) \(m\) is bungee jumping from a platform situated at a height \(h\) \(=\) \(25\) \(m\) above a lake. One end of an elastic rope is attached to his foot and the other end is fixed to the platform. He starts falling from rest in a vertical position. The length and elastic properties of the rope are chosen so that his speed will have been reduced to zero at the instant when his head reaches the surface of the water. Ultimately the jumper is hanging from the rope, with his head \(8\) \(m\) above the water. Find the unstretched length of the rope (in m) ?

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