3 A young woman wishes to make a bungee jump. One end of an elastic rope is attached to her safety harness. The other end is attached to the bridge from which she will jump. She calculates that the stretched length of the rope at the bottom of her motion should be 20 m , she knows that her weight is 576 N and the stiffness of the elastic rope is \(90 \mathrm { Nm } ^ { - 1 }\). She has to calculate the unstretched length of rope required to perform the jump safely. She models the situation by assuming the following.

• The rope is of negligible mass.
• Air resistance may be neglected.
• She is a particle.
• She moves vertically downwards from rest.
• Her starting point is level with the fixed end of the rope.
• The length she calculates for the rope does not include any extra for attaching the ends.
  1. (A) Show that the greatest extension of the rope, \(X\), satisfies the equation \(X ^ { 2 } = 256\).
     (B) Hence determine the natural length of rope she needs.
  2. To remain safe she wishes to be sure that, if air resistance is taken into account, the stretched length of the rope of natural length determined in part (i) will not be more than 20 m . Advise her on this point.