3 Ben has mass 60 kg and he is considering doing a bungee jump using an elastic rope with natural length 32 m . One end of the rope is attached to a fixed point O , and the other end is attached to Ben. When Ben is supported in equilibrium by the rope, the length of the rope is 32.8 m .
To predict what will happen, Ben is modelled as a particle B, the rope is assumed to be light, and air resistance is neglected. B is released from rest at O and falls vertically. When the rope becomes stretched, \(x \mathrm {~m}\) denotes the extension of the rope.
- Find the stiffness of the rope.
- Use an energy argument to show that, when B comes to rest instantaneously with the rope stretched,
$$x ^ { 2 } - 1.6 x - 51.2 = 0$$
Hence find the length of the rope when B is at its lowest point.
- Show that, while the rope is stretched,
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 12.25 x = 9.8$$
where \(t\) is the time measured in seconds.
- Find the time taken for B to travel between the equilibrium position \(( x = 0.8 )\) and the lowest point.
- Find the acceleration of \(\mathbf { B }\) when it is at the lowest point, and comment on the implications for Ben.