Bungee cords operate in a ballistic, gravity-forces dominated regime. The forces, momentums and energies involved are orders of magnitude higher than when operating in the drag- and friction-forces dominated regime of the anchor rodes discussed here.
For example, if we were to drop a 5-ton vessel on a bungee cord the gravity force generates ~1,000 horse power, as opposed to the ~3 horse power generated by the 900 lb drag available during the wind gust.
Remember, the kinetic rope is basically just an energy storage device, just like a battery. You can use it both as a power amplifier (by loading it up slowly and then letting it discharge quickly) or as a power reducer (by loading it up quickly and discharging it slowly).
After just 3 seconds, a 5 ton vessel on a bungee cord has a kinetic energy equal to about 612 Watt-hr (or 51 Amp hr in the case of a 12V battery). In other words, the loading up is going so incredibly fast that in the rebound, when the gravity force is no longer accelerating but decelerating, the bungee rope is going to be releasing its energy slower than it accumulated it.
In the case of the highly elastic anchor rode behavior calculated by Alain Fraisse, the elastic cord is clearly acting as a power amplifier.
This would all seem to be pretty straightforward, so I am going to stop keeping the mechanics 101 discussion alive.
I am grateful for the many thoughtful and informative responses received to this topic and, by way of a wrap-up, was hoping to focus on a few practical implications. For now, however, the local sewer line here in Summit Park (Wasatch Mountains, Utah) appears to have decided to start backing up into our home which requires my undivided attention. Do I smell a few puns coming in?
Happy flushes,
FD
Flying Dutchman