Equipment for bungee jumping
The vocabulary is new and multisyllabic - slingshotting, sandbagging, bodydipping, vinejumping, part-of-four, and more. The spelling is erratic - bungee, bungi, bungy. And terra firma types would say the trendy activity is a threat to body wholeness. But physicists, of course, calmly regard bungee jumping as a dramatic demonstration of the conversation of energy. They know that gravitational potential energy at the top of the jump is converted to elastic potential energy at the bottom. The basic equations involved have been used for years to describe events in which loads are suddenly applied to springs. The bungee cord is simply a very weak spring yielding large spring deflections and rather small force magnitudes. But for the benefit of those who didn't know this, we'll lay it all out here.
The origin of sport bungee jumping is quite recent, but the activity is related to the centuries-old ritualistic practices of "land divers" of Pentecost Island in the Pacific Archipelago of Vanuatu. There the men demonstrate their courage and offer their injuries to the gods for a plentiful harvest of yams. But it was members of the Oxford University Dangerous Sport Club who, inspired by a film about "vine jumpers, " plummeted off a bridge near Bristol, England, in April 1979 and thereby launched a new worldwide recreational activity. During the 1980s, the sport flourished in New Zealand and France and was brought into the United States by John and Peter Kockelman of California. In the early 1990s facilities sprang up all over this country with cranes, towers, and hot-air balloons serving as platforms. Thousands have now experienced that "ultimate adrenaline rush." Many have tried to describe that exhilaration. All share the post-jump elation and grin. In Fig. 1 the jumper, following countdown, has leapt backwards. In Fig. 2, the jumper is enjoying that motionless instant at the top of the rebound.
Bungee cords have some vague military origin, but today can be purchased from manufacturers who construct them specifically for jumping. They are soft and springy and may stretch to three or four times their free length. The harnesses are related to and derive from mountain climbing equipment, as does the carabiner, which is the principal link between the cord and the harness. Most present-day facilities use redundant connections, that is, double hookups to the jumper's body are provided as shown in Figs. 3 and 4. If a jumper chooses an ankle jump, the body harness is backup; if the body harness is primary, the chest/shoulder harness is secondary.
This article is not a commercial for bungee jumping, but familiarity with the equipment used and the forces involved may surprise some readers and may testify to the safety of this activity as a sport. The activity was banned in France after three deaths in 1989. The Australian government declared a hiatus after an accident in 1990, and the summer of 1992 saw a few accidents in the United States that were given major exposure by the media and caused several state governments to get involved. But the activity is clearly basically safe. All accidents can be traced to human error as related to improper attachment, mismatch between jumper and cord, total height of jump available, misunderstanding or miscalculation of the physics involved, and other lapses. This view is shared by Carl Finocchiaro, a registered professional engineer who operates Sky Tower Engineering Inc. and has been professionally involved in this sport for several years. He is a charter member of the North American Bungee Association and is the original and incumbent chairman of its safety committee. He has stated, "I have investigated many accidents and can confidently conclude that all are caused by human error and not faulty equipment."
Minor injuries such as skin burn, which is caused by gripping the cord, occur when jumpers act contrary to instructions. Some jumpers reported getting slapped in the face by the cord. But serious injury inflicted by the cord, such as strangulation, appears not to happen. This can be explained by a combination of factors, including:
- the cord's minimal torsional stiffness
- some pendulum motion, which tends to keep the cord away from the jumper
- the fact that any entanglement will occur when the cord is slack, and will be gradually and gently unwrapped and forgiven as the cord develops elongation and associated low tensile force.
Some daredevilish embellishments may tempt the adventurous participants. "Slingshotting" (from the ground up), "sandbagging" (jumping with extra weight), and "bodydipping" (over water) are examples. Extreme care and proper application of the physics involved are vitally important in these challenges.
Physics of Bungee Jumping
The principle components in the physics of this sport are the gravitational potential energy of the jumper and the elastic potential of the stretched cord.
Figure 5 depicts a jumper of mass m who is tethered by a bungee cord conveniently attached to the supporting structure on a level with her center of mass.
Figure 6 depicts the jumper just as she has fallen a distance equal to the free length (L) of the cord. This event terminates the free fall, which lasts between one and two seconds.
Figure 7 depicts the jumper at the bottom extremity of the jump. The jumper has fallen a total distance of L + d, the cord has stretched a distance d, and the velocity of everything at that instant is equal to zero.
Energy considerations dictate that the gravitational potential energy of the jumper in the initial state is equal to the elastic potential of the cord in the final state. Therefore:
If we allow the bungee cord to be a linear spring of stiffness K N/m, then
and from this the following quadratic equation is produced:
When a given cord (K, L) is matched with a given person (m), then the d will be determined byWhen a given jump height (L + d) is to be matched with a given person (m), then the stiffness (K) will be determined by
In many cases, the first match is made so that the total fall (L + d) will fit the facility, but in order to show the orders of magnitude involved, consider a hypothetical second match between a person (m) and a jump height (L + d). Suppose a person weighing 667 N is to jump using a 9-m cord which will stretch 18 m and use a jump height of 27 m:
This 200% elongation produces a maximum force three times the jumper's weight. A 300% elongation is a softer ride:
This requires a jump height of 36 m, and a maximum acceleration of 2.7 g's is produced.
Calculations Closer to Reality
In a more realistic vein, two factors must be considered:
- A given facility will have a limited number of cords of differing lengths and stiffnesses.
- Those cords have been found to demonstrate variable stiffness over their range of use.