I remember being amazed by the tower jumping of Vanuatu, when I saw it many years ago on a National Geographic episode, long before the modern incarnation of bungee jumping came on the scene. There's just something about jumping off of a tower with only a couple of vines to break your fall, inches above the ground, that's extremely appealing to a 12-year-old. Tower jumping and bungee jumping both rely on the same physical principles (although the tower jumpers have a lot less margin for error!) to reduce the accelerating forces to levels that are survivable, if not comfortable.
One of the major physics faux pas that you'll see in movies is what is known as "the sudden stop." It could be stated something like this: As long as you are caught before you hit the ground you'll be okay. Let's say Lex Luthor throws you off of the top of a 50-story building but Superman, who is waiting on the street below, catches you before impact. Although Superman's arms may slightly reduce the time in which you decelerate to a stop compared to the concrete sidewalk, you will still experience bone-shattering force upon impact. Walking away from the scene won't be an option. The important point is that because Fnet = ma (according to Newton's second law) we can see that a large acceleration must be associated with large net force. And because a = Δv/Δt, to reduce the acceleration for a given change in velocity you have to increase the time of impact.
Bungee jumping cords, (and Vanuatuan vines, to a lesser degree) rely on the principle of elasticity to reduce the forces acting on the jumper. An elastic object is one, like a rubber band or spring, that is able to store energy when it is compressed or extended. Also, an elastic object tends to exert greater force the further it is stretched from a relaxed position. Take a rubber band and stretch it, and you will see that it takes more force to hold it the further you extend it from its relaxed position. Because of this gradual increase of elastic force as a bungee cord extends, the average force for the entire acceleration is reduced and the time of acceleration is increased. Of course, vines are much less flexible than bungee cords and the force exerted to bring a jumper to a stop at the end of a 60-foot tower jump is much greater than the force a bungee cord exerts for a leap from three or four times the height. That's why vine jumpers commonly dislocate joints, and sometimes break bones, even if they never hit the ground.
Elastic objects also have the virtue of being good at storing energy. When vine or bungee jumpers leap from a great height they have a lot of gravitational potential energy. As they fall this is converted into energy of motion, or kinetic energy. According to the principle of conservation of energy, one of the fundamental principles of nature in the universe as we know it, this kinetic energy has to go somewhere when the jumper stops moving. If the stop is extremely abrupt, much of this energy will go into the body of the jumper -- and not in a good way. Conversely, an elastic cord is able to absorb a significant amount of kinetic energy in the form of elastic potential energy, saving the jumper a lot of internal damage and psychological trauma. Assuming, of course, that the cord doesn't break!
Adam Weiner is the author of Don't Try This at Home! The Physics of Hollywood Movies.