A group of 245 Brazilian daredevils recently set a record when they performed a harrowing feat: In one jump, all together, they launched themselves off the edge of a bridge and swung down toward the water. Of course, they were attached to swinging ropes, but even so, the group jumping achievement was not for the faint of heart.
Luckily, they had physics on their side. And while coordinating that many people to jump at once was tricky, the physics involved are relatively straightforward and related to pendulums.
Unlike bungee jumpers, those 245 daredevils weren't relying only on the elasticity of the ropes to absorb the kinetic energy, Carlos Torija Muñoz, a Spanish rope jumper and ski instructor, told Live Science. (Climbing rope has some stretch, but not nearly as much as bungee cord.) They were also relying on pendulum-like swinging, which kept the forces the jumpers experienced manageable. [The 18 Biggest Unsolved Mysteries in Physics]
Falling on a rope
If ropes like the average clothesline had been these jumpers' only energy absorber, they would've been in trouble, because when the rope played out, it would have tensed up suddenly, unlike the relatively gradual tension increase that occurs in a bungee cord.
For example, attached to a single rope, a falling person accelerates at about 32 feet (9.8 meters) per second squared. At the end of the rope, the person stops almost instantly — in a fraction of a second — and the change in velocity is huge. A person falling a distance of about 150 feet (46 m) would be moving at about 70 mph (113 km/h). Stopping in a tenth of a second means feeling about 32 times the acceleration of gravity, which is survivable, but not unlike getting hit by a car.
Newton's second law of motion says that force is equal to mass times acceleration, so a person weighing 154 lbs. (70 kilograms) stopping in 0.1 of a second would feel 21,910 newtons of force. That's about 4,922 lbs.' (2,233 kg) worth — the weight of a smallish Asian elephant. Climbing rope that has some stretch is often rated on "impact force," which is measured in thousands of newtons (kilonewtons) and uses a 176-lb. (80 kg) weight. A good climbing rope stretches enough — about 40 percent — to reduce the impact force to the order of 12,000 Newtons. That's about half the force as in the example above, but it's still a lot, and a heavier person would experience more force.
Another problem is that the rope is attached to the top point of your jump. Once the rope jerks, it swings you back in, like a pendulum. A pendulum's bob, measured from the center line, swings almost (but not quite) the same distance on each side. That means that, if a jumper gets a running start — or even makes a small leap off the edge — they will end up some distance in front of whatever they jumped off of when they get to the end of their rope (no pun intended). Like a pendulum, they swing right back and smack into the building wall or cliff (or bridge, if they do it near a piling or support). Because their momentum is conserved, they'd hit the wall hard. [8 Craziest Skydives of All Time]
Sergey Firsov, a rope jumper in Russia, said rope jumpers employ one of three strategies to avoid these life-threatening perils. From bridges, they often string the ropes under the span of the bridge, attaching the rope on one side, wrapping it under the span of the bridge and tying the other end to the person. This makes the jumpers like pendulums. And because they're jumping off a bridge and not a cliff face, they don't have to worry about hitting anything underneath it. The swinging rope is already under a little bit of tension — there's no hard jerking because the jumper isn't going far to the end of the rope, and it's more like a swing on a playground.
The other two methods involve what are called static lines and dynamic lines, according to Firsov. The static line is strung in one of two ways. One way is to string it from the point of the jump, downward at a shallow angle, to a point far in front of the cliff face or building, like a zip line; the other way is to string it horizontally in front of the jumper. In both cases, the rope that's directly attached to the jumper, called the dynamic line, is linked to the static line. Generally, the static line is less stretchy than the dynamic line.
Act like a pendulum
The bridge jumpers relied on pendulum physics for their jump. Here's how:
When you swing a pendulum, the distance the pendulum swings (or, in this case, the rope jumper) decreases with each swing, according to the laws of physics. This is because a small amount of kinetic energy is lost each time due to friction and air resistance. In addition, pendulums tend to swing on the same plane. They are so good at this, you can use one to prove Earth is rotating. Foucault pendulums are common sights in museums and art installations; the orientation of the swing changes slowly over the course of a day. This happens because the pendulum's swing stays facing the same direction and doesn't spin with Earth underneath it. How fast it makes a circuit depends on latitude. At the poles, Earth rotates underneath it in about 24 hours, and at lower latitudes, it takes longer — in New York, it would take 37 hours, and at the equator, it doesn't appear to rotate at all.
The bridge jumpers aren't swinging for long enough to see this phenomenon, and their ropes aren't hanging from anchors that are frictionless enough to keep swinging for an entire day. But even so, the physics means each jumper will tend to stay in the same "lane." As such, there's less danger that they'll hit each other, as long as they jump in the same exact direction: straight ahead. Jumpers who go at an angle relative to the others run the risk of tangling ropes with their jump mates precisely because their jump would tend to stay oriented in one direction, so their path would cross the others.
For the jumpers launching off cliffs and buildings, the system is set up differently. In that case, a rope called a static line absorbs the energy from the jumper, who is attached to a rope called a dynamic line.
Muñoz demonstrated on video a miniature version of the method a jumper might use in canyons or cliffs: The static line is strung nearly perpendicular to the dynamic line, which is attached to the middle of the static line. A third rope is connected to the jumper for safety and to haul the jumper back. As the jump begins, the ropes form a "T" shape. The jumper launches into the air, and free-falls until the dynamic line goes taut. [How a Skydiver Jumped Without a Parachute (and Survived)]
If the dynamic line were perfectly stiff and attached to something rigid, then the rope would jerk — hard — and injure the jumper. But neither line is completely rigid. The static line bends and absorbs some of the jumper's kinetic energy, and the dynamic line absorbs a bit as well. While not as stretchy as bungee cords, the ropes have enough give to make the transition from free-fall to a pendulum-like motion smoother; that's because the rope extends the time it takes for the jumper to decelerate. (Recall that the more seconds you spend slowing down, the lower your acceleration and the less force you experience.)
That's the point when the jumper starts acting like a pendulum. How much force they feel depends on how far they fall, how long the rope is and how much the static line bends to absorb the initial force of the transition. The bridge rope jumpers in Brazil launched about 100 feet (30 m) into the air; the ropes were about half that length.
Generally, pendulums' speed at the bottom of the swing depends only on the acceleration due to gravity and the length of the string. In the case of the bridge, there is little added energy from the person (or 245 people) falling, so it's a good estimate. Off of a cliff it differs somewhat, becausethere would be a bit of added energy from the initial fall, as there would be if you were to push a playground swing. This is because the cliff jumpers often have more slack on the rope initially.
Don't try this at home
It's very difficult to set up a rope-jumping system, and it's best left to experts. The changes in force on a rope are large, which requires good anchors. For the static and dynamic lines to absorb the force from a jumper swinging, Tarzan-like, and keep the jumper from suffering internal injuries or broken ribs, the anchors work in tandem with pulleys to adjust the tension on the rope. In videos from rope-jumping groups, the elaborate setup is difficult to see and can give the impression that it is simpler than it really is.
Dan Osman, an extreme-sports enthusiast credited with inventing rope jumping, died in 1998 when his rope broke. He was attempting a 1,100-foot (335 m) jump in Yosemite National Park. In Osman's case, an analysis showed that while his "rigging" — the arrangement of the ropes to hold him — was sound, he jumped in a way that, unbeknownst to him, dragged one rope against another. The friction burned the rope enough that it was weakened and snapped when he reached the end and the tension was at a maximum.
Rope jumping hasn't taken off in the United States, in part because many jurisdictions forbid jumping from bridges. Firsov noted that in Russia, the law is much murkier. Even so, he takes safety seriously. "Dan Osman used only a single rope," he said. Firsov uses at least two, because Newton's second law is rather unforgiving.
Originally published on Live Science.