The Physics of Bungee Jumping
A lack of energy or an excess of energy are expressions that are bandied about daily. But what do we mean precisely by the term?
In physics, energy is just the capacity of a system to do work. That’s still a fairly abstract definition, and this is because energy is a fairly abstract term. Nobody really knows what it is. And why would they? You can’t see or know energy directly, only properties that are related to it. For example: temperature. An object with a higher temperature has more energy than the same object at a lower temperature. But temperature isn’t energy – it’s just a measure of energy. Similarly, with velocity: an object that is moving quickly has more energy (kinetic energy to be precise) than the same object if it is stationary. Velocity is not energy – it is a measure of energy.
And, yet, despite this vagueness we can say some very definite things about energy. Energy may just be a property of a system, but it is a property that is conserved. This is the First Law of Thermodynamics. The total amount of energy in an isolated system (e.g. the universe) does not change. It can be neither created nor destroyed, only transformed from one form to another, from Kinetic to Potential say. We will use the example of a bungee-jumper to illustrate this.
Consider a bunger-jumper about to leap off a bridge. She has a slack rope attached to her body. She and the rope are a system. She is not yet moving so she has no Kinetic Energy (the energy associated with movement). But she does have Potential Energy – this is the energy associated with force (the force being gravity in this case).
The woman jumps. Now as she falls, she experiences a change in Gravitational Potential Energy proportional to the distance she is dropping (higher objects having more
Gravitational Potential Energy than lower ones). This decrease in Potential Energy becomes an increase in Kinetic Energy, which has a measure of velocity. So as the Kinetic Energy increases so too does her velocity.
This velocity continues to increase as she falls. However, after a certain distance the rope she is attached to becomes taut. Now, there is a tension force acting upwards from the rope to negate the gravitational force acting down. They are not equal (which is why she continues to accelerate), but the tension force grows the farther she falls. Consequently, some of the Gravitational Potential Energy that is being lost to Kinetic Energy goes to Elastic Potential Energy (in the rope) as well. The tension force from the rope increases until it equals and exceeds the force of gravity, thus reducing the woman’s speed until she slows down to nothing and starts to move in the opposite direction.
At this point, the Elastic Potential Energy starts decreasing as the ropes becomes slacker (she is moving upwards, after all). Since the rope is slacker, the tension force also starts to dwindle. The woman continues to move upwards, but her speed decreases due to the gravity that is acting downwards. This gravitational force is constant and easily overcomes the tensional force in the rope until the woman again reaches a standstill and starts falling. At this point there is no Elastic Potential Energy – it having been all transferred back to Gravitational Potential Energy.
The height at which the woman starts falling again (and the Kinetic Energy begins to increase) is not same height from which she first jumped, and it’s just as well. If the system were closed like this then any bungee-jumper would be bouncing up and down forever, her energy in continuous flux between Elastic Potential, Gravitational Potential and Kinetic. Instead, energy is lost to air-resistance and friction (in the rope). This dissipation is what ultimately causes the distance between each subsequent bounce (as the process repeats itself) to drop until finally when the tension force in the rope is just enough to equal the gravitational force and the bungee jumper simply hangs there. At this point there is no Kinetic Energy (as there is no movement), and the Gravitational Potential Energy that has been lost (and which hasn’t been converted to friction and air-resistance) is stored as Elastic Potential Energy in the rope, as it will be distended from the weight of the hanging woman.
One subtle point to note when visualising these sorts of examples (and one I often forget) is that just because the net force on an object is zero, doesn’t mean it isn’t moving. Force is proportional to acceleration (the famous F=ma) not speed or velocity. This means that at the moment when the tensional force in the rope equals the gravitational force (and the net force on the bungee jumper is zero) it does not mean that the woman’s speed is zero and immediately stops falling. It just means that there is no acceleration and she will continue to move at whatever speed she is at. Furthermore, even when the net force is upwards (in the favour of the tensional force) she could still be moving downwards for a while – it’s just that the acceleration is now upwards and so the woman’s speed starts decreasing to zero and then increasing in the opposite direction. She eventually moves up, just not immediately. Loading.
Force, Work, Energy in Bungee Jumping
Great question! Let’s look at this one part at a time. First, you asked about the forces on a bungee jumper.
The first force that the bungee jumper experiences is gravity, which pulls down on everything and makes the jumper fall. The gravitational force is almost exactly constant throughout the jump.
During the bungee jumper’s fall, he or she also experiences a force due to air resistance. The faster the jumper is falling, the more the air resistance pushes back opposite to the direction of motion through the air.
The third force the jumper experiences is a spring force due to the bungee cord. The amount that the bungee cord pulls back on the jumper depends on how far the cord has been stretched, i.e. the farther the jumper has fallen, the more the cord pulls back on him or her. Below a certain height, the spring force of the bungee cord pulling up on the jumper exceeds the force of gravity pulling down. In that range, even ignoring the air resistance, the fall slows down, and then starts to reverse, so the jumper heads back up.
Now that you know about the forces, let’s look at the work that is done on the jumper. Each little bit of work done on the jumper changes their kinetic energy, mv 2 /2, where m is their mass and v is their velocity. you calculate that work by multiplying the distance traveled times the component of the force in that direction. You can have negative work if the force and direction of motion are opposite to each other.
So now let’s look at the first fall that the jumper makes. As the jumper falls down, gravity does positive work because the force of gravity points in the same direction that the jumper falls in. The spring force of the bungee cord, however, does negative work on the jumper because the jumper is falling down while the cord is pulling up. The third force, air resistance, also does negative work during the fall since it pushes upwards. As the jumper reverses direction and starts to spring back up, gravity does negative work because the gravitational force pulls down while the jumper is moving up. The spring force does positive work this time because it is in the same direction as the jumper’s motion. However, air resistance still does negative work because now it pushes down on the jumper.
Now to finish off, let’s look at the energy in this situation. There are three types of energy here: potential energy of gravity and in the stretched cord, kinetic energy of the jumper, and thermal (heat) energy of the air and other things. Gravitational potential energy depends on how high of the ground you are, e.g. if you hold a book above your head, that book has more potential energy than a book that is sitting on the ground. The potential energy of the bungee cord depends on how much the cord has been stretched, i.e. the bungee cord has more potential energy when it is stretched out than when it is slack. Kinetic energy depends on how fast you are moving, as we mentioned. One of the most important equations in physics is the work-energy equation.
If there weren’t any air resistance, then we’d have a pretty simple result, because the potential + kinetic energy wouldn’t change. At the top of the fall the jumper isn’t moving, so the kinetic energy is zero. The gravitational potential energy there is large. At the bottom of the fall, again for an instant the jumper isn’t moving and the kinetic energy is zero. There the gravitational potential energy has gone down, but the bungee cord potential energy has gone up so much that the total potential energy is back to the starting value. In between, the jumper has kinetic energy, so the gain of potential energy by the cord in that range isn’t enough to make up for the loss of gravitational potential energy. Basically energy gets exchanged back and forth during the jump, and if air resistance were not present, the bungee jumper’s total energy would remain constant and he or she would continue boinging up and down forever.
However, you know very well that this is not the case! So now let’s take a look at air resistance and its effects on the bungee jumper. y. Air resistance is the main reason that the bungee jumper, on his or her way back up to the top, never quite reaches the place that he or she started the jump from. In fact, as the jumper bounces up and down, each time his or her maximum height gets less and less. This is similar to a ball that bounces lower and lower until it stops bouncing at all. This is because air resistance is working against the bungee jumper (and the bouncy ball) both on the way down and on the way up, i.e. it always does negative work and subtracts from the total energy of the bungee jumper. The jumper comes to rest right at the point where the cord pulls up just as much as gravity pulls down,
Does the energy just go away? Not really- air resistance, like all other friction forces, takes energy out of the big things you can see and dumps it into little jiggles of air molecules and similar small-scale stuff. In real life, there are some other types of friction here too. As the cord stretches and pulls back, there’s some friction inside the cord itself. So even without air, the energy would gradually get dumped into heat.
Thanks for your excellent question!
(published on 10/22/2007)
Follow-Up #1: bungee jumping physics
hi, I was Given a physics project recently in school and I’ve decided to do it on the physics of bungee jumping. I’ve been doing some research and I’m having a little difficulty understanding the equations behind freefall and the constant energy and hooke’s law, all of it really. please help if you can. thank you.
– Aisling Fiuza (age 16)
Ireland
That’s a pretty broad question. Perhaps you could break it down into specifics for us. Meanwhile, I’ve taken our old answer, which had a serious error, and fixed it up so that maybe it will be a little more useful to help get you started.
(published on 09/18/2011)
Follow-Up #2: Newton’s third bungee jump
Hello again. so here goes, could you please explain Newton’s third law in regards to bungee jumping. I’m a little confused on what exactly is the action and the reaction involved. Thank you so much.
– Aisling Fiuza (age 16)
Ireland
Let’s look at the forces on the jumper. There’s gravity from the earth pulling him down. So Newton’s third law says there’s equal gravity from him or her pulling the earth up. Of course the earth is so massive that that force doesn’t make a noticeable acceleration. There’s also bungee cord pull upward on the jumper. Therefore he or she is pulling downward on the cord.
(published on 11/11/2011)
Follow-Up #3: Elastic constant of bungee cord?
I have a little project, i need to construct a bungee cord. The cord will be made of simple materials such as sting and rubber bands. i need to calculate the spring constant of the rubber bands using hooke’s law. After I have determined the spring constant i need to build a mathematical model of the system and accurately determine how many extensions the spring can produce before failure. I also need to construct and simulate the system using computer programs as well do a physical model. If you can assist me in determining the spring constant of the rubber bands (i do not have access to a lab) and do you know any way to accurately determine how many uses i will be able to get from the bungee cable.
– kevin (age 19)
Trinidad
Hi Kevin
You don’t really need a lab in order to determine the elastic constant of a cord. All you need is a meter stick, a bucket, and a liter jar of water. You can do this at home.
Take your cord and tie the bucket to it’s end. Measure the height on a meter stick. Now pour a liter of water (1 kilogram) into the bucket and measure its new height. So 1 kg divided by the difference in the two heights gives you the spring constant in kilograms per meter. If you want to convert this to more conventional units, Newtons per meter, you need to multiply this number by the acceleration due to gravity; g = 9.8 Newtons per kilogram.
If you want to be a bit more accurate and explore the spring constant as a function of weight you should get a bigger bucket and pour a number of liters of water into it, measuring the height each time. A plot of the differences vs the number of liters should be a straight line.
As far as the number of uses before failure, I don’t have any advice other than to determine it empirically. The answer depends on many variables such as type of material, length of extension, etc.
(published on 03/20/2012)
https://adrenalinenow.com/skydiving/articles/the-physics-of-bungee-jumpinghttps://van.physics.illinois.edu/ask/listing/350
