One of the boys came with a question. Apparently, he had to model a bungee jumper with differential equations, and wanted to know how to model air resistance. I told him that, by my calculations, compared to the other forces in the system, air resistance would probably be neglibible.
A bit later, he showed up again; apparently his model was fine, but the jumper would be bounding forever. Which, of course, can’t be right. “So, if it’s not air resistance, where does the energy goes?”, he asked, and I had to fess up: I don’t know.
Checking online, I saw a couple of explanations (it turns into heat, it goes into deforming the rope) which also don’t seem convincing to me (It seems to me that a 160 pounds guy jumping from 80-100 feet would generate too much energy to disipate in heat or friction). So, I ask you: when the jumper finally stops bouncing, where did the energy go?
Lack of perpetual motion machine.
Air resistance isn’t zero. Even in the absence of air, energy is lost as heat to the bungee cord, so it can’t yank you back up 100% to the height from which you came.
Seems to be internal friction/heat losses in the cord and air resistance according to this video. GCSE Science Revision Physics "Energy Transfers: Bungee Jumper" - YouTube
When the guy has stopped bouncing and is just dangling motionless say 40 feet below where he started, the lost energy – originally gravitational potential energy, later converted to kinetic + potential energy of the elastic, has been transformed to a combination of thermal energy plus the remaining potential energy in the stretched elastic. There is nowhere else the energy could have gone.
Think of throwing a 160-pound rock from a height of 100 ft. It hits the ground with a thump. All the energy is gone. If it seems like “a lot” of energy, why doesn’t the rock feel hot? Most of the answer is that the thermal capacity of 160 pounds of rock is very high. The rock (and some of the surrounding soil) absolutely did get warmer, but not nearly enough to be easily detected by touch. Also, a small amount of thermal energy was dissipated to the air due to air resistance.
The conversion to thermal energy is obvious only when very high speeds are involved, such as a meteorite flaming through the atmosphere and crashing to earth.
In the case of the bungee jumper, I suspect that the vast majority of the energy loss is converted to thermal energy within the bungee cord itself, due to internal friction arising from elastic inefficiency. If you’ve ever broken a piece of metal by wiggling it back and forth, you’ll have noticed that it gets quite warm at the break point. It’s the same idea.
There will also be some small thermal conversion due to air resistance on the jumper’s body, but since the speeds involved here are quite low, the thermal conversion due to air resistance would probably be very minimal.
This is the classic problem in engineering design of shock absorbers for cars. It falls under damped vibrations.
Here are the equations and special cases :
https://tutorial.math.lamar.edu/classes/de/vibrations.aspx
Objects can store thermal energy more easily than you think. We can estimate how much the cord would heat up if all the jumper’s energy went into heating up the cord:
- A 160-pound object dropping 100 feet needs to dissipate about 22 kJ worth of energy.
- The specific heat of rubber compounds is somewhere around 1.5 kJ/kg/°C.
- I would expect the weight of the cords themselves to be somewhere in the range of 5 kg (≈ 10 lbs or so.)
Putting these all together, we find if all of the jumper’s energy went into heating up the cord, it would heat up by about 3°C. This is not a huge difference.
The ground if the cord isn’t long enough.
On the youtube channel “How Ridiculous” they seem to like to throw things from great heights, in case imagination isn’t enough. They’ve branched out into slightly different things, but it all comes down to converting potential energy into kinetic.
Anyway, on several occasions, especially when they are dropping a heavy metal object onto another metal object, they would remark at how warm or even painfully hot the objects had become.
To the OP, everything eventually becomes heat, one way or another, which is why everything eventually comes to a stop.
Imagine a bungee jumper on the Moon. There is no air resistance to slow them, so all the lost energy is being dissipated as heat in the bungee cord.
Something to keep in mind if anyone does try this, that heat will not be absorbed conductively by the atmosphere as it is on Earth, the bungee cord may get hot enough to fail. Safety first!
Heat transfer in this case (Earth’s atmosphere) is primarily by convection (forced convection) and is not “absorbed conductively”
Fair enough, and I waffled between them, as the point is not that it is transfering the heat to another object with the atmosphere as the intermediate fluid, but just to the atmosphere itself (which then would radiatively cool to space). I usually think of convection as a moving fluid carrying heat between different objects, but I guess conduction applies specifically to solids in contact?
Anyway point is, things don’t cool the way we expect them to in a vacuum.
Agreed and fair points.
The reason I know that a solid falling in a gas has convective heat transfer, is from Engineering design of Evaporators (like powder milk evaporators) and Urea Prilling tower, and NASA/Dippin Dots icecream. In these cases, a liquid is sprayed that changes phase partly due to convection heat transfer (as it falls).
Gases have thermal conductivity too which become evident in some cases like thin films (relevant to semiconductor manufacture). Hydrogen has one of the highest thermal conductivity and is used in big power generators to cool the cores. Since hydrogen is a small molecule, it has the added advantage of going into every nook and cranny and cooling it.
Band name!
Yep. Other energy losses include deformation of the rock, noise from the rock hitting the ground, deformation of the surface, heat created in the elastic due to internal friction, permanent deformation of the bungee if any, and heat applied to the atmosphere from friction. And once the mass stops bouncing up and down, some of the energy is still retained in the bungee as it is still being stretched by the mass even if it’s not moving.
Air resistance may be a significant factor depending on the shape of the mass being dropped and the distance. The effect would be energy loss through the heating of air molecules, mostly affecting the first bounce cycle or two. I think over 100m you could lose 5% of velocity or so to,air resistance. That’s a significant damping factor. I’m not sure offhand how it compares to other energy absorbing aspects of the setup.
It’s also going to be hard to figure dynamically, as air resistance goes up with the square of velocity, and when you are moving at the fastest is when the cord is doing the least deformation. When you are at your slowest is when the cord is at its greatest.
Then it gets interesting, as when you come back up, the cord will actually lower in temperature, possibly even going to a lower temperature than it initially was, so when you are heading back up at the fastest rate, the cord could actually be absorbing heat from the atmosphere.
But yeah, in the end, all of it gets converted to heat, whether it be from air friction, cord deformation, even hitting a bird on the way down, all of it contributes to converting your initial potential energy into high entropy heat.
Differential equations to the rescue!
You are at correct, at high speeds resistance can be modeled fairly accurately as just as a constant multiplied by the squared of the velocity, and those factors like deformation of the rope and transfer of kinetic energy into heat, can be safely ignored (as they are so small compared to other factors) and still produce an accurate result. And yeah this will get close to zero but never reach zero.
But when it does get velocity (or rather the amplitude of oscillation) does get close to zero then its no longer possible to ignore those factors as they are no longer trivially small compared to the other factors, and they will be what brings the jumper to rest.
There are two places where mechanical energy could be converted to thermal energy: In dissipation in the cord, and in air resistance. So the question is, which one is responsible for the largest share of the heat loss? Air resistance is easy to model, but the dissipation of the cord, not so much. We’d need experiments: Ideally, of a bungee jumper in a vacuum, but failing that, comparing a real bungee jump in air to the models of bungee jumping with air resistance but a perfectly-elastic cord.
There’s a third, actually: through the refrigeration cycle.
As k9bfriender noted, a rubber band actually gets cooler as it’s released. That’s because the force on a rubber band is entropic in nature. The rubber band heats up because the long chains of molecules get stretched, and a long linear molecule has fewer degrees of freedom than the unstretched, jumbled-up state. That creates a force on the band, with the extra energy going to heat, but it’s not friction–it’s reversible (like compressing a piston).
If energy is lost, it’s because it was transferred to the atmosphere, but that’s an external process. The result of the cord cooling is that there’s less force on the upstroke than there was on the downstroke.
Wow, this has been really enlightening. I was really underestimating heat and air resistance. Consider my ignorance fought. (Also, the boy sends his thanks; your links were really helpful, octopus and am77494)
PS: I just showed my wife Chronos’ “We’d need experiments: Ideally, of a bungee jumper in a vacuum”. He response: “Yeah, they’re your people all right”.
I think you mean “… if the cord isn’t short enough.” Excess length is bad and produces sudden stoppages at the bottom. 