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, mv2
/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!
(republished on 07/13/06)