The Physics of Bungee Jumping

What is energy?

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.

Dr Rush.