Changing mass phenomena like the motion of a fallin g chain, the behaviour of a falling elastic bar or spring, and the motion of a bungee jumper su rprise many a physicist. In this paper we discuss the first phase of bungee jumping, when the bungee jumper falls down, but the bungee rope is still slack. In instructional material this phase is often considered a free-fall, but when the mass of the bungee rope is taken into account, the bungee jumper reaches acceleration greater than g. This result is contrary to the usual experience w ith free falling objects and therefore hard to believe by many a person, even by an experienced physicist. It is often a starting point for heated discussions about the qua lity of the experiments and the physics knowledge of the experimentalist, or it may even pr ompt complaints about the quality of current physics education. But experiments do revea l the truth and students can do this supported by ICT tools. We report on a research pro ject done by secondary school students and use their work to discuss how measurements with sensors, video analysis of self-recorded high-speed video clips, and computer modelling allo w studying physics of bungee jumping.
Thrilling physics of bungee jumping A thrilling experience is the leap from a tall structure such as crane or a bridge to which the jumper is attached from his or her ankles by a large rubber band. This event, better known as bungee jumping, can also serve as an intriguing context for physics lessons and practical work [1,2]. Physics can help to give answers to safety questions like “How do I know that the rubber band has the right length and strength for my jump?” and “How am I sure that the g-forces are kept low enough so that bungee jumping does not hurt?”. A simple energy model of a bungee jump can be used to generate strain guidelines and practical design equations for the sizing of an all-rubber bungee rope [3]. In many studies (e.g., [1,4-6]), the motion is considered one-dimensional, the rope is modelled as a massless elastic, the jumper is replaced by a point mass, aerodynamic effects are ignored, and the stress-strain curve of the rope is assumed linear (i.e., Hooke’s law applies). The bungee jump can then be divided into three phases: (i) a free-fall (with acceleration of gravity g) of the jumper, when the rope is still slack; (ii) the stretch phase until the rope reaches its maximum length; and (iii) the rebound phase consisting of a damped oscillatory motion. Several assumptions in this model of bungee jumping can be removed so that the results of models and experiments are in better agreement. Kockelman and Hubbard [7] included effects of elastic properties of the rope, jumper air drag, and jumper pushoff. Strnad [8] described a theoretical model of a bungee jump that takes only the mass of the bungee rope into account. The first phase of bungee jumping can also be related to other phenomena such as the dynamics of a falling, perfectly flexible chain suspended at one end and released with both ends nearby to each other at the same vertical elevation [9-14]. Experiments, numerical simulations, and analytical models discussed in the literature (also for discrete models of chains) point at the paradoxical phenomenon that the tip of a freely falling, tightly folded chain with one end suspended from a rigid support moves faster than a free falling body under gravity. This phenomenon is the main subject of this paper, but we place it in the context of a research project of secondary school students and discuss how technology can contribute to the realization of such challenging practical investigations. A secondary school student project In the Dutch examination programme of senior secondary education, which is organized in so-called profiles consisting of fixed subject combinations, students are required to build up an examination portfolio by carrying out some small practical investigation tasks and one rather large (80 hours), cross-disciplinary research or design assignment. In the Nature & Health and Nature & Technology profiles, usually teams of two students collaborate in creating their piece of work as an independent experimental research in a topic of their own choice. In 2003, Niek Dubbelaar and Remco Brantjes, who were two secondary school students of the Bonhoeffercollege, teamed up to investigate the physics of bungee jumping, triggered by their own interest and an article [4] on www.bungee.com. In particular, they were intrigued by the alleged ‘greater-thang acceleration’ of a bungee jumper and they contacted during their experimental work one of the authors of a published paper on this subject [14] for more information. The students formulated the following research question: “How large is the acceleration at a bungee jump and to what degree is this acceleration influenced by the relative mass of the rope and the jumper?”. Using the analogy of the motion of a bullwhip, they hypothesized that the acceleration would be greater than g and that this effect would be more dramatic in case the rope is relatively heavier compared to the jumper. They collected position-time data through video measurements on a dropped scale model (an Action Man toy figure) and on dropped wooden blocks of various weight attached to ropes of various stiffness. Figure 1 is a sketch of the experimental setting, taken from the students’ report. Figure 1. Sketch of the experimental setting. The velocity and acceleration of the dropped object were computed by numerical differentiation. Soon the students realized that the mass ratio between rope and objects was too low to see an outstanding result and they repeated the experiment with objects of larger mass ratio. The graph of the acceleration at the moment that the block has fallen a distance equal to the rest length of the elastic as a function of the mass ratio of elastic and block is shown in Figure 2, together with the graph of the following theoretical result: ( ) 4 1 8 a g μ μ + = + , (1) where μ is the mass ratio of the elastic and the wooden block. This formula can be found in [14] and on Internet [15]. The students noted that the graphs obtained by measurement and theory are alike, with the theoretical values just a bit higher. They attributed the difference mainly to the development of heat during the motion. Figure 2. Graphical display of experimental results (blue) and computed values (purple). Not knowing that a Dutch physics teacher had published around the same time about an experimental verification of the physics of bungee jumping [16], the students wrote an article about their work that was published in the journal of the Dutch Physics Society [17]. It triggered quite a number of reactions in the journal and for almost a year on Internet. It seemed that a major part of the physics community, at all levels of education, was suddenly playing with ropes, chains, elastics, and so on. There were complaints about the quality of physics teaching in the Netherlands, arguing that obviously(!) g a ≤ and that the students’ work proved that the level of physics education in the Netherlands had decreased in the last decades. The editorial commentary was subtle, but to the point: “The students who wrote the paper may consider it a compliment that scepticism overcame professional physicists and physics teachers. That’s how (or maybe it is just the point that) experienced intuition can be wrong.” In the same issue, two theoretical physicists [18] agreed with the findings of the students and they explained that physics intuition is easily fooled, as everyone is taught the Galilean paradigm of the motion of constant masses, according to which every acceleration must be produced by a force. A launched rocket and a falling chain or slinky are important counterexamples to this line of thought. Actually, as we will see in the theoretical section, believing the statement a g > means giving up or generalizing the law a m F ⋅ = . Other experiments on bungee jumping An in-service training module on bungee jumping has been developed in the framework of the European project Information Technology for Understanding Science (IT for US). All teaching and learning activities, which can be downloaded from the project ́s website [19], are based on the use the COACH environment [20,21] for data logging, video analysis, and for computer modelling, simulation, and animation. One of the laboratory experiments is the measurement of the force during stretching of the elastic with different masses and of the force encountered by jumpers on different bungee ropes. Another bungee jumping related experiment is the measurement of the acceleration of a dropped, chained wooden block through an attached accelerometer. Figure 3 shows a measurement result (a) and the experimental setting (b). Without any doubt, the acceleration is greater than g and reaches maximum value when the chain is completely stretched.
