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Understanding the physics of bungee jumping

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2010 Phys. Educ. 45 63

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Understanding the physics ofbungee jumpingAndre Heck1, Peter Uylings1,2 and Ewa Kedzierska1

1 AMSTEL Institute, University of Amsterdam, Amsterdam, The Netherlands2 Bonhoeffercollege, Castricum, The Netherlands

E-mail: A.J.P.Heck@uva.nl, P.H.M.Uylings@uva.nl and E.Kedzierska@uva.nl

AbstractChanging mass phenomena like the motion of a falling chain, the behaviourof a falling elastic bar or spring, and the motion of a bungee jumper surprisemany a physicist. In this article we discuss the first phase of bungee jumping,when the bungee jumper falls, but the bungee rope is still slack. Ininstructional material this phase is often considered a free fall, but when themass of the bungee rope is taken into account, the bungee jumper reachesacceleration greater than g. This result is contrary to the usual experiencewith free falling objects and therefore hard to believe for many a person, evenan experienced physicist. It is often a starting point for heated discussionsabout the quality of the experiments and the physics knowledge of theexperimentalist, or it may even prompt complaints about the quality ofcurrent physics education. But experiments do reveal the truth and studentscan do them supported by information and communication technology (ICT)tools. We report on a research project done by secondary school students anduse their work to discuss how measurements with sensors, video analysis ofself-recorded high-speed video clips and computer modelling allow study ofthe physics of bungee jumping.

M This article features online multimedia enhancements

The thrilling physics of bungee jumping

Leaping from a tall structure such as a crane ora bridge to which the jumper is attached by hisor her ankles by a large rubber band is a thrillingexperience. This event, better known as bungeejumping, can also serve as an intriguing context forphysics lessons and practical work [1, 2]. Physicscan help to give answers to safety questions like‘How do I know that the rubber band has the rightlength and strength for my jump?’ and ‘How amI sure that the g-forces are kept low enough thatbungee jumping does not hurt?’

A simple energy model of a bungee jump canbe used to generate strain guidelines and practicaldesign equations for the sizing of an all-rubberbungee rope [3]. In many studies (e.g., [1, 4–6]),the motion is considered one dimensional, the ropeis modelled as a massless elastic, the jumper isreplaced by a point mass, aerodynamic effects areignored, and the stress–strain curve of the rope isassumed linear (i.e., Hooke’s law applies). Thebungee jump can then be divided into three phases:(i) a free fall (with acceleration of gravity g) of thejumper when the rope is still slack; (ii) the stretchphase until the rope reaches its maximum length;

0031-9120/10/010063+10$30.00 © 2010 IOP Publishing Ltd P H Y S I C S E D U C A T I O N 45 (1) 63

A Heck et al

and (iii) the rebound phase, consisting of a dampedoscillatory motion.

Several assumptions in this model of bungeejumping can be removed so that the results ofmodels and experiments are in better agreement.Kockelman and Hubbard [7] included the effectsof the elastic properties of the rope, jumper airdrag, and jumper push-off. Strnad [8] describeda theoretical model of a bungee jump that takesonly the mass of the bungee rope into account. Thefirst phase of bungee jumping can also be related toother phenomena such as the dynamics of a falling,perfectly flexible chain suspended at one end andreleased with the two ends near to each other atthe same vertical elevation [9–14]. Experiments,numerical simulations, and analytical modelsdiscussed in the literature (also for discrete modelsof chains) point to the paradoxical phenomenonthat the tip of a freely falling, tightly folded chainwith one end suspended from a rigid supportmoves faster than a free falling body under gravity.This phenomenon is the main subject of thisarticle, but we place it in the context of a researchproject of secondary school students and discusshow technology can contribute to the realizationof such challenging practical investigations.

A secondary school student projectIn the Dutch examination programme of seniorsecondary education, which is organized inso-called profiles consisting of fixed subjectcombinations, students are required to build upan examination portfolio by carrying out somesmall practical investigation tasks and one ratherlarge (80 h), cross-disciplinary research or designassignment. In the ‘Nature and Health’ and‘Nature and Technology’ profiles, teams of twostudents may collaborate in creating their piece ofwork as independent experimental research on atopic of their own choice.

In 2003, Niek Dubbelaar and Remco Brantjes,who were two secondary school students fromthe Bonhoeffer college in Amsterdam, teamedup to investigate the physics of bungee jumping,triggered by their own interest and an article [4]on www.bungee.com. In particular, theywere intrigued by the alleged ‘greater than gacceleration’ of a bungee jumper and, during theirexperimental work, they contacted one of theauthors of a published paper on this subject [14]for more information.

The students formulated the following re-search question: ‘How large is the acceleration ina bungee jump and to what degree is this acceler-ation influenced by the relative mass of the ropeand the jumper?’ Using the analogy of the mo-tion of a bullwhip, they hypothesized that the ac-celeration would be greater than g and that this ef-fect would be more dramatic if the rope was rel-atively heavy as compared with the jumper. Theycollected position–time data through video mea-surements on a dropped scale model (an ActionMan toy figure) and on dropped wooden blocks ofvarious weights attached to ropes of various stiff-nesses. Figure 1 is a sketch of the experimentalsetting, taken from the students’ report.

The velocity and acceleration of the droppedobject were computed by numerical differentia-tion. Soon the students realized that the mass ra-tio between rope and objects was too low to see aclear result and they repeated the experiment withobjects of larger mass ratio. The graph of the ac-celeration at the moment that the block has fallena distance equal to the rest length of the elastic asa function of the mass ratio of the elastic and theblock is shown in figure 2, together with the graphof the following theoretical result:

a = g

(1 + µ (4 + µ)

8

), (1)

where µ is the mass ratio of the elastic and thewooden block. This formula can be found in [14]and on the Internet [15]. The students noted thatthe graphs obtained by measurement and theoryare alike, with the theoretical values just a bithigher. They attributed the difference mainly tothe development of heat during the motion.

Not knowing that a Dutch physics teacherhad published around the same time on anexperimental verification of the physics of bungeejumping [16], the students wrote an article abouttheir work that was published in the journal of theDutch Physics Society [17]. It triggered quite anumber of reactions in the journal and for almost ayear on the Internet. It seemed that a major part ofthe physics community, at all levels of education,were suddenly playing with ropes, chains, elastics,and so on. There were complaints about the qualityof physics teaching in the Netherlands, arguingthat obviously(!) a � g and that the students’work proved that the level of physics education

64 P H Y S I C S E D U C A T I O N January 2010

Understanding the physics of bungee jumping

elastic

points of attachment

weight

platform

ruler

camera

Figure 2. Graphical display of experimental results and fit (purple) and computed values (red).

1.0

4.5

4.0

3.5

3.5

3.0

2.5

2.0

1.5

fit of acceleration (g)

+

3.02.52.01.5

81

in the Netherlands had degraded in the last fewdecades.

The editorial commentary was subtle, but tothe point: ‘The students who wrote the papermay consider it a compliment that scepticismovercame professional physicists and physicsteachers. That’s how (or maybe it is just thepoint that) experienced intuition can be wrong.’

In the same issue, two theoretical physicists [18]agreed with the findings of the students and theyexplained that physics intuition is easily fooled, aseveryone is taught the Galilean paradigm of themotion of constant masses, according to whichevery acceleration must be produced by a force.A launched rocket and a falling chain or slinky areimportant counterexamples to this line of thought.Actually, as we will see in the theoretical section,believing the statement a > g means giving up orgeneralizing the law F = ma.

Other experiments on bungee jumpingAn in-service training module on bungee jumpinghas been developed in the framework of theEuropean project ‘Information Technology forUnderstanding Science’ (IT for US). All teachingand learning activities, which can be downloadedfrom the project’s website [19], are based onthe use of the COACH environment3 [20] for data

3 COACH 6 is a versatile computer learning and authoringenvironment that provides integrated tools for MBL-basedmeasurement, control activities, digital image and videoanalysis, and computer modelling. It has been translated intomany languages, is used in many countries, and the CMAFoundation distributes it. For more information, see www.cma.science.uva.nl.

January 2010 P H Y S I C S E D U C A T I O N 65

A Heck et al

0 0.40

time (s)

a (m

s–2

)

0.3

(b)(a)

0.20.1

4

8

12

16

20

24

28

32

0.5

logging, for video analysis, and for computermodelling, simulation and animation. One of thelaboratory experiments is the measurement of theforce during stretching of the elastic with differentmasses and of the force encountered by jumpers ondifferent bungee ropes. Another bungee jumpingrelated experiment is the measurement of theacceleration of a dropped, chained wooden blockusing an attached accelerometer. Figure 3 showsa measurement result (a) and the experimentalsetting (b). Without doubt, the acceleration isgreater than g and reaches its maximum valuewhen the chain is completely stretched.

Originally, the students made video record-ings of chained objects falling from a height ofabout 4 m with a webcam operating at a speedof 30 frames s−1. This corresponds to a data setof 15 measured positions. The size of this dataset is too small for computing reliable accelera-tions through numerical differentiation. Much bet-ter results could be obtained with a high-speedcamera. However, at the time that the studentsdid their project such cameras were very expen-sive. Nowadays point-and-shoot cameras that canrecord videos at a speed up to 1000 frames s −1 areavailable at a consumer level price.

We tried this out in the following experiment(see figure 4): two identical wooden blocks aredropped at the same time from a height of a coupleof metres. One block is in free fall and the

other block is chained. The chained block touchesthe ground earlier than the block that is in freefall, which can be observed with the naked eyeand can be recorded with a common camcorder.This implies that the chained block must haveacceleration greater than the acceleration of freefall. The motion of the blocks is recorded with ahigh-speed camera at a speed of 300 frames s−1 (avideo clip is available in the online version of thejournal at stacks.iop.org/physed/45/63/mmedia).In the video analysis tool of COACH [20], thevertical position of the blocks can be automaticallymeasured via point tracking. Manual datacollection would be too time consuming.

Figure 5 shows the graphs of the measureddistances of the blocks, relative to the points wherethey were released (i.e., we select a coordinatesystem with a positive vertical coordinate inthe downward direction), and the velocity–timegraphs of the blocks. These graphs havebeen obtained with a numerical differentiationalgorithm that is based on a penalized quinticspline smoothing technique (for details aboutthe point tracking and numerical differentiationalgorithms in COACH, we refer the reader to [21]).The blue velocity–time graph, which is almost astraight line, belongs to the free falling block. Thered graphs, where the cross-hairs in scan modemeet, belong to the chained block that has already

66 P H Y S I C S E D U C A T I O N January 2010

Understanding the physics of bungee jumping

Figure 4. Dropping two wooden blocks simultaneouslyfrom a height of a few metres, while one of the blocksis chained and the other is in free fall.

travelled at the selected moment a greater distance

than the free falling object.

Theoretical underpinning of a > g

Kagan and Kott [14] derived equation (1) byapplying the law of conservation of energy. Thisis correct but it does not give much insight intowhat is really going on. In a more direct approach,Pasveer and de Muynck [15] applied the followingequation of motion:

∑F = dp

dt, (2)

where the left-hand side is the sum of forces Facting on the object and the right-hand side isthe derivative of the momentum p of the movingobject. However, they did not reproduce the resultof Kagan and Kott. We resolve this in the nextsection.

In the case of the chained block we do not dealwith a falling rigid body, but instead with an objectof changing mass, not unlike the moving end of alion tamer’s whip. Therefore, the traditional formof Newton’s second law F = ma is not suitablehere and should be replaced by the followinggeneralized form:

∑F = dpobj

dt= dmobj

dtvobj + mobjaobj, (3)

where mobj, vobj, aobj, and pobj represent themass of the object (changing in time), andthe velocity, acceleration and momentum of theobject, respectively, and F represents a forceacting on the object.

The most interesting object is in this case thewooden block together with its attached chain.The picture of the experimental setting shown infigure 3(b) illustrates that the moving part on theright-hand side diminishes during the fall because

Figure 5. Video analysis of two dropped blocks. The red position and velocity–time graphs relate to the chainedblock and the blue curves belong to the free falling block.

January 2010 P H Y S I C S E D U C A T I O N 67

A Heck et al

M

y

part of the chain ‘moves’ to the left-hand side. Thisimplies

dmobj

dt< 0. (4)

Because∑

F = mobjg when only gravitationalforce is taken into account and v > 0 in thedirection of motion, a > g must hold!

A detailed mathematical model

With the goals in mind of being able to comparetheoretical results with experimental results andbeing able to understand the graphical computermodel shown in the next section, we give a detailedderivation of the equation of motion. Figure 6 is asketch of the situation of a falling chained block.The following symbols are used (numerical valuesapplicable in the experiment and the computermodel are in brackets):

M = mass of the block (0.125 kg);m = mass of the chain (0.68 kg);µ = m/M = the chain : block mass ratio;L = length of the chain (4.15 m);g = acceleration due to gravity (9.81 m s−2);a = acceleration of the chained block;v = speed of the chained block;y = distance travelled by the block.

The object under consideration is the right-hand side consisting of the chained block and the

moving part of the chain. We call this the free sideof the bend. Thus

mobj = M + 1

2(L − y)

m

L,

dmobj

dt= −mv

2L.

(5)

The left-hand side of equation (3) is not as simpleas it may seem at first sight. Of course agravitational force acts on the chain on the freeside of the bend and friction forces, but as Calkinand March [9] pointed out, there is also a nonzerotension on this part, which additionally pulls thechain down.

We consider in this article an alterna-tive perspective, similar to the viewpoint ofBiezeveld [16]: the free side of the bend falls withspeed v, the fixed side of the bend hangs still,and the bend, where links of the chain in mo-tion come to rest, moves at speed u = 1

2v. Inequation (3), vobj denotes the velocity with whichthe mass leaves the moving system. In our case,this velocity therefore almost instantaneously de-creases from v to 0 and is taken to be the aver-age value, i.e., the speed of the bend. We ignorefriction forces and only take the gravitational forceinto account:

∑F = mobjg, vobj = u = 1

2v, aobj = a.

(6)It is noted that Pasveer and de Muynck [18]erroneously used vobj = v. Substitution ofequations (5) and (6) into equation (3) gives

a = g +12µv2

µ (L − y) + 2L. (7)

Instead of considering the velocity v as a functionof time we can also consider it as a function of thevertical position y:

a = dv

dt= dv

dy

dy

dt= v

dv

dy= 1

2

dv2

dy. (8)

Combining equations (7) and (8) we get thefollowing linear, first-order ODE:

dv2

dy+ µv2

µ (y − L) − 2L= 2g. (9)

A person who has already a fair amount ofknowledge of calculus can easily solve the initial

68 P H Y S I C S E D U C A T I O N January 2010

Understanding the physics of bungee jumping

value problem with v2(0) = 0. Others may needa computer algebra system. The solution of thedifferential equation is

v2 = gy4L + µ (2L − y)

2L + µ (L − y). (10)

Substitution of equation (10) into equation (7)gives

a = g

(1 + µy (4L + µ (2L − y))

2 (µ (L − y) + 2L)2

). (11)

Taking y = L in equation (11) gives equation (1).An analytical formula for the time T needed

for the chained block to reach its lowest pointcan be found with a computer algebra systemlike MAPLE. As Strnad [8] showed, this formulaneeds the notion of elliptic functions and is beyondsecondary school level. However, two interestinglimiting cases for the falling time T are the freefall of an object over a distance L (µ ↓ 0) and thefalling chain fixed on one side and free on the otherside (µ → ∞):

limµ↓0

T =√

2L

g,

limµ→∞ T ≈ 0.847

√2L

g.

(12)

This illustrates that when an object and a chain oflength L that is fixed at height L on one side and isheld up on the other side are released from heightL at the same time, the chain reaches the groundearlier than the free falling object.

Computer modelling and simulationSecondary school students are most probably notable to solve the differential equation (9) by hand.But even if they have the knowledge of calculus,it still does not give formulae for the verticalposition, velocity, and acceleration as functions oftime. To obtain these, the nonlinear, second-order,ordinary differential equation (7) in y(t) must besolved for the initial values a(0) = v(0) = 0.It suffices to find a numerical solution and themodelling tool of COACH 6 brings this within reachof secondary school students.

Biezeveld [16] used the text-based version ofthe modelling tool, which is in fact programming

in a computer language that is dedicated tomathematics, science and technology education.The authors take the view that the systemdynamics-based graphical mode of modelling,which is similarly implemented in modelling toolssuch as STELLA and POWERSIM, is simpler forstudents and accessible at secondary school level(see also, for example, [22]). One of the argumentsis that this graphical representation symbolizesboth the system of equations and the numericalalgorithm used to solve it, which seems to makeit easier for students to build their own modelsand to achieve results of good quality. A user canexpress his or her thoughts about the behaviour ofa dynamic system in the graphical representation,and these ideas are then automatically translatedinto more formal mathematical representations.

The upper left corner of the screen shot infigure 7 is an example of a graphical model.It computes the motion of a free falling blockand a chained block according to the previouslypresented theory. For example, the second formulain equation (5) is behind the outflow dm dt , andthe formula g+0.5m ′

objv/mobj is behind the inflowa.

The graphical model in fact represents acomputer model, which provides in many casesan iterative numerical solution of a systemof differential equations, e.g., via a Runge–Kutta algorithm for integrating the correspondingdifferential equation.

In figure 7 also shown are the position andvelocity–time graphs of a simulation run and thegraph of the ratio a/g, which increases while thechained block is falling. Parameter values havebeen chosen such that the model-based graphsfor the chained block are in good agreementwith the graphs obtained through measurements.Prediction and measurement match very well: thetime that the chained block needs to reach itslowest position according to equation (12) forthe given masses and chain length is equal tothe measured time and to the time found in asimulation run within an error margin of 1%!

AnimationThe computer model can also be used to createan animation of the motion of the chained andfree falling block. The tool windows on the right-hand side of figure 7 are a slider and an animationwindow that displays the simulation results as

January 2010 P H Y S I C S E D U C A T I O N 69

A Heck et al

Figure 7. Screen shot of a COACH activity in which a graphical model implements the motion of a chained block (1)and a free falling block (2). The position and velocity–time graphs of a simulation run have been plotted.Parameter values are chosen such that the calculated plots for the chained block match well with the measured datashown as background point plots. The graphical model is connected with a slider and an animation window.

animations where model variables are presented asanimated graphics objects. A student can interactwith the model and the animation through a sliderbar, that is, select the value of the mass of thechain before the start of the simulation and alsoduring the model run. Animation allows studentsto concentrate on understanding a phenomenonwith the help of simulations before going intothe details of how the simulations have beenimplemented by means of computer models.

ConclusionAdmittedly, the mathematics and physics of thefalling chained block is more complicated thanusually is the case for problems in physicsschoolbooks. The main reasons are that (i)motion of a non-rigid body is involved; and (ii)the factor 1/2 for the velocity at which linksof the chain come to rest at the bend, whichis required in the extra term in the generalizedNewton law, is easily overlooked (as in [18]).Selling points of the students’ project are thatit is much more challenging work than common

practical work, and that it brings both physicsteachers and students down to earth as regards theindiscriminate application of Newton’s second lawF = ma.

Furthermore, theory and experiment supple-ment each other in the activities. We take theview that modelling is not just the understandingof the (computer) model with the hope and expec-tation that nothing went wrong during the theo-retical work; it includes sound understanding ofthe underlying physics principles and of the as-sumptions made in the modelling process, as wellas validation of the model on the basis of experi-ments. The latter point is in our opinion essentialin good physics education. The words of the No-bel Prize winner Martinus Veltman (cf. [23]): ‘Ifone removes experiments, physics becomes reli-gion. Then the facts do not count anymore, butthe opinions of someone who was appointed pope’also hold for physics education.

The main role of technology in the students’investigative work is to allow them to collect real-time data of good quality, to construct and usecomputer models of dynamics systems, and to

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Understanding the physics of bungee jumping

compare results from experiments, models, andtheory with each other. For measuring, data pro-cessing and analysis, modelling, and animation,several tools are available for education. For ex-ample, Sismanoglu et al [24] used a camcorderto record the motion of a falling chain. Usingthe freely available video tool VIRTUALDUB (www.virtualdub.org) they went frame by frame throughthe recorded video clip and manually did measure-ments on each frame. The spreadsheet program EX-

CEL was used for making tables and graphs, and forcomputing velocity and acceleration using finitedifference methods. In other words, these authorsused a set of rather disconnected tools. In such anapproach, in our opinion, one runs the risk that oneends up with a grab bag of tools that are not gearedup to work with each other and all require consid-erable time to familiarize oneself with. The com-puter modelling and construction of an animationdescribed in this article could also have been car-ried out in another computer modelling environ-ment, for example MODELLUS [25]. The drawbackfor education could then be that it is onerous in thisenvironment to compare modelling results with ex-perimental results. In contrast, COACH [20, 26] hasbeen designed with the vision of a hardware andsoftware environment in which tools for measur-ing (sensor based and through video capturing),data processing and analysis, control experiments,modelling and animation are integrated in a singlemultimedia authoring package that supports stu-dents’ learning in an enquiry-based approach toscience education. A learn-once, use-often philos-ophy of educational tools is more easily realizedin such an environment. Another advantage of asingle environment compared to a software suite isthe possibility of combining different tools in oneactivity.

In general, students have a positive attitudetoward the use of technology in science education,especially when they recognize that this allowsthem to do activities similar to those in which‘real’ scientists engage. The satisfaction ofICT-supported investigative work is highest whenexperiment, model and theory are in fullagreement, as is the case in the presented studyof achieving an understanding of the physics ofbungee jumping.

Received 30 July 2009, in final form 14 October 2009doi:10.1088/0031-9120/45/1/007

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real-life phenomena Modelling in Physics andPhysics Education, Proceedings GIREP Conf.2006 ed E van den Berg, T Ellermeijer andO Slooten (Amsterdam: University ofAmsterdam) pp 121–36 www.girep2006.nl/

[23] Mols B 2003 Een gevoelige snaar: Veltman vsDijkgraaf (Touching the right chord: Veltmanvs Dijkgraaf) Natuurwetenschap Techniek 7118–25

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[25] Teodoro T 2006 Embedding modelling in thegeneral physics course: rationale and toolsModelling in Physics and Physics Education,Proc. GIREP Conf. 2006 edE van den Berg, T Ellermeijer andO Slooten (Amsterdam: University ofAmsterdam) pp 66–77 www.girep2006.nl/

[26] Heck A and Ellermeijer T 2010 Mathematicsassistants: meeting the needs of secondaryschool physics education Acta DidacticaNapocensia at press

Andre Heck earned MSc degrees inmathematics and chemistry. He is projectmanager at the Faculty of Science of theUniversity of Amsterdam. His researcharea is the application of ICT inmathematics and science education,especially in practical work.

Peter Uylings graduated in physics andmathematics, and obtained his PhD intheoretical physics. He has a part-timejob as a teacher at a secondary school.His research area is ICT in scienceeducation and science curriculumdevelopment.

Ewa Kedzierska graduated in physics.Her working area has been ICT in scienceeducation and teacher training. She isinvolved in the development of the COACH

environment and of sample curriculummaterials for primary and secondaryscience education.

72 P H Y S I C S E D U C A T I O N January 2010