Understanding Physics Of Bungee Jumping
Understanding Physics of Bungee JumpingAndré Heck,1 Peter Uylings,1,2 and Ewa Kędzierska112AMSTEL Institute, University of Amsterdam, Amsterdam, The NetherlandsBonhoeffercollege, Castricum, The NetherlandsA.J.P.Heck@uva.nl, P.H.M.Uylings@uva.nl, E.Kedzierska@uva.nlAbstractChanging mass phenomena like the motion of a falling chain, the behaviour of a falling elasticbar or spring, and the motion of a bungee jumper surprise many a physicist. In this paper wediscuss the first phase of bungee jumping, when the bungee jumper falls down, but the bungeerope is still slack. In instructional material this phase is often considered a free-fall, but whenthe mass of the bungee rope is taken into account, the bungee jumper reaches accelerationgreater than g. This result is contrary to the usual experience with free falling objects andtherefore hard to believe by many a person, even by an experienced physicist. It is often astarting point for heated discussions about the quality of the experiments and the physicsknowledge of the experimentalist, or it may even prompt complaints about the quality ofcurrent physics education. But experiments do reveal the truth and students can do thissupported by ICT tools. We report on a research project done by secondary school studentsand use their work to discuss how measurements with sensors, video analysis of self-recordedhigh-speed video clips, and computer modelling allow studying physics of bungee jumping.Thrilling physics of bungee jumpingA thrilling experience is the leap from atall structure such as crane or a bridge towhich the jumper is attached from his orher ankles by a large rubber band. Thisevent, better known as bungee jumping,can also serve as an intriguing context forphysics lessons and practical work [1,2].Physics can help to give answers to safetyquestions like “How do I know that therubber band has the right length andstrength for my jump?” and “How am Isure that the g-forces are kept low enoughso that bungee jumping does not hurt?”.A simple energy model of a bungeejump can be used to generate strain guidelines and practical design equations for thesizing of an all-rubber bungee rope [3]. Inmany studies (e.g., [1,4-6]), the motion isconsidered one-dimensional, the rope ismodelled as a massless elastic, the jumperis replaced by a point mass, aerodynamiceffects are ignored, and the stress-straincurve of the rope is assumed linear (i.e.,Hooke’s law applies). The bungee jumpcan then be divided into three phases: (i) afree-fall (with acceleration of gravity g) ofthe jumper, when the rope is still slack;(ii) the stretch phase until the rope reachesits maximum length; and (iii) the reboundphase consisting of a damped oscillatorymotion.Several assumptions in this model ofbungee jumping can be removed so that theresults of models and experiments are inbetter agreement. Kockelman and Hubbard[7] included effects of elastic properties ofthe rope, jumper air drag, and jumper pushoff. Strnad [8] described a theoretical model of a bungee jump that takes only themass of the bungee rope into account. Thefirst phase of bungee jumping can also berelated to other phenomena such as the dynamics of a falling, perfectly flexible chainsuspended at one end and released withboth ends nearby to each other at the samevertical elevation [9-14]. Experiments, numerical simulations, and analytical modelsdiscussed in the literature (also for discretemodels of chains) point at the paradoxicalphenomenon that the tip of a freely falling,tightly folded chain with one end suspend-
ed from a rigid support moves faster than afree falling body under gravity. This phenomenon is the main subject of this paper,but we place it in the context of a researchproject of secondary school students anddiscuss how technology can contribute tothe realization of such challenging practical investigations.A secondary school student projectIn the Dutch examination programme ofsenior secondary education, which is organized in so-called profiles consisting offixed subject combinations, students are required to build up an examination portfolioby carrying out some small practical investigation tasks and one rather large (80hours), cross-disciplinary research or design assignment. In the Nature & Healthand Nature & Technology profiles, usuallyteams of two students collaborate in creating their piece of work as an independentexperimental research in a topic of theirown choice.In 2003, Niek Dubbelaar and RemcoBrantjes, who were two secondary schoolstudents of the Bonhoeffercollege, teamedup to investigate the physics of bungeejumping, triggered by their own interestand an article [4] on www.bungee.com. Inparticular, they were intrigued by the alleged ‘greater-than-g acceleration’ of abungee jumper and they contacted duringtheir experimental work one of the authorsof a published paper on this subject [14]for more information.The students formulated the followingresearch question: “How large is the acceleration at a bungee jump and to what degree is this acceleration influenced by therelative 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 thiseffect would be more dramatic in case therope is relatively heavier compared to thejumper. They collected position-time datathrough video measurements on a droppedscale model (an Action Man toy figure)and on dropped wooden blocks of variousweight attached to ropes of various stiff-ness. 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 thedropped object were computed by numerical differentiation. Soon the students realized that the mass ratio between rope andobjects was too low to see an outstandingresult and they repeated the experimentwith objects of larger mass ratio. Thegraph of the acceleration at the momentthat the block has fallen a distance equal tothe rest length of the elastic as a functionof the mass ratio of elastic and block isshown in Figure 2, together with the graphof the following theoretical result: µ (4 µ ) (1)a g 1 ,8 where µ is the mass ratio of the elastic andthe wooden block. This formula can befound in [14] and on Internet [15]. The students noted that the graphs obtained bymeasurement and theory are alike, with thetheoretical values just a bit higher. Theyattributed the difference mainly to the development of heat during the motion.Figure 2. Graphical display of experimentalresults (blue) and computed values (purple).
Not knowing that a Dutch physicsteacher had published around the sametime about an experimental verification ofthe physics of bungee jumping [16], thestudents wrote an article about their workthat was published in the journal of theDutch Physics Society [17]. It triggeredquite a number of reactions in the journaland for almost a year on Internet. It seemedthat a major part of the physics community, at all levels of education, was suddenlyplaying with ropes, chains, elastics, and soon. There were complaints about the quality of physics teaching in the Netherlands,arguing that obviously(!) a g and that thestudents’ work proved that the level ofphysics education in the Netherlands haddecreased in the last decades.The editorial commentary was subtle,but to the point: “The students who wrotethe paper may consider it a complimentthat scepticism overcame professionalphysicists and physics teachers. That’show (or maybe it is just the point that) experienced intuition can be wrong.” In thesame issue, two theoretical physicists [18]agreed with the findings of the studentsand they explained that physics intuition iseasily fooled, as everyone is taught theGalilean paradigm of the motion of constant masses, according to which everyacceleration must be produced by a force.A launched rocket and a falling chain orslinky are important counterexamples tothis line of thought. Actually, as we willsee in the theoretical section, believing thestatement a g means giving up or generalizing the law F m a .Other experiments on bungee jumpingAn in-service training module on bungeejumping has been developed in the framework of the European project InformationTechnology for Understanding Science (ITfor US). All teaching and learning activities, which can be downloaded from theproject s website [19], are based on the usethe COACH environment [20,21] for datalogging, video analysis, and for computermodelling, simulation, and animation. Oneof the laboratory experiments is the meas-urement of the force during stretching ofthe elastic with different masses and of theforce encountered by jumpers on differentbungee ropes. Another bungee jumping related experiment is the measurement of theacceleration of a dropped, chained woodenblock through an attached accelerometer.Figure 3 shows a measurement result (a)and the experimental setting (b). Withoutany doubt, the acceleration is greater thang and reaches maximum value when thechain is completely stretched.(a)(b)Figure 3. Acceleration of the chained block (a)and the experimental setting (b).Originally, the students made video recordings of chained objects falling from aheight of about 4 m with a webcam operating at a speed of 30 frames per second.This corresponds with a data set of 15measured positions. The size of this dataset is too small to compute reliable accelerations through numerical differentiation.Much better results could be obtained witha high-speed camera. However, at the timethat the students did their project suchcameras were very expensive. Nowadayspoint-and-shoot cameras that can recordvideos at a speed up to 1000 fps are available at consumers’ price level.We tried this out in the following experiment (See Figure 4): Two identicalwooden blocks are dropped at the sametime from a height of a couple of meters.One block is in free fall and the other blockis chained. The chained block touches theground earlier than the block that is in free
fall, which can be observed with the nakedeye and can be recorded with a commoncamcorder. This implies that the chainedblock must have acceleration greater thanthe acceleration of free fall. The motion ofthe blocks is recorded with a high-speedcamera at a speed of 300 frames per second. In the video analysis tool of COACH[20,21], the vertical position of the blockscan be automatically measured via pointtracking. Manual data collection would betoo time consuming.Figure 4. Dropping twowooden blocks simultaneously from a height of afew meters, while one ofthe blocks is chained andthe other is in free fall.Figure 5 shows the graphs of the measured distances of the blocks, relative to thepoints where they were released (i.e., weselect a coordinate system with a positivevertical coordinate in the downward direction), and the velocity-time graphs of theblocks. These graphs have been obtainedwith a numerical differentiation algorithmthat is based on a penalized quintic splinesmoothing technique (for details about thepoint tracking and numerical differentiation algorithms in COACH, we refer to[22]). The blue velocity-time graph, whichis almost a straight line, belongs to the freefalling block. The red graphs, where thecross hairs in scan mode meet, belong tothe chained block that has already travelledat the selected moment a further distancethan the free falling object.Theoretical underpinning of a gKagan and Kott [14] derived Eq. (1) by applying the conservation law of energy. Thisis correct but it does not give much insightin what is really going on. In a more directapproach, Pasveer and de Muynck [15] applied the following equation of motion:dp(2) F dt ,where the left-hand side is the sum offorces F acting on the object and the righthand side is the derivative of the momentum p of the moving object. However, theydid not reproduce the result of Kagan andKott. We repair this in the next section.In the case of the chained block we donot deal with a falling rigid body, but instead with an object of changing mass, notunlike the moving end of a lion tamer’swhip. Therefore, the traditional form ofNewton’s second law F m a is not suitedhere and should be replaced by the following generalized form:dpdm F dtobj dtobj vobj mobj aobj , (3)where mobj, vobj, aobj, and pobj represent themass of the object (changing in time), thevelocity, acceleration, and momentum ofthe object, respectively, and F represents aforce acting on the object.Figure 5. Video analysis of two dropped blocks. The red position and velocity-time graphs belong tothe chained block and the blue curves belong to the free falling block.
The most interesting object is in thiscase the wooden block together with itsattached chain. The picture of the experimental setting shown in Fig. 3(b) illustratesthat the moving part on the right-hand sidediminishes during the fall because part ofthe chain ‘moves’ to the left-hand side.This implies:dmobj 0.(4)dtBecause F mobj g when only gravitational force is taken into account and v 0in the direction of motion, a g must hold!A detailed mathematical modelWith the goals in mind of being able tocompare theoretical results with experimental results and being able to understandthe graphical computer model shown in thenext section, we give a detailed derivationof the equation of motion. Figure 6 is asketch of the situation of a falling chainedblock. The following symbols are used(numerical values applicable in the experiment and the computer model are in brackets):M mass of the block (0.125 kg)m mass of the chain (0.68 kg)µ m /M, mass ratio chain : blockL length of the chain (4.15 m)g acceleration of gravity (9.81 m s-2)a acceleration of the chained blockv speed of the chained blocky distance travelled by the blockFigure 6. Sketch of the block of mass Mattached to a chain of length L that has alreadyfallen a distance y and is travelling at a speed vand acceleration a.The object under consideration is theright-hand side consisting of the chainedblock and the moving part of the chain. Wecall this the free side of the bend. Thus:mobj M 12 ( L y ) dmobjm,L(5)mv .dt2LThe left-hand side of Eq. (3) is not as simple as it may seem at first sight. Of coursea gravitational force acts on the chain onthe free side of the bend and frictionforces, but as Calkin and March [9] pointed out, there is also a nonzero tension onthis part, which extra pulls down the chain.We consider in this paper an alternativeperspective, similar to the viewpoint ofBiezeveld [16]: The free side of the bendfalls with speed v, the fixed side of thebend hangs still, and the bend, where linksof the chain in motion come to rest, movesat speed u 12 v . In Eq. (3), vobj denotes thevelocity by which the mass leaves themoving system. In our case, this velocitytherefore almost instantaneously decreasesfrom v to 0 and is taken to be the averagevalue, i.e., the speed of the bend. Weignore friction forces and only take thegravitational force into account: F mobj g , vobj u 12 v, aobj a . (6)It is noted that Pasveer and de Muynck[18] erroneously used vobj v . Substitutionof Eq. (5) and (6) into Eq. (3) gives:212 µva g .(7)µ ( L y ) 2LInstead of considering the velocity v as afunction of time we can also consider it asa function of the vertical position y:dv dv dydv 1 dv 2a v . (8)dt dy dtdy 2 dyCombining Eq. (7) and (8) we get the following linear, first-order ODE:dv 2µ v2 2g .(9)dy µ ( y L ) 2 LA person who has already a fair amount ofcalculus knowledge can easily solve theinitial value problem with v 2 (0) 0 . Othersmay need a computer algebra system. Thesolution of the differential equation is:4L µ ( 2L y )v 2 gy.(10)2L µ ( L y )
Substitution of Eq. (10) into Eq. (7) gives: µ y ( 4L µ ( 2L y )) . (11)a g 1 2 2L y 2Lµ()() Taking y L in Eq. (11) gives Eq. (1).An analytical formula for the time Tneeded for the chained block to reach itslowest point can be found with a computeralgebra system like MAPLE. As Strnad [8]showed, this formula needs the notion ofelliptic functions and is beyond secondaryschool level. However, two interesting limiting cases for the falling time T are thefree fall of an object over a distance L( µ 0 ) and the falling chain fixed on oneside and free on the other side ( µ ):lim T µ 02L,g(12)2Llim T 0.847.µ gThis illustrates that when an object and achain of length L that is fixed at height Lon one side and is held up on the other sideare released from height L at the sametime, the chain reaches the ground earlierthan the free falling object.Computer modelling and simulationSecondary school students are most probably not able to solve the differential equation (9) by hand. But even if they have thecalculus knowledge, it still does not giveformulas for the vertical position, velocity,and acceleration as functions of time. Tothis end, the nonlinear, second-order, ordinary differential equation (7) must be solved for the initial values a (0) v (0) 0 . Itsuffices to find a numerical solution andthe modelling tool of COACH 6 brings thiswithin reach of secondary school students.Biezeveld [16] used the text-based version of the modelling tool, which is in factprogramming in a computer language thatis dedicated to mathematics, science, andtechnology education. The authors take theview that the system dynamics, graphicalmode of modelling, which is similarlyimplemented in modelling tools such asSTELLA and POWERSIM, is simpler for stu-dents and accessible at secondary schoollevel (See also, for example, [23]). One ofthe arguments is that this graphical representation symbolizes both the system ofequations and the numerical algorithm tosolve it, which seems to make it easier andless error-prone for students to build theirown models. A user can express his or herthoughts about the behaviour of a dynamicsystem in the graphical representation, andthese ideas are then translated into moreformal mathematical representations.The upper-left corner of the screen shotin Figure 7 is an example of a graphicalmodel. It computes the motion of a freefalling block and a chained block according to the previously presented theory. Forexample, the second formula in Eq. (5) isbehind the outflow dm dt, and the formula′ v mobj is behind the inflow a.g 0.5 mobjThe graphical model represents in facta computer model, which provides in manycases an iterative numerical solution of asystem of differential equations, e.g., via aRunge-Kutta algorithm for integrating thecorresponding differential equation.In Figure 7 are also shown the positionand velocity-time graphs of a simulationrun and the graph of the ratio a g , whichincreases while the chained block is falling. Parameter values have been chosensuch that the model-based graphs for thechained block are in good agreement withthe graphs obtained through measurements.Prediction and measurement match verywell: the time that the chained block needsto reach its lowest position according toEq. 12 for the given masses and chainlength is equal to the measured time and tothe time found in a simulation run withinan error margin of one percent!AnimationThe computer model can also be used tocreate an animation of the motion of thechained and free falling block. The toolwindows at the right-hand side of Figure 7are a slider and an animation window thatdisplays the simulation results as animations where model variables are presentedas animated graphics objects. A student
can interact with the model and the animation through a slider bar, that is, select thevalue of the mass of the chain before thestart of the simulation and also during themodel run. Animation allows students tofirst concentrate on understanding a phenomenon with the help of simulations before going into the details of how the simulations have been implemented by meansof computer models.Figure 7. Screen shot of a COACH activity in which a graphical model implements the motion of achained block (1) and a free falling block (2). The position and velocity-time graphs of a simulationrun have been plotted. Parameter values are chosen such that the calculated plots for the chained blockmatch well with the measured data shown as background point plots. The graphical model is connected with a slider and animation window.ConclusionAdmittedly, the mathematics and physicsof the falling chained block is more complicated than usually is the case for problems in physics schoolbooks. The mainreasons are that (i) it is about a motion of anon-rigid body; and (ii) the factor ½ for thevelocity at which links of the chain cometo rest at the bend, which is required in theextra term in the generalized Newton’slaw, is easily overlooked (as in [18]). Selling points of the students’ project are thatit is much more challenging work thancommon practical work, and that it bringsboth physics teachers and students down toearth concerning the indiscriminate application of Newton’s second law F m a .Furthermore, theory and experimentsupplement each other in the activities. Wetake the view that modelling is not only theunderstanding of the (computer) modelwith the hope and expectation that nothingwent wrong during the theoretical work. Itincludes sound understanding of the underlying physics principles and of the assumptions made in the modelling process, aswell as validation of the model on the basisof experiments. The latter point is in ouropinion essential in good physics education. The words of the Nobel Prize winnerMartinus Veltman (cf., [24]), “If one removes experiments, physics becomes religion. Then the facts do not count anymore, but the opinions of someone whowas appointed pope,” also hold for physicseducation.The main role of technology in the students’ investigative work is to allow them
to collect real-time data of good quality, toconstruct and use computer models of dynamics systems, and to compare resultsfrom experiments, models, and theory witheach other. For measuring, data processingand analysis, modelling, and animationseveral tools are available for education.For example, Sismanoglu et al [25] used acamcorder to record the motion of a fallingchain. Using the freely available video toolVIRTUALDUB (www.virtualdub.org) theywent frame by frame through the recordedvideo clip and manually did measurementson each frame. The spreadsheet programEXCEL was used for making tables andgraphs, and for computing velocity and acceleration by finite difference methods. Inother words, these authors used a set ofrather disconnected tools. In such an approach one runs in our opinion the risk thatone ends up with a grab bag of tools thatare not geared with each other and all require considerable time to familiarize with.The computer modelling and constructionof an animation described in this papercould also have been carried out in anothercomputer modelling environment such asfor example MODELLUS [26]. Drawback ineducation could then be that it is cumbersome in this environment to compare modelling results with experimental results. Incontrast, COACH [21,27] has been designedwith a vision of a hardware and softwareenvironment in which tools for measuring(sensor-based and through video capturing), data processing and analysis, controlexperiments, modelling, and animation areintegrated in a single multimedia authoringpackage that support students’ learning inan inquiry-based approach of science education. A learn-once, use-often philosophyof educational tools is in such an environment more easily realized. Another advantage of a single environment comparedto a software suite is the possibility tocombine different tools in one activity.In general, students have a positiveattitude toward the use of technology inscience education, especially when theyrecognize that this allows them to dosimilar activities in which ‘real’ scientistsengage. The satisfaction of ICT supportedinvestigative work is highest when experiment, model, and theory are in full agreement, as is the case in the presented studyof understanding the physics of bungeejumping.References[1] Horton P 2004 Elastic experiment islicensed to thrill Physics Education 39(4) 326-328[2] Turner R and Taylor B 2005 Physicsfairs in the classroom: Bungee ropes& killer tomatoes Physics Education40 (6) 515-516[3] Kockelman J and Hubbard M 2004Bungee jumping cord design using asimple model Sports Engineering 7(2) 89-96[4] Menz P 1993 The physics of bungeejumping The Physics Teacher 31 (8)483-487[5] Palffy-Muhoray P 1993 Problem andsolution: Acceleration during bungeecord jumping American Journal ofPhysics 61 (4) 379 and 381[6] Martin T and Martin J 1994 The physics of bungee jumping Physics Education 29 (4) 247-248[7] Kockelman J and Hubbard M 2005Bungee jump model with increasingstrain-prediction accuracy SportsEngineering 8 (3) 89-96[8] Strnad J 1997 A simple theoreticalmodel of a bungee jump EuropeanJournal of Physics 18 (5) 388-391[9] Calkin M and March R 1989 The dynamics of a falling chain AmericanJournal of Physics 57 (2) 154-157[10] Schagerl, M, Steindl, A., Steiner, Wand Troger, H. 1997 On the paradoxof the free falling folded chain ActaMechanica 125, 155-168[11] Tomaszewski W and Piernaski P 2005Dynamics of ropes and chains: I. Thefall of the folded chain New Journalof Physics 7, article no. 45[12] Tomaszewski W, Piernaski P andGeminard, J-C 2006 The motion of afreely failing chain tip AmericanJournal of Physics 74 (9) 776-783
[13] Wong C and Yasui K (2006) American Journal of Physics 74 (6) 490-496[14] Kagan D and Kott A 1996 Thegreater-than-g-accelerationofabungee jumper The Physics Teacher34 (9) 368-373[15] http://www.darylscience.com/Demos/Bungee.html[16] Biezeveld H 2003 The bungeejumper: A comparison of predictedand measured values. The PhysicsTeacher, 41 (4) 238-241[17] Dubbelaar N and Brantjes R 2003 Devalversnelling bij bungee-jumping[gravitational acceleration in bungeejumping] Nederlands Tijdschrift voorNatuurkunde 69 (10) 316-318[18] Pasveer F and de Muynck W 2003Wat is het verrassende aan bungeejumping [What is surprising aboutbungee jumping?] Nederlands Tijdschrift voor Natuurkunde 96 (12) 394[19] http://www.itforus.oeiizk.waw.pl[20] Coach 6 is a versatile computer learning and authoring environment thatprovides integrated tools for MBLbased measurement, control activities,digital image and video analysis, andcomputer modelling. It has been translated into many languages, it is usedin many countries, and the CMAFoundation distributes it. For moreinformation, see:http://www.cma.science.uva.nl[21] Heck A, Kędzierska E and EllermeijerT 2009 Design and implementation ofan integrated computer working environment Journal of Computers inMathematics and Science Teaching 28(2) 147-161[22] Heck A and Ellermeijer T 2009Giving students the run of sprintingmodels American Journal of Physics77 (11) 1028-1038[23] D’Anna M 2006 Modeling in theclassroom: Linking physics to otherdisciplines and to real-life phenomena. In van den Berg E, Ellermeijer Tand Slooten O (eds.) Modelling inPhysics and Physics Education, Proceedings GIREP Conference 2006(pp. 121-136), Amsterdam: Universityof Amsterdam.http://www.girep2006.nl[24] Mols B 2003 Een gevoelige snaar:Veltman vs Dijkgraaf [Touching theright chord: Veltman vs Dijkgraaf]Natuurwetenschap & Techniek 71 (9)18-25[25] Sismanoglu B, Germano J and Caetano R 2009 A utilização da filmadoradigital para o estudo do movimentodos corpos [Using the camcorder tostudy bodies movement] Revista Brasileira de Ensino de Física 32 (1)article no 1501[26] Teodoro T 2006 Embedding modelling in the general physics course:Rationale & tools. In van den Berg E,Ellermeijer T and Slooten O (eds.)Modelling in Physics and PhysicsEducation, Proceedings GIREP Conference 2006 (pp. 66-77), Amsterdam:University of Amsterdamhttp://www.girep2006.nl[27] Heck A and Ellermeijer T (2009)Mathematics assistants: Meeting theneeds of secondary school physicseducation Acta Didactica Napocensia(accepted for publication)Key words: Video Analysis, NewtonianMechanics, Modelling, Simulation, AnimationPACS: 01.50, 02.60, 45.20
Biographic information of the authorsAndré Heck earned MScdegrees in mathematics andchemistry. He is project manager at the Faculty of Scienceof the University of Amsterdam. His research area is theapplication of ICT in mathematics and science education,especially in practical work.Peter Uylings graduated inphysics and mathematics, andhe obtained his PhD in theoretical physics. He workspart-time as teacher at a secondary school. His researcharea is ICT in science education and science curriculumdevelopment.Ewa Kędzierska graduatedin physics. Her working areahas been ICT in science education and teacher training.She is involved in the development of the Coach environment and of sample curriculum materials for primary andsecondary science education.
Understanding Physics of Bungee Jumping André Heck, 1 Peter Uylings, 1,2 and Ewa K ędzierska 1 1 AMSTEL Institute, University of Amsterdam, Amsterdam, The Netherlands 2 Bonhoeffercollege, Castricum, The Netherlands A.J.P.Heck@uva.nl, P.H.M.Uylings@uva.nl, E.Kedzierska@uva.nl Abstract Changing mass phenomena like the moti
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bungee rope [3]. In many studies (e.g., [1, 4–6]), the motionis 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
Physics 20 General College Physics (PHYS 104). Camosun College Physics 20 General Elementary Physics (PHYS 20). Medicine Hat College Physics 20 Physics (ASP 114). NAIT Physics 20 Radiology (Z-HO9 A408). Red River College Physics 20 Physics (PHYS 184). Saskatchewan Polytechnic (SIAST) Physics 20 Physics (PHYS 184). Physics (PHYS 182).
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Lesson Three: Students fold origami jumping frogs and determine their frog’s jumping rate. Students create tables, graphs, and equations based on their frog's jumping rate. Each students should be given a 3'' x 5'' card and instructions for folding an origami jumping frog. Students should be allowed to help each other with the folding.
Advanced Placement Physics 1 and Physics 2 are offered at Fredericton High School in a unique configuration over three 90 h courses. (Previously Physics 111, Physics 121 and AP Physics B 120; will now be called Physics 111, Physics 121 and AP Physics 2 120). The content for AP Physics 1 is divided
General Physics: There are two versions of the introductory general physics sequence. Physics 145/146 is intended for students planning no further study in physics. Physics 155/156 is intended for students planning to take upper level physics courses, including physics majors, physics combined majors, 3-2 engineering majors and BBMB majors.
us88685734 agma 1003-g 1993 us88685738 agma 2008-b 1990 us88685800 agma 6004-f 1988 us88685801 agma 6017-e 1986 us88685804 agma 6033-b 1998 us88685818 agma 9001-a 1986 de88686927 tgl 18790/03 1972-09 us88687103 a-a-20079 1984-04-16 us88687140 a-a-1953 1982-08-31 us88687157 a-a-1669 1982-10-18 us88687212 a-a-55063 1992-08-13 us88687305 a-a-59606 2002-08-23 us88687309 a-a-59606/2 2002-08-23 .
