Examining student ability to interpret and use potential energy diagrams for classical systems
aa r X i v : . [ phy s i c s . e d - ph ] S e p Examining student ability to interpret and use potentialenergy diagrams for classical systems
Brian M. Stephanik and Peter S. Shaffer
Department of Physics, University of Washington, Seattle, WA 98195-1560
Abstract.
The Physics Education Group at the University of Washington is examining the extent to which students are ableto use graphs of potential energy vs. position to infer kinematic and dynamic quantities for a system. The findings indicatethat many students have difficulty in relating the graphs to real-world systems. Some problems seem to be graphical in nature( e.g., interpreting graphs of potential energy vs. position as graphs of position vs. time). Others involve relating the graphs tototal, kinetic, and potential energies, especially when the potential energy is negative. The results have implications beyondthe introductory level since graphs of potential energy are used in advanced courses on classical and quantum mechanics.
Keywords: physics education research, student understanding, potential energy, potential energy diagram
PACS:
INTRODUCTION
Potential energy offers a powerful framework withwhich to characterize and understand complex systems.However, this concept also introduces a significant con-ceptual hurdle for many students. Some specific diffi-culties with this concept have been documented in priorstudies [1–4]. Other papers describe instructional strate-gies for teaching energy and potential energy [5–7].This paper describes preliminary results from an in-vestigation by the Physics Education Group at the Uni-versity of Washington (UW) into student thinking aboutpotential energy diagrams ( i.e. , graphs of potential en-ergy of a system vs. position of a particle). The focusis on the ability of introductory students to use potentialenergy diagrams to determine kinematic quantities andto reason about total, kinetic, and potential energies.This investigation was motivated, in part, by researchthat our group and others have been conducting into stu-dent understanding of basic quantum mechanics [8], atopic that is increasingly being covered in introductoryphysics courses. We are interested in probing the ex-tent to which students understand the underlying clas-sical analogues and in using the results to guide the de-sign of instructional materials [9]. A goal is to help stu-dents be able to compare and contrast the predictions ofthe two models. As part of this investigation, we had de-signed questions that asked students to draw potential en-ergy diagrams as a step in helping them sketch the cor-responding classical probability distributions. We found,however, that many students had difficulty with this firststep, even for simple systems [10]. This observation ledus to examine the ability of introductory students to usepotential energy diagrams in classical mechanics.
CONTEXT FOR INVESTIGATION
This investigation has involved more than 500 studentsat the UW in the three-quarter introductory calculus-based sequence for scientists and engineers: Phys 121(mechanics), 122 (E&M) and 123 (waves, optics, mod-ern physics, and quantum mechanics). Most of the stu-dents were enrolled in the regular sections, but somewere in an honors version that covers a greater amountof material in greater depth. (Table 1 summarizes thestudent populations.) The textbooks for these courses in-clude potential energy diagrams [11, 12], although thetime spent in lecture on this topic varied by instructor. Ineach course, students had completed the relevant lectureand tutorial [9] instruction on mechanical energy by thetime the questions were administered.
QUESTIONS USED FOR RESEARCH
Several different questions were administered to stu-dents in a variety of formats, including individual studentinterviews and multiple choice exams. (Table 1 summa-rizes the various formats.) Most questions were designedto probe student ability to relate potential energy dia-grams of classical systems to real-world motions. In eachcase, students are asked to consider a particle that is partof a one-dimensional system in which the energy can betreated as consisting only of potential energy of the sys-tem and translational kinetic energy of the particle. A fewexamples (questions 1–3) are shown in Fig. 1.In questions 1a and 1b, students are shown a potentialenergy diagram and asked about the directions of theacceleration and velocity of the particle when it is at aparticular position. To answer, they can recognize that
ABLE 1.
Student populations and the formats of thequestions that were administered.
Population Question format N
121 Multiple choice final exam 165;183123 Individual student interviews 8123 Online, ungraded quiz 152123 Honors Written, ungraded quiz 34 the kinetic energy is given by the difference betweenthe total and potential energies. Thus, if the particlewere traveling in the positive x –direction it would beslowing down, while if the particle were traveling inthe negative x –direction it would be speeding up. Bothmotions correspond to acceleration in the negative x –direction. Alternatively, students can use the relation F = − dU / dx and Newton’s second law. The direction of thevelocity of the particle cannot be determined.In question 2, students are shown two additionalgraphs and asked which, if either, can represent the samemotion as the original graph. To answer, students can rec-ognize that only graph II has the same kinetic energy ateach point as the original graph. Thus, only graph II cancorrespond to the same motion.In question 3, which was used only during the inter-views, students are given a potential energy diagram andtold the particle is released from rest at x =
23 m. Theyare then asked to describe the subsequent motion of theparticle. In order to answer, students can recognize thatsince the particle is released from rest, the total energyis equal to the potential energy at that position ( i.e., x =
15 m theparticle would obtain a maximum kinetic energy of 5 J (= − ( − )) before turning around at x = IDENTIFICATION OF SPECIFICDIFFICULTIES
Many students had difficulty in answering each of thequestions discussed in the previous section. Some of theproblems seemed to be associated with their ability to ex-tract kinematic information from a potential energy di-agram. Others were related to student ideas about totalenergy or about potential energy, especially when it isnegative. In this section, we discuss some of the morecommon difficulties. The level of instruction and the for-mats of the questions varied significantly, therefore wedo not compare the percentages of correct and incorrectresponses among the various populations. x E tot [ j ou l e s ] Question 1a:
What is the direction of the acceleration of the particle at x = 6 cm? Question 1b:
What is the direction of the velocity of the particle at x = 8 cm? x E tot I x II Question 2:
Which, if either, of these two graphscould represent the same physical system as that in question 1 and have the particle undergo the samemotion ( i.e., have the same speed at every position)? E n e r gy [ j ou l e s ] Position [meters]Potential energy U ( x ) x Question 3:
A particle is released from rest at x = 23 m.Describe the subsequent motion of the particle. FIGURE 1.
Examples of questions administered to students.
Difficulties related to kinematic quantities
Many of the errors that students made on each ofthe questions seemed to be related to their ability to in-fer kinematic quantities from potential energy diagrams.These are discussed below. Difficulties related to studentunderstanding of kinematic concepts ( e.g., confusing ve-locity and acceleration) are not discussed since they aredocumented extensively elsewhere [13].
Belief that potential energy diagrams represent mo-tion only in the positive x –direction Early in the investigation, we asked question 1a, aboutthe direction of acceleration of the particle, but notthe corresponding question about velocity (question 1b).Many of the explanations were based explicitly on mo-tion only in the positive x –direction. For example, “[theacceleration is in the] negative-x direction. It is movingin + x direction, but gaining Potential E, [therefore] los-ing KE, [e.g.,] slowing down.” Although students oftenobtained the correct answer, many seemed to believe that
IGURE 2.
Potential energy diagram drawn by a student foran Earth-ball system near the surface of Earth. potential energy diagrams only represent motion in a sin-gle direction.In order to probe student thinking in greater detail,we later added question 1b. On a version given in thewaves and optics course, about 15% of the students statedexplicitly that the particle travels only in the positivedirection. The reasoning was often circular ( e.g., theparticle speeds up since the kinetic energy increases,therefore the particle must be moving in the positive x –direction). This line of reasoning is consistent withthe results on a multiple-choice version given in themechanics course in which about 55% of the studentsgave a similar answer. Tendency to treat potential energy vs. position as po-sition vs. time
Between 10% and 30% of the students found the ve-locity of the particle from the slope of the potential en-ergy diagram or the acceleration from the second deriva-tive. Typical explanations included “[the acceleration iszero since the] double derivative of the graph at x = iszero.” Some students also described the local minima ofpotential energy diagrams as turn-around points, consis-tent with interpreting the slope as the velocity.In most cases, these responses did not seem to bedue simply to students misreading the question or thelabels on the axes. On the written questions and in theinterviews, students sometimes switched between correctand incorrect interpretations.It is interesting to note that a similar difficulty arosein a sophomore-level quantum mechanics class (not in-cluded in Table 1). These students were asked a differ-ent question in which they had to sketch a potential en-ergy diagram for an Earth-ball system near the surface ofEarth. Roughly 20% of the students drew curved graphsas shown in Fig. 2. (The flat line represents total energy.)A common justification was “[t]he ball starts with a po-tential energy of mgh. As it falls, it accelerates due togravity and the value h of mgh decreases at an increas-ing rate.”
These students appeared to be conveying in-formation about the increasing rate of change of poten-tial energy with respect to time through the slope of thegraph.
Difficulties related tonegative potential energy
Some of the questions used in this study involve po-tential energy diagrams in which the potential energy isnegative in some regions. Roughly 35% of the studentsstruggled in making sense of the negative values. Manyof the errors seemed to reflect a lack of understanding ofthe arbitrary choice of reference value for potential en-ergy [14] as well as the general relationship between thetotal, kinetic, and potential energies. Two common diffi-culties are discussed.
Belief that potential energy cannot be negative
Many students seemed to believe that potential energycannot be negative. This idea was elicited, for example,in question 3, which was used during the interviews.Some students simply stated that the potential energydiagram was not possible: “... you cannot have negativepotential energy in a system.”
Others argued that theexpressions mgh and kx are always positive. (For thesestudents, h was strictly a positive quantity.) They did notseem to recognize that these expressions are a result of aparticular reference value for potential energy.Some students who had studied quantum mechanicsused a different argument. About 10% of the students ona sophomore-level written final exam argued that “clas-sically, there is no way to achieve a negative potentialenergy.” These students regarded negative potential en-ergy as a feature unique to quantum mechanics [10].
Belief that kinetic energy cannot exceed total energy
Some students seemed to recognize that potential en-ergy can be negative but had difficulty in relating thetotal, kinetic, and potential energies when the potentialenergy was negative. For example, one student on ques-tion 3 stated: “[Y]ou placed it there and let it go, whichtells me you didn’t give it any kinetic energy at first. SoI actually know all of the energy [...,] it’s 2 [J]. And at23 [m] it’s 2 [J] of potential energy and 0 [J] of kineticenergy. So it’s only going to move until down at 15 [m]and it has 2 [J] of kinetic, and back up to 2 [J] of poten-tial.”
This student incorrectly stated that the maximumkinetic energy of the particle is 2 J rather than 5 J. Thisresponse and many others revealed a strong belief thatthe numeric value for the kinetic energy cannot exceedthe numeric value for the total energy. Common justifi-cations included “energy cannot be created from noth-ing” and “the total energy of the system encompassesits kinetic energy as well as its PE.”
These students didnot seem to understand how to interpret the relationshipbetween the numeric values for the total, kinetic, and po-tential energies when the potential energy was negative. ifficulties related to total energy
Some of the student responses revealed insight intotheir thinking about the total energy of a system. Thiswas particularly the case for question 2, which asked stu-dents to identify two different potential energy diagramsthat might correspond to the same motion. Between 10%and 25% of student answers were consistent with the twoways of reasoning described below.
Tendency to associate the motion of the particle in agiven system with only the total energy
In comparing the potential energy diagrams in ques-tion 2, some students focused only on the total energy.For example, one student, who chose graph I, stated, “[t]he total energy would [need] to be the same as theprevious graph or else the motion of the particle is notthe same ...”
This response and others suggested a ten-dency to associate the motion of a particle with only thetotal energy of the system. These students failed to rec-ognize that equal shifts in the total and potential energiescould represent the same motion.
Misapplication of conservation of energy
In answering question 2, many students based their ex-planations on conservation of energy. For example, “[thesystem] would have to have the same total energy be-cause energy cannot be created or destroyed ...” or “bythe law of con[s]ervation of energy, all energy is con-served.” These students did not seem to realize that thearbitrary choice of reference value for potential energyresults in an arbitrary total energy. They instead usedconservation of energy, which states that the total energydoes not change with time, to account for differences dueto different reference values for potential energy.
DISCUSSION
This preliminary investigation has revealed a varietyof difficulties that students encounter in interpreting po-tential energy diagrams. These include determining kine-matic quantities as well as relating the total, kinetic, andpotential energies, especially when the potential energyis negative. Underlying many of the responses is a failureto understand how different reference values for potentialenergy do and do not impact the formal description of theenergy and motion of a particle. Moreover, instruction onmore advanced topics may result in additional complica-tions, such as a belief that negative potential energy isonly allowed for quantum mechanical systems.The results of this research have implications for in-struction. For example, on problems involving gravitynear the surface of Earth, potential energy is commonlychosen to be zero at the lowest point of the motion of aparticle. This choice can result in convenient statementssuch as “all of the energy is kinetic at the bottom,” but may hide difficulties that can arise when the potential en-ergy is negative, which is common or required of manysystems. The prevalence of the errors and their persis-tence to the sophomore level, together with observationsof students during the interviews, suggest that the under-lying difficulties are strongly held and are not likely to beeasily addressed. There is a need for additional researchto probe student thinking in greater detail and to identifyinstructional strategies that are effective at helping themdeepen their understanding of the abstract concept of en-ergy.
ACKNOWLEDGMENTS
The authors would like to thank current and past mem-bers of the Physics Education Group and the facultymembers who welcomed this research into their classes.This research would not have been possible without thesupport of the National Science Foundation under grantsDUE-0618185 and DUE-1022449.
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