European Journal of Physics

The higher derivatives of motion are rarely discussed in the teaching of classical mechanics of rigid bodies; nevertheless, we experience the effect not only of acceleration, but also of jerk and snap. In this paper we will discuss the third and higher order derivatives of displacement with respect to time, using the trampolines and theme park roller coasters to illustrate this concept. We will also discuss the effects on the human body of different types of acceleration, jerk, snap and higher derivatives, and how they can be used in physics education to further enhance the learning and thus the understanding of classical mechanics concepts.

The paper aims to fulfil three main functions: (1) to serve as an introduction for the physics education community to the functioning of large language models (LLMs), (2) to present a series of illustrative examples demonstrating how prompt-engineering techniques can impact LLMs performance on conceptual physics tasks and (3) to discuss potential implications of the understanding of LLMs and prompt engineering for physics teaching and learning. We first summarise existing research on the performance of a popular LLM-based chatbot (ChatGPT) on physics tasks. We then give a basic account of how LLMs work, illustrate essential features of their functioning, and discuss their strengths and limitations. Equipped with this knowledge, we discuss some challenges with generating useful output with ChatGPT-4 in the context of introductory physics, paying special attention to conceptual questions and problems. We then provide a condensed overview of relevant literature on prompt engineering and demonstrate through illustrative examples how selected prompt-engineering techniques can be employed to improve ChatGPT-4's output on conceptual introductory physics problems. Qualitatively studying these examples provides additional insights into ChatGPT's functioning and its utility in physics problem-solving. Finally, we consider how insights from the paper can inform the use of LLMs in the teaching and learning of physics.

Language is an important resource for physicists and learners of physics to construe physical phenomena and processes, and communicate ideas. Moreover, any physics-related instructional setting is inherently language-bound, and physics literacy is fundamentally related to comprehending and producing both physics-specific and general language. Consequently, characterizing physics language and understanding language use in physics are important goals for research on physics learning and instructional design. Qualitative physics education research offers a variety of insights into the characteristics of language and language use in physics such as the differences between everyday language and scientific language, or metaphors used to convey concepts. However, qualitative language analysis fails to capture distributional (i.e. quantitative) aspects of language use and is resource-intensive to apply in practice. Integrating quantitative and qualitative language analysis in physics education research might be enhanced by recently advanced artificial intelligence-based technologies such as large language models, as these models were found to be capable to systematically process and analyse language data. Large language models offer new potentials in some language-related tasks in physics education research and instruction, yet they are constrained in various ways. In this scoping review, we seek to demonstrate the multifaceted nature of language and language use in physics and answer the question what potentials and limitations artificial intelligence-based methods such as large language models can have in physics education research and instruction on language and language use.

The double slit experiment was the first demonstrative proof of the wave nature of light. It was expounded by the English physician-physicist Thomas Young in 1801 and it soon helped lay to rest the then raging Newton–Huygens debate on whether light consisted of a fast-moving stream of particles or a train of progressive waves in the ether medium. In the experiment, light is made to pass through two very narrow slits spaced closely apart. A screen placed on the other side captures a pattern of alternating bright and dark bands called fringes which are formed as a result of the phenomenon of interference. In prior work by the same author, it was shown that the conventional analysis of Young's experiment that is used in many introductory physics textbooks, suffers from a number of limitations in regards to its ability to accurately predict the positions of these fringes on the distant screen. This was owing to the adoption of some needless and paradoxical assumptions to help simplify the geometry of the slit barrier-screen arrangement. In the new analysis however, all such approximations were discarded and a hyperbola theorem was forwarded which was then suitably applied to determine the exact fringe positions on screens of varied shapes (linear, semi-circular, semi-elliptical). This paper further builds on that work by laying down the mathematical framework necessary for counting fringes and then comparing their distributions on differently shaped screens, using MATLAB software package for numerical–graphical simulation. In addition, a pair of equivalent laws of proportionality are predicted that govern the distribution of fringes independent of the shape of the detection screen employed.

A typical undergraduate course in mechanics does not cover the fascinating and important gravity-assist manoeuvre that allows satellites or other spacecrafts to navigate through our solar system on efficient and desired paths. Instead, it usually remains a mystery to students how energy is conserved when a spacecraft gains speed as it flies past a planet. Indeed, one might be led to believe that the curved path of the planet is the root cause for the gain in speed, requiring consideration of gravity-assist within the framework of the restricted three-body problem. This contribution will emphasize that this extension is not required to explain the gain in kinetic energy. Instead, a simple, scaffolded analysis of the planet-satellite system alone, using elementary physics, two reference frames and analytical methods, provides a sufficient explanation. Our simplified analysis is successfully validated against mission data from Voyager 2's gravity-assist manoeuvre around Jupiter.

The 'ball on a string' demonstration is a common tool used in physics education to illustrate the concept of conservation of angular momentum. However, various confounding factors can cause significant deviations from the idealized case, particularly under extreme conditions or when using low-stiffness pivots or high coefficients of friction. These factors include air resistance, contact friction at the pivot point, the mass of the ball and string, the angle of the string due to gravity, and the wobbling of the pivot point due to the centrifugal forces acting on it. In this work, we critically review by means of accurate simulations the adequateness of the 'ball on a string' demonstration in view of these confounding factors and provide recommendations for instructors on how to maximize the educational value of the demonstration while minimizing potential confusion for students. Our analysis suggests that a stiff pivot and avoiding extreme conditions are key to obtaining results that are in good agreement with the idealized case. We also caution instructors against using the demonstration without at least mentioning the confounding factors, as this may lead to a questionable understanding of the underlying physics principles.

The geodynamo usually appears as a somewhat intimidating subject. Its understanding seems to require knowledge of the intricate theory of magnetohydrodynamics. The solution of the corresponding equations can only be achieved numerically. It seems to be a subject for the specialist. We show that one can understand the basics of the functioning of the geodynamo solely by using the well-known laws of electrodynamics. The topic is not only important for geophysicists. The same physics is the cause for the magnetic fields of Sun-like stars, of the very strong fields of neutron stars, and also of the cosmic magnetic fields.

In this research, we investigated the impact of peer collaboration and changes from individual to group performance of graduate students on the conceptual survey of electricity and magnetism (CSEM) without any guidance from the instructor. We define construction of knowledge as a case in which the group answered the question correctly but in the individual administration of the survey before the group work, one member gave the correct answer and the other gave incorrect answer. We find that there was a significant improvement in the performance of students after peer interaction, which was mostly attributed to construction of knowledge. However, students had very few opportunities to co-construct knowledge as there were hardly any situations in which neither student in a group provided a correct answer. We analyzed the effect size for improvement from individual to group scores for each CSEM item to understand the characteristics of these questions that led to productive group interaction. We also compared the group performance of the graduate students to the introductory physics students in a prior study using the CSEM to get insight into the concepts that showed differences for the two groups and those that were challenging for both groups of students before and after collaboration with peers. Our findings can motivate physics instructors to incorporate group interactions both inside and outside of the classroom even without instructor's involvement so that students at all levels can learn from each other and develop a functional understanding of the underlying concepts.

A long-standing controversy concerning the causes of the magnetic field in and around a parallel-plate capacitor is examined. Three possible sources of contention are noted and detailed. The first is the ambiguous initial impression given by the calculation of the magnetic field using the integral form of the Ampere–Maxwell law which incorporates the displacement current density. The second is misinterpretation of this law as a cause–effect formula. The third is insufficient recognition of the fact that the electric field in Maxwell's equations represents the sum of the well-distinguished irrotational and divergence-free fields, which are independently responsible for conservation of charge and the existence of the electromagnetic waves, respectively.

The first historical steps of radioactivity research offer an excellent opportunity to teach a key concept of modern physics: non-deterministic phenomena. However, this opportunity is often wasted because of historical misconceptions and of the irrational fear of radioactive effects. We propose here a lecturing strategy - primarily for undergraduate students - based on interesting historical facts. In particular, on a key conceptual contribution by Marie Curie, an attractive figure for the young women and men of today. Paradoxically, this milestone is almost unknown, whereas it should contribute to her immortal fame -- perhaps as much as the discovery of radium.

The work is devoted to the calculation of the self-inductance of the Möbius strip (MS), assuming that a self-contained surface current flows on its surface. Subsequently, the vector potential corresponding to this situation is expressed in cases where: (a) the surface current is constant (b) the surface current is inversely proportional to the length of the line along which it flows. The self-inductance of the MS is determined by the integration of the vector potential. From the derived relations, the inductance of the MS is determined by computer simulation at different values of the ratio of width and radius of the MS. The reference value to the results for MS is the calculated and shown inductance of the cylindrical surface with a surface current flowing around the circumference of its shell. In conclusion, simple relations are derived that enable quick calculation of the inductances of both the MS and the cylindrical surface from their geometrical parameters. The article is intended for students of mathematical-physical and technical faculties as well as for graduates of these faculties dealing with the issue of (meta) materials.

Conditionally convergent series are infinite series whose result depends on the order of the sum. One of the most famous examples of conditionally convergent series of interest in Physics is the calculation of Madelung's constant α in ionic crystals. The appearance of this type of series is quite disturbing to students and often causes misunderstandings. In this work we analyze the physical meaning of the conditional convergence from a pedagogical point of view. The problem is posed using a toy model of ionic crystal in which the lattice sums can be calculated explicitly for various forms of expansion of the crystal about a central core. It is seen directly how the Coulomb series does not converge to α when there are charge accumulations on the surfaces. Therefore, it becomes clear what the appropriate strategy should be when choosing the order of summation to arrive at the correct value of α.

We investigate the question discussed in the literature as to whether the magnetic field can perform work using two models that describe interacting magnetic dipoles. In the first model, the dipoles are realized by rigidly rotating charge clouds, whereas in the second model, one of the two dipoles is described by a real macroscopic spin density. The theoretical foundations of the second model are formulated in a recently published paper. We derive equations of motion and detailed energy balance for both models when the initial magnetic moments are parallel to the connecting line of the initial dipole positions. In this scenario the 'large' dipole remains stationary and generates an external magnetic field in which the 'small' dipole is accelerated. The answer to the title question depends on the choice of criteria for 'work of the magnetic field'.

We study various physical quantities of objects with petal shapes. N-petal shapes exhibit N-fold rotational symmetry. Furthermore, they might have an additional characteristic: the equation defining their boundaries could be represented by F(Nθ). We will show that physical quantities of objects with these characteristics may show strange properties. By 'physical quantities', we refer to aspects such as electric potential and electric field due to a charged petal-shaped plate or cylinder on the rotation axis, their mass and moment of inertia. We are going to show that for such objects, these physical observables do not depend on the number of petals, N. This intriguing result has a simple reason.

We present, for the non-specialist, an experimental implementation of Shor's factoring algorithm. We are unaware of any other single reference that explains, in a beginner-friendly way, complete circuits that implement Shor's algorithm. We perform the experiment with IBM quantum processors, which are remotely accessible online for free. We reproducibly factor 15 with one 7-qubit processor (ibm_perth), while four other quantum processors exhibit excessive error.

Although synchronization effects play an important role in many areas of basic and applied science, their treatment in undergraduate physics courses requires more attention. Based on acoustic experiments with a driven organ pipe, the article proposes analytical, numerical and qualitative approaches to this universal phenomenon, suitable for introductory teaching. The Adler equation is developed, a first-order nonlinear differential equation describing the phase dynamics of driven self-sustained oscillations in the weak coupling limit. Analytical solutions, intuitive mechanical analogues and properties of the resulting comb spectra are discussed. The underlying phase model is paradigmatic for synchronization-based self-organization phenomena in a wide range of fields, from physics and engineering to life and social sciences.

The use of augmented reality (AR) allows for the integration of digital information onto our perception of the physical world. In this article, we present a comprehensive review of previously published literature on the implementation of AR in physics education, at the school and the university level. Our review includes an analysis of 96 papers from the Scopus and Eric databases, all of which were published between 1st January 2012 and 1st January 2023. We evaluated how AR has been used for facilitating learning about physics. Potential AR-based learning activities for different physics topics have been summarized and opportunities, as well as challenges associated with AR-based learning of physics have been reported. It has been shown that AR technologies may facilitate physics learning by providing complementary visualizations, optimizing cognitive load, allowing for haptic learning, reducing task completion time and promoting collaborative inquiry. The potential disadvantages of using AR in physics teaching are mainly related to the shortcomings of software and hardware technologies (e.g. camera freeze, visualization delay) and extraneous cognitive load (e.g. paying more attention to secondary details than to constructing target knowledge).

Language is an important resource for physicists and learners of physics to construe physical phenomena and processes, and communicate ideas. Moreover, any physics-related instructional setting is inherently language-bound, and physics literacy is fundamentally related to comprehending and producing both physics-specific and general language. Consequently, characterizing physics language and understanding language use in physics are important goals for research on physics learning and instructional design. Qualitative physics education research offers a variety of insights into the characteristics of language and language use in physics such as the differences between everyday language and scientific language, or metaphors used to convey concepts. However, qualitative language analysis fails to capture distributional (i.e. quantitative) aspects of language use and is resource-intensive to apply in practice. Integrating quantitative and qualitative language analysis in physics education research might be enhanced by recently advanced artificial intelligence-based technologies such as large language models, as these models were found to be capable to systematically process and analyse language data. Large language models offer new potentials in some language-related tasks in physics education research and instruction, yet they are constrained in various ways. In this scoping review, we seek to demonstrate the multifaceted nature of language and language use in physics and answer the question what potentials and limitations artificial intelligence-based methods such as large language models can have in physics education research and instruction on language and language use.

This pedagogical article elucidates the fundamentals of trapped-ion quantum computing, which is one of the potential platforms for constructing a scalable quantum computer. The evaluation of a trapped-ion system's viability for quantum computing is conducted in accordance with DiVincenzo's criteria.

Mass–energy equivalence (MEE) has become a basis of modern physics. In spite of the current educational trends highlighting modern physics education, it has been pointed out that interpretations of MEE are still not in general agreement. In addition, the derivations of MEE found in textbooks gloss over some logical oversights. MEE is often introduced with only a declarative knowledge that mc² represents the rest energy of a particle, making the learning process difficult for students. To resolve the instructional challenges, distinguished papers on MEE are analyzed. By specifying common features of derivations in each paper, it is found that there are at least three types of MEE. By identifying the entire hierarchical structure of each one, a type of MEE is suggested that can potentially be useful in the establishment of the connection between the particle and field.

We present, for the non-specialist, an experimental implementation of Shor's factoring algorithm. We are unaware of any other single reference that explains, in a beginner-friendly way, complete circuits that implement Shor's algorithm. We perform the experiment with IBM quantum processors, which are remotely accessible online for free. We reproducibly factor 15 with one 7-qubit processor (ibm_perth), while four other quantum processors exhibit excessive error.

Research suggests that interacting with more peers about physics course material is correlated with higher student performance. Some studies, however, have demonstrated that different topics of peer interactions may correlate with their performance in different ways, or possibly not at all. In this study, we probe both the peers with whom students interact about their physics course and the particular aspects of the course material about which they interacted in six different introductory physics courses: four lecture courses and two lab courses. Drawing on social network analysis methods, we replicate prior work demonstrating that, on average, students who interact with more peers in their physics courses have higher final course grades. Expanding on this result, we find that students discuss a wide range of aspects of course material with their peers: concepts, small-group work, assessments, lecture, and homework. We observe that in the lecture courses, interacting with peers about concepts is most strongly correlated with final course grade, with smaller correlations also arising for small-group work and homework. In the lab courses, on the other hand, small-group work is the only interaction topic that significantly correlates with final course grade. We use these findings to discuss how course structures (e.g. grading schemes and weekly course schedules) may shape student interactions and add nuance to prior work by identifying how specific types of student interactions are associated (or not) with performance.

A simple experiment is described where a metal ring was rotated by hand on a horizontal rod. The ring rotated about 100 times before coming to a stop, so the friction force on the ring remained very small. However, measurements of the rotation frequencies of the ring around the rod and around its centre of mass indicated that the ring was sliding rather than rolling, with an unusually low coefficient of sliding friction. The results can be explained if any given contact point on the ring slides to a stop when it first contacts the rod.

Lens phenomena, such as caustics, image distortions, and the formation of multiple images, are commonly observed in various refracting geometries, including raindrops, drinking glasses, and transparent vases. In this study, we investigate the ball lens as a representative example to showcase the capabilities of Berry's eye caustic as an optical tool. Unlike the conventional paraxial approximation, the eye caustic enables a comprehensive understanding of image transformations throughout the entire optical space. Through experimental exploration, we establish the relationship between the eye caustic and traditional light caustics. Furthermore, we provide mathematical expressions to describe both the caustic and the image transformations that occur when viewing objects through the ball lens. This approach could be of interest for optics education, as it addresses two fundamental challenges in image formation: overcoming the limitations of the paraxial approximation and recognizing the essential role of the observer in comprehending lens phenomena.

In this research, we investigated the impact of peer collaboration and changes from individual to group performance of graduate students on the conceptual survey of electricity and magnetism (CSEM) without any guidance from the instructor. We define construction of knowledge as a case in which the group answered the question correctly but in the individual administration of the survey before the group work, one member gave the correct answer and the other gave incorrect answer. We find that there was a significant improvement in the performance of students after peer interaction, which was mostly attributed to construction of knowledge. However, students had very few opportunities to co-construct knowledge as there were hardly any situations in which neither student in a group provided a correct answer. We analyzed the effect size for improvement from individual to group scores for each CSEM item to understand the characteristics of these questions that led to productive group interaction. We also compared the group performance of the graduate students to the introductory physics students in a prior study using the CSEM to get insight into the concepts that showed differences for the two groups and those that were challenging for both groups of students before and after collaboration with peers. Our findings can motivate physics instructors to incorporate group interactions both inside and outside of the classroom even without instructor's involvement so that students at all levels can learn from each other and develop a functional understanding of the underlying concepts.

The first historical steps of radioactivity research offer an excellent opportunity to teach a key concept of modern physics: non-deterministic phenomena. However, this opportunity is often wasted because of historical misconceptions and of the irrational fear of radioactive effects. We propose here a lecturing strategy - primarily for undergraduate students - based on interesting historical facts. In particular, on a key conceptual contribution by Marie Curie, an attractive figure for the young women and men of today. Paradoxically, this milestone is almost unknown, whereas it should contribute to her immortal fame -- perhaps as much as the discovery of radium.

Upper-division undergraduate physics coursework necessitates a firm grasp on and fluid use of mathematical knowledge, including an understanding of non-cartesian (specifically polar, spherical and cylindrical) coordinates and how to use them. A limited body of research into physics students' thinking about coordinate systems suggests that even for upper-division students, understanding of coordinate system concepts is emergent. To more fully grasp upper-division physics students' incoming understanding of non-cartesian coordinates, the prevalence of non-cartesian content in seven popular Calculus textbooks was studied. Using content analysis techniques, a coding scheme was developed to gain insight into the presentation of coordinate system content both quantitatively and qualitatively. An initial finding was that non-cartesian basis unit vectors were absent in all but one book. A deeper analysis of three of the calculus textbooks showed that cartesian coordinates comprise an overwhelming proportion of the textbooks' content and that qualitatively the cartesian coordinate system is presented as the default coordinate system. Quantitative and qualitative results are presented with implications for how these results might impact physics teaching and research at the middle and upper-division.

An inverted pendulum can be stabilised by hand or by a high frequency sinusoidal vertical oscillation of the bottom end or by feedback control if a horizontal force is applied at the bottom end. The pendulum is unstable if a sinusoidal force is applied in a horizontal direction at the bottom end. It is shown in the present paper that an inverted pendulum can be stabilised if a low frequency horizontal force is applied at the bottom end to right the pendulum after it falls through a small angle. The technique requires a measurement of the fall angle but is not sensitive to the actual fall angle. The technique represents a simple example of feedback control and is more easily understood than vertical oscillation of the bottom end.

Researchers generally agree that physics experts use mathematics in a way that blends mathematical knowledge with physics intuition. However, the use of mathematics in physics education has traditionally tended to focus more on the computational aspect (manipulating mathematical operations to get numerical solutions) to the detriment of building conceptual understanding and physics intuition. Several solutions to this problem have been suggested; some authors have suggested building conceptual understanding before mathematics is introduced, while others have argued for the inseparability of the two, claiming instead that mathematics and conceptual physics need to be taught simultaneously. Although there is a body of work looking into how students employ mathematical reasoning when working with equations, the specifics of how physics experts use mathematics blended with physics intuition remain relatively underexplored. In this paper, we describe some components of this blending, by analyzing how physicists perform the rearrangement of a specific equation in cosmology. Our data consist of five consecutive forms of rearrangement of the equation, as observed in three separate higher education cosmology courses. This rearrangement was analyzed from a conceptual reasoning perspective using Sherin's framework of symbolic forms. Our analysis clearly demonstrates how the number of potential symbolic forms associated with each subsequent rearrangement of the equation decreases as we move from line to line. Drawing on this result, we suggest an underlying mechanism for how physicists reason with equations. This mechanism seems to consist of three components: narrowing down meaning potential, moving aspects between the background and the foreground and purposefully transforming the equation according to the discipline's questions of interest. In the discussion section we highlight the potential that our work has for generalizability and how being aware of the components of this underlying mechanism can potentially affect physics teachers' practice when using mathematics in the physics classroom.

Calculations are presented on the trajectory of a golf ball that rolls across the inclined surface of a golf green. The ball follows a curved path and comes to a stop at a point displaced at an angle to the initial launch direction. It is shown that the displaced angle is independent of the launch speed but depends on the launch angle and the ratio of the incline angle to the coefficient of rolling friction. The stopping distance is proportional to the launch speed squared. A simple experiment is described to check the calculations.