Préprint : Physics Education (1994).
IMAGES AS A BASIS FOR COMPUTER MODELLING D Beautils, J-C Le Touzé and F-M Blondel
New computer technologies such as the graphics data tablet, video digitisation and numerical or graphic methods can be used for measurement, data processing and mathematical modelling in physics. At the Institut National de Recherche Pédagogique in France we have been studying these new opportunities in science teaching for several years. We have developed software programs on Newtonian mechanics and related scientific activities for A-level students (16-18 years old).
Computer and camera: tools for solving physics problems Computers are used in A-level science teaching mainly as laboratory tools. Hardware and software are designed to collect, save, display and analyse data. These tools enable experimental studies to be made in various areas of physics and chemistry (TERC 1989, BUP 1991, Barton 1991). However, in some circumstances, experimental probes can perturb the phenomenon or reduce the range of experimental conditions available. This affects the teaching of Newtonian mechanics for example: potentiometric sensors used to measure rotation angles add unwanted friction to the oscillatory movement of the pendulum. By contrast, in the study of free fall, light-gates must be placed on the known trajectories of specific objects. Let us note here that questions also arise about the pedagogical uses of these tools. For instance, using computers in order to increase the amount of data and to discover fundamental relationships (such as energy or momentum conservation) through an inductive process would neither be valid epistemologically nor realistic, considering average students’ abilities. In the same way, displaying multiple and unusual graphs could be more a source of misunderstanding than a help to students, and using advanced numerical methods could be meaningless when applied to linear relations or equations with simple analytic solutions (Beaufils 1991). Initially, the limitations on the use of probes led us to consider all cameras, whatever their output (photo, movie or video) as particularly interesting sensors, because they do not introduce friction and allow us to study non-linear trajectories of any point of any object moving within a plane. This relates to the work of Muybridge or Marey, but also to more recent techniques that have been developed in research laboratories dealing with the physics of fluids and sports science (Smith, 1982). It is our opinion that these image-processing techniques could also be applied in school science laboratories. We note here that the choice of technology is obviously related to the nature of the expected data. Photographs make it possible to study plan figures (hanging cable, water nozzle, etc.) or certain paths (time exposure of a moving luminous point). Films allow us to split a movement up into successive states, and thus give direct information about the sequence of co-ordinates. The multiple-flashes photograph (taken using a stroboscope) is somewhere between the two, because it yields at the same time a breaking-up of the movement and a plot of the trajectory. The use of images, and in particular strobe photographs, has already been promoted in the HPP and PSSC lessons (PSSC 1973) and introduced into French physics textbooks. But there are several drawbacks: the difficulty of taking and developing photographs and the time-consuming, rather inaccurate manual measurement. At present, new technologies, especially computer technologies, allow us to overcome these practical difficulties and enable these approaches to be updated (Glover et al 1989, Graham 1991, Beaufils and Le Touzé 1992).
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Préprint : Physics Education (1994).
Analysing photographs with a graphics data tablet We first designed specific software (TABLE) to analyse multiple-flash photographs of movements. The image is fixed on the active area of a data tablet connected to the computer (figure 1). Using the cursor, the student sets the origin and the scale of the reference system. Then successive images are located with the cursor: at each click, the co-ordinates are displayed and the corresponding point is plotted on the screen in a graph of co-ordinates versus times (figure 2). After measurement, three sets of functions (available from a menu bar) can be used for data processing and mathematical modelling. The first one allows calculation of other quantities such as speed or energy and the second performs basic calculations on initial data, such as smoothing and origin modification. With the third function, the student can compare or elaborate mathematical models by the leastsquares method, graph plotting and numerical simulation. One command, named ‘FONCTION’, can be used to obtain graphs of mathematical functions, which can be superimposed on the experimental points; this is suitable for the elaboration of an empirical model, which describes the experimental data quantitatively. Another command, named ‘SIMULATION’, can be used to solve (by iterative numerical methods) differential equations drawn from Newton’s second law (figures 3 and 4) and draw the simulated evolution whatever the physical quantities plotted on the axis (see example below). This is geared to the elaboration of a physical model, which gives a theoretical interpretation of the phenomenon. In spite of its direct and simple pointing facilities, this process is limited to certain images. Because of the overlap of successive positions, it is impossible to photograph a moving object whose size is larger than its 'mean free path'. And if it is technically possible to use a film that can be projected frame by frame on the tablet (a technique that has been used in some studies in sports science), it is not very practical for school laboratory work.
Modelling with digital images One possibility that emerges from these last remarks is to use digital images. This takes into account the availability of new peripherals such as scanners and digitising boards, which allow users to display digital images on a computer screen with adequate definition. The fundamental advantage of using a digital image, which is determined more by teaching constraints than by technical ones, is the possibility of ‘working on’ the image. With some numerical functions, it is possible to modify the colour map either to make some unnecessary background details invisible or to heighten the contours of other features. Our aim is obviously not to 'doctor’ images but to make the location of objects easier and to highlight the transformation of a recognisable object into a schematic shape. Ibis last transformation, which we can call ‘graphic modelling’, is central to the whole modelling process, even if it is not always made explicit. A second important feature is that it is possible (i) to size and to shape the graphic cursor to the object, and (ii) to superimpose points, outlines, curves, etc, on the image itself so that the image and the drawing will be in the same 'format'. Because it is possible to superimpose a theoretical drawing onto the image itself and not just onto the data points plotted beforehand (and thus cut off), the screen is the location of not just data / model comparisons but also phenomenon / model comparisons. In order to put these ideas into practice, we designed a second piece of software (named Image)1. The user can load and display on the computer screen a frame or a sequence of frames. Then he/she can select the mouse-controlled graphic cursor from predefined geometric shapes (cross, circle, square), or
1
Both software (with strobe photographs or image files) are commercially available (each for a price of less than 400 FF) from CNDP-SIE, 31 rue de la Vanne, 92120 Montrouge, France. Software and instruction manuals are at present only in French, but a summary of the reference manual in English is available from the authors, at INRP, 91 rue G Péri, 92120 Montrouge, France.
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Préprint : Physics Education (1994). choose the object itself as a cursor with an inner 'hot point', i.e. the point (not necessarily the centre) whose co-ordinates will be recorded at every click.
Figure 1. The strobe photograph is need on a data tablet connected to the computer.
Figure 2. Each measurement of co-ordinates is plotted on x(t) and y(t) graphs.
Figure 3. The 'MODELISATION’ menu and the 'SIMULATI0N’ command.
Figure 4. The ‘SIMULATION’ command: differential equations of the theoretical model.
The measurement can then be carried out with the mouse, according to a 'classical' scheme: choice of the origin of co-ordinates, calibration of the scale and then location of the successive positions of the chosen point. Results can be stored in a standard format file for further analysis with TABLE software or other modelling software. In the modelling option, the user can super-impose on the image itself any theoretical drawing derived from either a mathematical description of the trajectory (plotting function) or a theoretical modelling of movement (numerical solution of differential equations). The drawing can be plotted either with dashed or dotted points or with the image-cursor of the object itself! This last option enhances the difference between a general movement (including rotation) and a single translation movement (which is not necessary a linear one).
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A case in point modelling air friction When studying the movement of a light object thrown into the air (Ping-Pong bail or shuttlecock for example, figure 5), the following questions arise. What kind of empirical model can be chosen to interpret the deceleration? Is the magnitude of the friction proportional to the velocity (kV) or to the square of the velocity kV²)? What empirical value can be given to the parameter k?
Figure 5. Strobes photograph of the movement of a shuttlecock.
The study can be worked out as follows. First, the user needs a multiple-flash or video camera to collect the data. Then, following the first hypothesis (f = -kV), he/she will plot speed components against time: vx(t) and vy(t). In this case, the solution is known and an exponential function will be tested. But the user will also consider the differential equations of motion. Their numerical solutions can be calculated and their graphs plotted (even if the model is quadratic) as ax(vx) and ay(vy) (figure 6). With the same menu command, the user will also compare the models and the experimental graphs with another physical quantity plotted on the axis: total energy versus time, for example (figure 7).
Figure 6. Solution of the differential equations in the graphs of acceleration versus velocity.
Figure 7. Solution of the differential equations in the graph of total energy versus time.
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Préprint : Physics Education (1994). If the strobe photograph bas been digitised, the theoretical movement, calculated from the differential equations including air friction, can be superimposed as a curve or represented by the object itself (figure 8).
Figure 8. An image of the object is used to superimpose the theoretical trajectory on the digitised photograph displayed on the computer.
Conclusions and prospects The main purpose of these programs is to facilitate, with the help of numerical and graphical methods, the study of non-elementary’ phenomena such as the motion of various solids (not just points) or the modelling of air friction. The second idea is that these studies are performed in a less abstract way by using images: measurements are made on recognisable objects and the results of the mathematical modelling can be readily compared with the phenomenon. The students do not work on materials specially designed for physics classes but on phenomena closer to reality. As regards the future, the first thing to note is that digital recording will spread rapidly in several domains. Digitisation devices are becoming more widely available and users will be able to obtain and analyse digitised images by themselves, using their own peripherals. As a consequence, software bas to be ‘open ended’, i.e. it bas to be capable of importing general files formats as BMP, PCX and 11F. The actual limitation on image storage comes from the size of the corresponding files, particularly when a study requires a series of 20 or 30 frames. But the general increase in bard disk capacities, the expansion of CD-ROM and the efficiency of data compression techniques will soon enable us to go beyond these limitations. The main issue will be the impact of image processing techniques on the modelling process. Image enhancement and perhaps edge detection may prove to be new and effective methods of analysing series of rough images.
References Barton R 1991 Data logging in A-Level physics Phys.Educ. 26 124-6 Beaufils D 1991 Ordinateur outil de laboratoire dans l’enseignement des sciences physiques, propositions pour la construction d’activités, première analyse des difficultés et des compétences requises chez les élèves de lycée, Thesis Paris VII University Beaufils D and Le Touzé J C 1992. Learning physics with image based modelling software Proc. 91/tint. Conf. on Technology and Education (Paris) pp 20-22 Glover DM, Graham GR and Macdonald RM 1989. The CCAT videodisc - a new resource for physics education Phys. Educ. 24304—8 Graham G R 1991 Let’s see it for real - a new medium for an old message Phys. Educ. 26355-8 PSSC 1973 Physics 3rd Ed (Massachusetts: Heath and Co) Smith T 1982 Gymnastics: A Mechanical Understanding (London: Hodder and Stoughton Educational) TERC 1989 Hands on! - innovation and issues, 12(1) Images as a basis for computer modelling, Physics Education 29, 89-93.
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