# Student Programming in the Introductory Physics Course: M.U.P.P.E.T.Part 2

## Part 2

### Professional skills may be introduced at an earlier stage than is usual

As discussed above, approximation plays a critical role in understanding how physics works. We are forever making simplified models of real-world systems. Yet for our undergraduates, especially at the introductory level, because of their lack of mathematical sophistication, we are forced to treat most problems as mathematics rather than as physics problems.[11] The presence of the computer lets us introduce corrections and their sometimes striking effects.

#### Example: The large amplitude pendulum

One of the best examples of an approximate equation is the large amplitude pendulum. An excellent model equation for an idealized pendulum is derived in most texts and is within the reach of most of our students:

This equation, unfortunately, requires advanced special functions for its analytic solution.[12] For small amplitudes, the equation becomes

directly equivalent to the simple harmonic oscillator equation. Essentially all introductory texts give both these equations. The first is ignored except for the construction of the correct form of the energy. Dynamics problems are done with the second equation.

In the M.U.P.P.E.T. class, we are able to consider the large amplitude equation in more detail. Because we are solving Eq. (1) numerically, we have the situation shown in Fig. 3.

Many students assume that, because the analytic expression can be expressed in closed form, it is the "better" solution. We can bring them to a dramatic contradiction of this viewpoint by asking them to consider the analytic and numerical solutions for the cases:

In the first case, the correct (numerical) solution goes "over the top", spinning round and round the pivot. The analytic solution goes over the top some distance, stops in midair, turns around, goes back over the top. The second case is a small angle oscillation, but the "analytic" solution doesn't recognize this. Instead of falling and oscillating through a small angle, it rises over the top, oscillating back and forth nearly a full circle each time.

The explanation of these strange results is fairly simple. The small angle approximation doesn't hold for large angles, so the analytic form shouldn't be applied. But these simple examples illustrate a principle of great importance.

When approximate solutions are extrapolated beyond their realm of validity they can give results that are qualitatively wrong.

Again, this result is obvious to the professional physicist, but there is essentially no example of this important result anywhere in the traditional introductory curriculum. If our students are to learn the art of approximation, they must have simple touchstone examples that clearly illustrate the possible pitfalls.

One additional point is important and illustrative about this case. Note that the mathematics of solving the approximate equation is simpler. But from the students' point of view, the ideal pendulum requires one extra logical step to set up its equation of motion and is therefore conceptually more difficult than the realistic one. Indeed, many students have a poor understanding of the small angle approximation. With M.U.P.P.E.T., we can discuss the exact case first.

### More realistic problems may be treated than in the traditional approach

The traditional curriculum is largely restricted to those problems and methods that are analytically tractable. The powerful tools that permit the solution of almost any problem are usually touched only in passing since they require numerical methods. In a computer-based course, they can be restored to their rightful importance.

#### Example: Projectile motion with air resistance

An example of the possibilities opened up by including the power tools is the discussion of air resistance in the M.U.P.P.E.T. course. The importance of including this topic goes far beyond the issue of simply making our description of motion more realistic. With this example, a large number of valuable reasoning tools can be introduced that are usually ignored until much later in the curriculum.

A combination of dimensional reasoning and symmetry principles can be used to construct the Newton drag law:

where etais a dimensionless parameter. This has interesting consequences[13] and further discussion can be given during the section on kinetic theory.

We then use a M.U.P.P.E.T. program to study the behavior of an object under the influence of this force. The total amount of programming required from the student is to put in the equation for the force law into the program Proj1D. The output of this program is shown in Fig. 2.

The students can be asked to carry out an interesting mix of qualitative and quantitative analyses. In studying the qualitative behavior we can ask the student the following questions:

• What is the effect of including air resistance for an object thrown straight up? When there is no air resistance, it takes a projectile the same time to go up as to come down. Does this change when air resistance is added? Give a qualitative argument to show which way it should work and use the program to demonstrate the correctness of your reasoning.

• What is the effect of including air resistance for an object thrown at an angle? When there is no air resistance it travels the same horizontal distance while rising to its maximum height as it does descending. Does this change when air resistance is added? Give a qualitative argument to show which way it should work and use the program to demonstrate the correctness of your reasoning.

• When the object is very light or the air resistance coefficient is large, something strange happens to the velocity of a falling object. Use the program to determine what this is.

Once the student has observed the phenomenon of terminal velocity qualitatively on the computer, we can ask them to derive the expression for terminal velocity analytically. With the computer program in hand, we can ask the students to study some interesting realistic cases. Here are two sample homework problems that can be done at the end of the unit:

Sample problem 1: A 10 gm sheet of paper is crumpled up into a compact ball with a radius of 4 cm. When dropped, it takes 1.0 sec to fall a distance of 2 meters. Use this to determine the air resistance coefficient b in the force law . If a wooden and a steel ball are dropped from the same height, how long would they each take to fall? What accuracy would you need in your measurements to see the difference in the rates of fall between the wooden and steel balls?

Sample problem 2: A ball of mass 0.14 kg is thrown straight up with a speed of 20 m/s. It comes down 1 second earlier than expected, if air resistance is ignored. Find the air resistance coefficient b for this object if the force has the form

Find the coefficient g if the force has the form

Design a simple experiment (with numbers!) using this ball to determine which force gives a better description of the real world.

The example shows that the actual computational work and the programming involved is a fairly small part of the unit. But having it present permits us to bring in scale analysis, dimensional analysis, and to demonstrate approximation techniques and ways of extracting physics from computer programs. These are all skills that the professional must know, but which we have had little opportunity to teach in undergraduate courses.

### Contemporary topics may be introduced

Once power tools are put in the student's hands, a much wider variety of problems can be addressed. These include more realistic problems than are usually handled, as well as ones of contemporary interest.

#### Example: Chaos theory

Recent developments in the theory of classical mechanics have stressed the sensitivity of non-linear classical problems to initial conditions. Although this sensitivity has been known to workers in the field, especially Lagrange and Poincare, when we teach Newtonian mechanics we tend to tell only half the story. We traditionally emphasize that:

In principle, classical systems are totally predictable once starting conditions are specified.

However, it is equally important that the student understand the contemporary lesson of chaos theory which emphasizes that:

In practice, it is usually impossible to predict the long-term motions of any classical system with a finite calculation since they are highly sensitive to the starting conditions.

In the M.U.P.P.E.T. course, the presence of the computer allows us to include a segment on chaos theory as a natural extension of Newtonian dynamics at the end of the first semester. The students find this topic of great interest, and many of them choose some non-linear problem as a research topic in the second semester. A screen from the M.U.P.P.E.T. sample program Iterate is shown in Fig. 5. This program illustrates the phenomena of bifurcation, period doubling, and repeatable randomness. The code for this program contains only 60 lines, many of which can be clipped from the basic sample programs.

### Students may begin research at an early stage

Introducing the computer to students at an early stage gives them the power to investigate a wide variety of complex problems in an open-ended inquisitive fashion and thereby begin to get a real exposure to how science is done in practice. We have had considerable success with first and second year physics majors performing independent projects in the M.U.P.P.E.T. course.

When one of us (EFR) taught the M.U.P.P.E.T. course, he required project work of every student in every semester. An earlier attempt to require projects of sophomore physics majors in 1970-72 had not been successful. Only about 15% of the students were able to do projects that had any characteristics of normal scientific research. Most students were severely hampered by not having sufficiently strong analytic and mathematical skills to carry through an open-ended investigation. However, in the M.U.P.P.E.T. environment with freshman majors in 1986-1989, when each student had access to a computer and the M.U.P.P.E.T. tools, the results were strikingly different. About two thirds of the students were able to do valuable and interesting projects.

The students were told to seek a topic they were interested in and would like to know more about. It had to have some relation to the content of the course, although we tended to be flexible if a student showed strong interest in some other topic. They were told that this was not to be a project where they read, organized, and replayed other people's materials. They were supposed to design their own project, carry out an investigation, and write a report.

In an ideal project, we believe that the student should carry out the following activities:

• Formulate a question of broad general interest.

• Perform library research on the subject.

• Reformulate the question more sharply so it is amenable to modeling or an experiment.

• Set up the calculation or experiment and run it.

• Check the results for consistency, accuracy, and correctness.

• Extract some physical insight from the results.

• Propose further research based on the results.

• Present the results both in written and oral form.

The professional researcher does all of these. Consider for your own undergraduate majors' curriculum: At what point in their undergraduate training do each of your students get experience with each of these kinds of tasks? Of course few of our freshman projects succeeded in accomplishing all these goals. We were surprised and delighted that some did, but we considered a project successful if a student demonstrated five of the eight activities.

Some of the subjects investigated by students in conjunction with our calculus-based physics course include:

• Colliding galaxies
• Shepherd moons and Saturn's rings
• Tethered satellites
• Capture of a planet by a wandering star
• The flight of the frisbee
• Grain boundary growth in crystals
• The evolution of light from galaxies
• The motion of a spinning ping-pong ball
• Pumping a pendulum swing
• The Lyapunov exponent in the Sinai billiard problem
• Diffractive lens design
• Period doubling and chaos in the van der Pol oscillator
• Designing an airplane wing
• The planet Nemesis and its effect on the Oort cloud
• The motion of a golf ball
• The effect of the backboard in basketball
It would have been a substantial burden on the teacher of the class to advise semester long research projects for even a class of 25 students. Fortunately, we had the support of the Maryland physics department. In most cases, students were sent to other faculty with particular relevant expertise for advice.[14] This turned out to have a substantial benefit. The undergraduate majors met faculty on a personal basis at an early stage in their careers. This resulted in students making valuable contacts that often developed into research projects when they became juniors and seniors. Since the start of project work in the introductory courses, the number of upperclass majors seeking to do independent research projects has grown from essentially zero (1-2 per year) to a substantial fraction of the class (10-12 per year).[15]

At universities with very high admission standards, it is well known that freshman can do independent research (viz. winners of the Westinghouse competition). What was not widely appreciated before M.U.P.P.E.T. was that freshmen with a wide range of abilities and backgrounds can begin to design their own research projects and carry them out successfully if the physics course is tied to the computer and the students empowered in its use.

Perhaps the most surprising result was the distribution of good projects. Students who would have been identified as mediocre students by their exam grades occasionally did outstanding projects. A careful analysis of these students showed that they had "stylistic" rather than content problems. They did not perform well under exam pressure, but preferred to work slowly and carefully. Some of them had extraordinary intensity and persistence when they were interested and involved in a project. Others who did well on exams could find no topic at all to interest them and turned in very poor research projects. These observations raise the question whether it is a good idea to use a student's performance on traditional timed hour exams as a "first cut filter" to weed out those who should not be physicists. This automated hoe may be chopping some valuable flowers!

#### M.U.P.P.E.T. can be used in association with the Laboratory

The M.U.P.P.E.T. approach has been used by one of us (JMW) to provide modeling and analysis tools in conjunction with the laboratory that accompanies the first semester introductory physics course (mechanics).[16] (M.U.P.P.E.T. can also be used in the laboratory directly to serve as a user interface with the computer to accumulate data directly from analog to digital converters.[17]) The laboratory was taught by alternating between a traditional "hands-on" laboratory one week and a computer modeling/data analysis session the next. The students used M.U.P.P.E.T. to calculate means and standard deviation and performing linear least square fits. They also developed computer models and compared the model's predictions with experimental observations.

For example, in one of the laboratories, students observed and took data on the motion of a pendulum using a stroboscope and a Polaroid camera. They then used their graphing and data analysis package to plot the observed angle, and to calculate and plot the angular velocity, angular acceleration, kinetic energy, potential energy, and total energy vs. time. They then used M.U.P.P.E.T. to build a mathematical model of the system on a template we provided. With their program, they were able to compare the prediction of a mathematical model with their observations and make an estimate of the damping. With this model, students were able to extend their analysis of their investigations to consider large angle corrections, driving forces, and resonance.

Instead of listening to lectures about modeling and error analysis (as had been the previous practice -- even in the lab section of the class!) the students performed the activities themselves using our computer tools. The students' projects were significantly improved in quality and showed a better understanding of the phenomena than when they worked in the traditional mode.

### V. What did we have to leave out?

We are certain that many of our readers will respond to our claims with skepticism. Everyone who has taught the traditional introductory calculus-based physics course has had the problem of having too much material to cover. How could we possibly teach programming, add more realistic problems, include the large amplitude pendulum, air resistance, random walks, and integrals over non-simple charge distributions (among others)? What did we leave out?

Indeed, some materials were left out that are included in the traditional course. Rigid body motion was suppressed, as was fluid dynamics and much discussion of sound. In our three semester sequence we did not do any modern physics or relativity, in part since the sequence is followed immediately by a full two semester sequence on modern physics.

The additional materials, however, did not take very much time to include. The programming handouts were read in parallel with the standard reading, a few pages per week. Less than 5% of the lecture time was used to discuss programming. Two to three lecture hours per semester were actually spent in the microcomputer laboratory with the students getting started on some of the more computer oriented homework assignments.

From this small basis of computer instruction, we were able to include one to two homework problems per week that were somehow associated with the computer. (These were occasionally estimation, analytic, or qualitative problems.) These were done at the cost of reducing the number of standard "plug-and-chug" problems the students were assigned.

### VI. Is M.U.P.P.E.T. transferable?

One question to consider is whether the M.U.P.P.E.T. environment is transferable. Many educational developments work well as long as they are taught by their creators, but do not succeed when taught by anyone else. There is good evidence that M.U.P.P.E.T. is usable elsewhere.

The University of Maryland is a reasonably typical university environment. The College Park campus of the University of Maryland is a large state university with a large student body having a wide range of interests, backgrounds, and levels of ability. Materials developed and tested at Maryland should be usable at many campuses across the nation.

M.U.P.P.E.T. at Maryland has not been restricted to its developers. The M.U.P.P.E.T. course for physics majors has been and is being taught at Maryland by faculty not involved in the development of M.U.P.P.E.T.

New courses have been developed with M.U.P.P.E.T. materials in Australia.

Prof. Ian Johnston at the University of Sydney became acquainted with the M.U.P.P.E.T. idea at the Raleigh Conference on Computers in Physics Instruction in the summer of 1988.[18] Since then, he and his colleagues have developed materials for second year physics majors on numerical methods and quantum mechanics. In 1989-90, he used the M.U.P.P.E.T. environment to integrate computational physics into the undergraduate courses at Sydney University.[19]

Johnston tested M.U.P.P.E.T. in three successive semesters. In the first test, 18 volunteers, chosen from a class of 200 second-year students, were given six four-hour microlab sessions in addition to the normal work in a course in quantum mechanics. In the second test, this was repeated with 92 students out of a class of 202. The third test involved 24 students in the third year class. These students were asked to work through four computer modeling problems in diverse areas of physics including solid state, kinetic theory, plasma physics, and Fourier transforms. Johnston and McPhedran conclude:

(1) Students do not need to be able to program before they can handle these materials. Students who had no previous programming experience (about 25% and 16% of the students in the first two trials) had to work harder at first, but had little trouble once they got started.

(2) The students' understanding of a number of traditional subjects was significantly improved by adding computer modeling problems as shown by grades in a comparison of the students in the test and traditional groups on traditional tests. This was because the computer programs allowed students to explore many more cases than they could by hand. For example, because the students in these trials had seen the shapes of many different wave functions, they could easily answer questions in ordinary texts dealing with the geometrical property of eigenfunctions and performed significantly better than students in the traditional group on such questions in exams.

(3) In the last microlab, students were asked to investigate broadly posed problems. They were able to successfully design and complete projects in a one week period, thanks in a large part to the ease of building models with the M.U.P.P.E.T. software package. This confirms our experience with projects at Maryland.

We conclude from these experiences that the M.U.P.P.E.T. environment is robust and survives being transferred to other users.

### VII. Conclusions

#### Summary

We have reported on the development of the M.U.P.P.E.T. utilities -- a flexible and powerful computing environment that permits introductory students to add programming to their tools for solving physics problems quickly and easily and with minimal overhead. When these tools are added to the traditional calculus-based introductory physics course, the students' power to solve problems expands enormously. This opens many possibilities for changing the curriculum. Elements may be rearranged in a more natural order; professional skills may be introduced at an earlier stage than is traditional; contemporary topics such as chaos and quantum theory may be introduced; and students may begin research immediately.

Our conclusion is that M.U.P.P.E.T. works well for majors in small classes. We have not yet tested whether these methods can be extended to large classes with other scientists such as chemists and engineers. It may be possible if the infrastructure exists to provide students with sufficiently accessible networked computer resources. Success in this environment could also be aided substantially by good coordination with other departments.

#### Future developments

As a result of its strength as an open environment capable of growing as the student's strength grows (and as the power of the computer grows), the M.U.P.P.E.T. utilities have been adopted as the basis for two multi-university projects of national scale -- CUPS and CUPLE.

The Consortium for Undergraduate Physics Software (CUPS) is a project based at George Mason University and funded by the NSF to add computers to upperclass physics courses. A group of 27 physics faculty with software design experience are developing six manuals to accompany upperclass physics courses. Each manual contains nine simulations, each of which will add an element of new physics, not easily includable without the computer.

The Comprehensive Unified Learning Environment[20] (CUPLE) is a project to bring together in a single unified computer environment some of the successful attempts to reach more introductory physics students and to train them more effectively and professionally. CUPLE is bringing together sophisticated tools for handling graphing, student programming, laboratories, and video with modularized text materials and a database of information. The M.U.P.P.E.T. environment is being upgraded for this project to an object-oriented approach now called Window on Physics (or WinPhys for short). WinPhys is built on Turbo Pascal for WindowsTM and takes full advantage of the Graphical User Interface (GUI) Microsoft WindowsTM 3.

#### Acknowledgments

M.U.P.P.E.T. was supported by the Fund for the Improvement of Postsecondary Education. The project has involved a large number of faculty, students, and visitors who played various roles throughout the years. We particularly acknowledge the collaboration and efforts of those faculty members who participated in the development of the original conception of M.U.P.P.E.T. and who have contributed extensively to its development: Profs. Charles Misner, Bill MacDonald, and Jordan Goodman for concepts and fundamental ideas; James Harold, Ken Hennacy, Gerhard Norkus and Madhura Nirke for developing programs and utilities. Visitors to the program who made important contributions include Ian Johnston (Sydney), Gordon Aubrecht (Ohio State), Ed Taylor (MIT), Pat Cooney (Millersville), Steve Hanzely (Youngstown), and Gunther Kurz (Esslingen).

#### Endnotes

[1] W. M. MacDonald, E. F. Redish, and J. M. Wilson, "The M.U.P.P.E.T. Manifesto", Computers in Physics, 2(4), 23-30 (July/Aug 1988).

[2] F. Verbrugge, "Conference on Introductory Physics Courses", Amer. J. of Phys., 25, 127-128 (1957) ; "Improving the Quality and Effectiveness of Introductory Physics Courses", ibid. 417-424; F. Bitter et al., "Report of Conference on the Improvement of College Physics Courses", ibid., 28(1960) p. 568-578.

[3] R. L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley, 1989).

[4] J. S. Brown, A. Collins, and P. Duguid, "Situated cognition and the culture of learning", Educational Researcher, p. 32-42 (Jan-Feb 1989) .

[5] The most precise comparison of theory and experiment occur in quantum electrodynamics, where the g-2 value of the muon and the binding energy of the helium atom can be calculated to more than 10 significant figures. At greater than this level of accuracy, one runs into the problem of virtual production of strongly interacting particles where the theory does not yet exist to permit further improvements.

[6] We are greatful to John Risley for a discussion of this idea.

[7] E. F. Redish, J. M. Wilson, and I. P. Johnston, The M.U.P.P.E.T. Utilities (Physics Academic Softwarem 1994).

[8] We have focussed in this work on the course for physics majors. Some preliminary testing of the use of student programming in large classes with engineering students was begun in the fall of '91. This effort is to focus more on conceptual problems and building up a strong view of how one does physics than on developing professional skills.

[9] These last two libraries were developed by I. P. Johnston at the University of Sydney.

[10] E. F. Redish and Edwin Taylor, "Impulse Mechanics", AAPT Announcer 17(4), 82 (Dec. 1987)

[11] This tends to hold largely for textbook problems. The classic laboratory in which the student measures the value of g to a high accuracy with a pendulum and calculates many corrections provides one of a number of excellent counter-examples to this statement.

[12] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Fourth Edition (Cambridge U. Press, 1952).

[13] E. F. Redish, "The impact of the computer on the physics curriculum", in Computers in Physics Instruction, E. F. Redish and J. S. Risley, eds. (Addison-Wesley, 1990), p. 15-22.

[14] In the sections taught by EFR, students were instructed: "Your paper has to teach the teacher something he doesn't know in order to earn an A." This had the effect of encouraging them to seek advising elsewhere in the department.

[15] The total number of upperclass majors has remained constant at about 50.

[16] J. M. Wilson, "Combining computer modeling with traditional laboratory experiences in the introductory mechanics laboratory for physics majors", AAPT Announcer, 17(2), 80 (May, 1987).

[17] Ian Johnston, Sydney University, private communication.

[18] The Conference on Computers in Physics Instruction, Proceedings, E. F. Redish and J. S. Risley, eds. (Addison-Wesley, 1990)

[19] I. D. Johnston and R. C. McPhedran, "Computational Physics in the Undergraduate Curriculum", submitted to the The Australian Physicist.

[20] J. M. Wilson and E. F. Redish, "The Comprehensive Unified Physics Learning Environment: Part I. Background and system operation", Computers in Physics, 6(2) (Mar/April 1992), 202-209; "..: Part II. The basis for integrated studies", ibid. 6(3) (May/June 1992), 282-286 .

©1994, American Association of Physics Teachers.

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