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Register a new userThe Brachistochrone: An excellent problem for all levels of physics students
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Authors
John A. Milsom
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The classic brachistrochrone problem is standard material in intermediate mechanics. Many variations exist including some accessible to introductory students. While a quantitative solution isnt feasible in introductory classes, qualitative discussions can be very beneficial since kinematics, Newtons Laws, energy conservation and motion along curved trajectories all play a role. In this work, we describe an activity focusing on a qualitative understanding of the brachistochrone and examine the performance of freshmen, juniors and graduate students. The activity can be downloaded at https://w3.physics.arizona.edu/undergrad/teaching-resources .
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As part of a larger research project into massively open online courses (MOOCs), we have investigated student background, as well as student participation in a physics MOOC with a laboratory component. Students completed a demographic survey and the Force and Motion Conceptual Evaluation at the beginning of the course. While the course is still actively running, we have tracked student participation over the first five weeks of the eleven-week course.
In an Introductory Physics for Life Science (IPLS) course that leverages authentic biological examples, student ideas about entropy as disorder or chaos come into contact with their ideas about the spontaneous formation of organized biological structure. It is possible to reconcile the natural tendency to disorder with the organized clustering of macromolecules, but doing so in a way that will be meaningful to students requires that we take seriously the ideas about entropy and spontaneity that students bring to IPLS courses from their prior experiences in biology and chemistry. We draw on case study interviews to argue that an approach that emphasizes the interplay of energy and entropy in determining spontaneity (one that involves a central role for free energy) is one that draws on students resources from biology and chemistry in particularly effective ways. We see the positioning of entropic arguments alongside energetic arguments in the determination of spontaneity as an important step toward making our life science students biology, chemistry, and physics experiences more coherent.
We provide a simple approach for the evaluation of inverse integral transforms that does not require any knowledge of complex analysis. The central idea behind the method is to reduce the inverse transform to the solution of an ordinary differential equation. We illustrate the utility of the approach by providing examples of the evaluation of transforms, without the use of tables. We also demonstrate how the method may be used to obtain a general representation of a function in the form of a series involving the Dirac-delta distribution and its derivatives, which has applications in quantum mechanics, semi-classical, and nuclear physics.
Four sections of introductory physics for physical scientists and engineers (about 180 students each) are compared. One section, treatment group, was organized so that students worked to learn the classical ideas connecting forces and motion over the first 6 weeks of the 10 week quarter and then used the final 4 weeks to apply those principles to algebraically complicated problems. The other sections learned ideas at essentially the same time as calculations over the entire 10 weeks of the quarter. The treatment group and one of the control sections were taught by the same instructor, had identical curricular materials and this instructor was blind to the comparison measure, the final exam. After controlling for GPA as well as for incoming conceptual understanding, the treatment group was found (with greater than 99% confidence) to perform better on the final exam than the control group taught by the same instructor and, by a similar measure, the treatment group performed significantly better than any other section. The treatment group also had higher conceptual learning gains and so should be better prepared for later learning.
An important goal of introductory physics for the life sciences (IPLS) is for those students to be prepared to use physics to model and analyze biological situations in their future studies and careers. Here we report our findings on life science students ability to carry out a sophisticated biological modeling task at the end of first-semester introductory physics, some in a standard course (N = 34), and some in an IPLS course (N = 61), both taught with active learning and covering the same core physics concepts. We found that the IPLS students were dramatically more successful at building a model combining multiple ideas they had not previously seen combined, and at making complex decisions about how to apply an equation to a particular physical situation, although both groups displayed similar success at solving simpler problems. Both groups identified and applied simple models that they had previously used in very similar contexts, and executed calculations, at statistically indistinguishable rates. Further study is needed to determine whether IPLS students are more expert problem-solvers in general or solely in biological settings.
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