# Conceptual Understanding in Introductory Physics XXV

This question emphasizes geometry and should be done without use of a coordinate system. It should also be done using only symbolic manipulation of vectors. Here it is. Consider a particle moving with a constant, non-relativistic velocity. Starting with a general expression for kinetic energy in terms of either velocity or momentum, prove that the […]

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# Help Me! Question About Tensors and Projections

I need some help. I am working hard to find ways to bring more geometry into introductory calculus-based physics (and conceptual physics as well). By geometry, I mean specifically the geometry associated with vectors and tensors, and the information encoded therein. Yes, I said tensors. I have been heavily influenced by these notes by Kip […]

# Conceptual Understanding in Introductory Physics XX

This post is inspired by the October 2015 AstroNotes in The Physics Teacher. I have sometimes introduced vectors into my introductory astronomy course and students were able to do most of the things described below. We never discussed angular momentum or the Laplace-Runge-Lenz vector, but the other quantities were familiar. I was not permitted in […]

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# Conceptual Understanding in Introductory Physics XVIII

The Maxwell equations contain everything mentioned in an introductory calculus-based electromagnetic theory course, and then some. They contain detailed mathematical structure and deep insight into electromagnetic fields. They are a magnificent playground for learning the various theorems of vector calculus, and applying these theorems to the fields of particles naturally leads to things like divergence, […]

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# Conceptual Understanding in Introductory Physics XVI

Superposition is a powerful principle in physics, especially in introductory electromagnetic theory. A charged particle’s electric field exists independently of the fields of any other particles. A thorough understanding of superposition can prevent the frequent misunderstanding that fields can somehow be “blocked,” which is subtly different from saying that the net field at a point […]

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# Conceptual Understanding in Introductory Physics XV

Symmetry is part of the foundation of contemporary physics, but it is seldom emphasized in introductory physics in proportion to its significance. There may be some value in discussing how symmetry applies to otherwise traditional introductory problems rather than just replicating numerical examples from a textbook (even a good textbook). These questions illustrate symmetry in electromagnetic […]

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# Conceptual Understanding in Introductory Physics X

For this post, I decided to ask what I think is a very simple question. It is simple at first, but it also gets to the heart of the meaning of vector quantities, at least as they are typically presented in introductory physics. It also emphasizes the fact that vector quantities have an existence all their own, […]

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# Conceptual Understanding in Introductory Physics IX

The question in this post continues the thread related to vectors. We really need to do a much better job of treating vectors effectively, and accurately, in introductory. I have been heavily influenced by mathematics colleagues, and in particular by Keith Devlin, in that I have come to see that we should focus more on […]

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# Conceptual Understanding in Introductory Physics VIII

This post’s question reinforces the previous question in this series, which asks the reader to articulate why division by a vector is undefined in Gibbsian vector analysis. Here’s the question: Solve the following vector equation for a: Don’t be deceived. It’s not as straightfoward as it looks. I would really love hearing from colleagues who give […]

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# Conceptual Understanding in Introductory Physics VII

This post begins a series of questions centered on vector analysis. I have always felt that vectors get insufficient treatment in introductory physics courses. They’re presented too quickly, with too little attention paid to consistent notation and practically no attention paid to their analytical and geometrical properties. I try very hard to correct these oversights […]

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