In this post, I present a conceptual question that is so important, I feel that no undergraduate student should complete an introductory calculus-based physics course without being able to correctly answer it. Given the way introductory calculus and introductory physics are traditionally taught, I just don’t think there is nearly enough carryover from the calculus to the physics. This is especially true for setting the stage for vectors. Ironically, whether or not vectors were “covered” in students’ introductory calculus courses, students have in fact seen vectors all along, but they just didn’t call them that. They called them partial derivatives. With that in mind, here’s the question:
Discuss the relationship between the concepts of “conservative force” from introductory physics and “exact differential” from introductory calculus. Go get started, if you need a hint, consider an arbitrary function f(x,y,z) and write its total derivative.
My prediction is that most student will not be able to answer this question correctly. As always, I’d be interested in hearing from you and your students if you show them this question.
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, independent of our arbitrary coordinate systems.
A particle moves through space with momentum 5 SI units. Resolve the particle’s momentum vector into components in each of the following ways: (a) the momentum vector has one nonzero component (b) the momentum vector has two nonzero components (c) the momentum vector has three nonzero components.
Be aware that there are infinitely many “correct” responses to this question. If you don’t like momentum, you can use any other vector quantity. Go!
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 the properties of vectors that allow us to manipulate them mathematically (algebraically really). Students should understand vectors in terms of their commutative, associative, distributive, and linear properties. Fortunately, these properties bring with them a lot of rich and beautiful geometry. It is this beautiful geometry that makes classical vector analysis so useful in the absence of coordinate systems. We need to do more to exploit this!
So here’s the new question:
Consider the vector equation A • B = C (C is a scalar). Consider B and C to be given. (a) Carefully draw a diagram using arrows to represent all possible vectors A that satisfy the equation. In other words, you’re attempting to solve the equation for A and drawing the solution set. (b) What if you now restrict your solution only to vectors A with a fixed (constant) magnitude? Draw arrows representing all such vectors that satisfy the equation. (c) What if you now restrict your solution only to vectors A making a fixed angle with B? Draw arrows representing all such vectors that satisfy the equation. Now consider the vector equation A × B = C (this time C is a vector), with B and C to be given. (d) Carefully draw a diagram using arrows to represent all possible vectors A that satisfy the equation. (e) What if anything, changes if you now specify that all three vectors are mutually orthogonal? Draw arrows representing all such vectors A that satisfy the equation. (f) Revisit the previous question on vector division and answer it again and see if your answer has changed as a result of this question.