A derivation of the conservation equations for fixed bed tubular reactors in cylindrical coordinates is presented and a differential operator for the substantial derivative in porous beds is introduced. The emphasis on the distinction between the functions of the void and volume fractions in the derivation of the conservation equations sets the…

This work gathers in one place what is pertinent about the connections between metric coefficients, basis- and unit vectors in a four-dimensional relativistic manifold. Some of this material can be found scattered elsewhere; its collection into one place reveals connections that either are not known or are obscure, for example that the metric…

Most textbooks and lecturers present Michaelis-Menten kinetics using the equation v = V[subscript max][S]/(K[subscript m] + [S]). There are advantages to presenting this relationship in a slightly different form, namely v = V[subscript max]/{1 + (K[subscript m]/[S])}. We articulate advantages for single-substrate reactions and extend the formalism…

The relativistic addition of velocities is usually introduced early in the study of Einstein's special theory of relativity. The equations are simple enough, but randomly chosen velocities lead to unwieldy calculations that can dishearten the student. This paper presents tables of velocity components in two dimensions, composed of five or fewer…

Transfer of structural math knowledge to physics is difficult for students. While research suggests various improvement techniques, enhancing parallelism of algebraic structures used in physics to those studied in mathematics courses seems underrepresented. This paper proposes an alternative way of introducing wave functions as a set of parametric…

The question of the sources of electric and magnetic fields and their causes has been discussed extensively in the literature over the last decades. In this article, we approach this problem from the unified treatment of electromagnetic fields emphasizing the role of their sources in accordance with the cause-effect relationships. First, we…

Oblique angle collisions of two penny coins on a smooth, horizontal surface were filmed with a video camera to investigate the physics of the collision process. If one of the coins is initially at rest, then the two coins emerge approximately at right angles, as commonly observed in billiard ball collisions and in puck collisions on an air table.…

Problems involving chains, cables, or ropes that are dropped, folded, or pass around pulleys attract ongoing interest, in part because they can become variable-mass situations if the chain is partitioned into sections for analysis. Less attention has been paid to trying to intentionally project the end of a string as far as possible. Here we…

Calculations are presented showing that the usual 'faster than g' demonstration has a surprising property. That is, a rod hinged at its bottom end rotates at an exponentially increasing rate until it falls with maximum vertical acceleration, unlike an object that falls freely by gravity alone. If the rod is hinged at its top end and released from…

The Young-Laplace (Y-L) equation relates the pressure difference across the interface of two fluids (such as air and water) to the curvature of the interface. The pressure rises on crossing a convex interface such as a rain drop and falls on crossing a concave interface such as the meniscus of water in a glass capillary. The relation between…

Background: Blended mathematical sensemaking in science ("Math-Sci sensemaking") involves deep conceptual understanding of quantitative relationships describing scientific phenomena and has been studied in various disciplines. However, no unified characterization of blended Math-Sci sensemaking exists. Results: We developed a theoretical…

Redox reaction is a core concept in teaching and learning chemistry. This article explores a new method for balancing organic redox reactions that requires the balancing of both atoms and charges. The H+, O, H[subscript 2]O, and e- are used as balanced vehicles in two half reactions. A non-oxidation number approach can be applied to both molecular…

The cross diagram is a way to illustrate and plan the solutions for all kinds of constant-acceleration kinematics problems--including projectile motion, multistage, and multiple-object problems. The cross diagram arranges the kinematic variables in a logical way, making the kinematic relationships between them easy to identify. Combined with a…

Much of physics involves the construction and interpretation of equations. Research on physics students' understanding and application of mathematics has employed Sherin's symbolic forms or Fauconnier and Turner's conceptual blending as analytical frameworks. However, previous symbolic forms analyses have commonly treated students' in-context…

Despite its crucial importance in physics, there are not many studies focusing on student difficulties and teaching strategies related to the (1D) wave equation in the PER literature. In order to contribute to fill this gap, we conducted a study with university students which focused on specific aspects that are crucial for understanding this…