Enrico Fermi is remembered for his many contributions to theoretical and experimental physics, but from an educational point of view he also popularized the use of the kind of questions we now call "Fermi problems" (or "Fermi questions"). Fermi problems (FPs) are back-of-envelope problems that arose from the need to make order…

We will present a problem-solving method for the dynamics of a projectile that has two perpendicular acceleration vectors through rotation of the axes. This methodology of reparameterizing the two-dimensional system simplifies the speed optimization calculus.

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…

Graphs are an important component in physics laboratory reports. We have noticed that our students have difficulty selecting the "scale" and the "starting value" for an axis while generating graphs on graph paper. Recommendations concerning convenient scales and starting values are found in textbooks, laboratory manuals, and…

What does it mean to "make sense" of physics? It's not a simple question. Most people have an intuitive feeling for when things do (or do not) make sense to them. But putting this feeling into words--especially actionable words--is another task entirely. Generally speaking, there are two ways in which we commonly use the term "make…

In a typical first physics class, homework consists of problems in which numerical values for physical quantities are given and the desired answer is a number with appropriate units. In contrast, most calculations in upper-division undergraduate physics are entirely symbolic. Despite the need to learn symbolic manipulation, students are often…

When authoring physics problems, professors may develop an intuition for how much information they need to provide such that the problem has a unique answer and is not over constrained. It is an open question as to whether using intuition leads to a sufficiently broad range of problems. In this paper we discuss a systematic way of authoring…

Students in introductory college physics often have wide ranges of preparation, making it challenging to deliver effective instruction for all. While numerous educational products and strategies have been shown to be broadly effective, there is a need for research-based online materials that support personalized learning and focus specifically on…

On August 16, 1960, at 7:12 a.m., United States Air Force Captain Joseph W. Kittinger Jr. looked out from his balloon gondola at a height of over 19 miles and… jumped. Kittinger's altitude and free-fall speed record lasted until Felix Baumgartner's jump from a height of nearly 24 miles on Oct. 14, 2012. Baumgartner broke several records, including…

Are you frustrated when students focus on "getting the right answer" without understanding why the physics works that way, or even why their own brains came to that conclusion? If so, this might be less about the student and more about the pedagogy--you might be stuck on the "One True Path," a way of thinking "that assumes…

A common homework problem in many introductory physics courses is similar to the following. "A car drives at constant speed over a hill on a road in the shape of a vertical circular arc. What is the maximum speed the car can have and not lose contact with the road at the crest of the hill?" Unfortunately this problem is flawed, because…

The distant magnetic field of a magnetic dipole is usually derived via the magnetic vector potential and substantial vector calculus. This paper presents an alternate proof that is less mathematically intensive, and that ties together various problem-solving tricks (the principle of virtual work, observation that only instantaneous quantities…

In a recent submission to "The Physics Teacher," we related how trigonometric identities can be used to find the extremes of several functions in order to solve some standard physics problems that would usually be considered to require calculus. In this work, the functions to be examined are polynomials, which suggests the utilization of…

For years there has been an acknowledged interest in having students assess the rationality of their solutions to physics problems. In fact, many textbooks now routinely include end-of-problem assessments as part of the authors' detailed solutions to examples. Over the past two decades, I have experimented with various forms of end-of-problem…

When teaching a general education introductory astronomy course that includes a mathematics prerequisite, it is important to convey that formulas and equations physically represent properties of actual objects and processes that occur that can actually be observed. Students with varying comfort levels with mathematics often need to be convinced…