The law of conservation of momentum is a fundamental law of nature. It states that the momentum of an isolated system is conserved. In high school or introductory-level physics courses, for simplicity, teachers and textbooks always use collisions in one dimension as the examples to introduce the concept of conservation of momentum. To solve simple…

Symmetry and invariance play a fundamental role in modern physics and related disciplines, and students entering a STEM field need practice with symmetry-based arguments as part of the preparation for their future careers. We refer to symmetry based solutions as those that either make use of the existing geometric symmetry in a problem to simplify…

Over the past several years, we have contributed articles to "TPT" that focus on a forensics-style reexamination of significant historical events. The purpose of these articles is to afford students the opportunity to apply basic principles of physics to unsolved mysteries and potentially settle the historical debate. We assembled the…

An important step in learning to use math in science is learning to see symbolic equations not just as calculational tools, but as ways of expressing fundamental relationships among physical quantities, of coding conceptual information, and of organizing physics knowledge structures. In this paper, I propose "anchor equations" as a…

The aim of this paper is to propose a method for solving head-on elastic collisions, without algebraic complications, to emphasize the use of the fundamental conservations laws. Head-on elastic collisions are treated in many physics textbooks as examples of conservation of momentum and kinetic energy.

The use of systems in many introductory courses is limited and often implicit. Modeling two or more objects as a system and tracking the center of mass of that system is usually not included. Thinking in terms of the center of mass facilitates problem solving while exposing the importance of using conservation laws. We present below three…

Our high school optics course finishes with an assignment that students usually appreciate. They must take pictures of everyday situations representing optical phenomena such as reflection, refraction, or dispersion, and post them on Instagram. When the photos were presented to the class, one student revealed an intriguing photo, similar to Fig.…

By describing the motion of a charged particle in the well-known nonuniform field of a current-carrying long straight wire, a variety of teaching/learning opportunities are described: 1) Brief review of a standard problem; 2) Vector analysis; 3) Dimensionless variables; 4) Coupled differential equations; 5) Numerical solutions.

Consider a skier who goes down a takeoff ramp, attains a speed "V", and jumps, attempting to land as far as possible down the hill below (Fig. 1). At the moment of takeoff the angle between the skier's velocity and the horizontal is [alpha]. What is the optimal angle [alpha] that makes the jump the longest possible for the fixed magnitude of the…

The following kinematics problem was given to several students as a project in conjunction with a first-semester calculus-based physics course. The students were asked to keep a journal of all their work and were encouraged to keep even their scrap paper. The goal of the project was to expose the students to the process of doing theoretical…

Newton's second law is one of the cornerstones of the introductory physics curriculum, but it can still trouble a large number of students well after its introduction, hobbling their ability to apply the concept to problem solving and to related concepts, such as momentum, circular motion, and orbits. While there are several possibilities for…

Halfway through the 2015 AFC Championship game between the New England Patriots and Indianapolis Colts, game officials discovered that the Patriots were using underinflated footballs on their offensive snaps. A controversy ensued because the Patriots had actually supplied these balls to the game's referee just hours before kickoff. Athletes and…

The problem of a ladder leaning on a wall has been a staple of introductory physics for years. It is discussed in numerous physics textbooks and in journals. Now, it even has an Internet presence. Postings from students seek help for "ladder on a wall" problems. A quick review of those postings would show that they all deal with frictionless…

A typical textbook problem in rotational dynamics involves calculating the angular acceleration of a massive pulley due to a string, such as in the example shown in Fig. 1. The string is assumed to be massless and to move without slipping over the pulley, which is mounted on a frictionless axle. If T[subscript L] and T[subscript R] are the…

The Lorentz velocity addition formula for one-dimensional motion presents a number of problems for beginning students of special relativity. In this paper we suggest a simple rewrite of the formula that is easier for students to memorize and manipulate, and furthermore is more intuitive in understanding the correction necessary when adding…