In this note, the classical Doppler shift for some accelerated mechanical systems is considered under a common graphical approach. In one dimension, we study uniform accelerated motion and simple harmonic motion. In two dimensions, uniform circular motion and pendular motion are considered. In each case, an elementary treatment shows that the…

At the undergraduate level, the composite motion of simple harmonic vibrations has always been the main content of physics as well as several other scientific disciplines. Many textbooks tell us clearly that when the frequency ratio [omega][subscript 1]/[omega][subscript 2] of two perpendicular vibrations is simple integer ratio n[subscript…

Traditional school laboratory exercises on a system of moving objects connected by strings involve deriving expressions for the system acceleration, a = (?F)/m, and sketching a graph of acceleration vs. force. While being in the form of rational functions, these expressions present great opportunities for broadening the scope of the analysis by…

The goal of this paper is to illustrate different ways that cardsorting activities (or "card stacks") can be implemented in the introductory physics classroom, along with various tips and resources for getting started. My first attempt at developing a card stack came about from simply wanting to try out a fun way to change student…

Kinematics is a topic students are unknowingly aware of well before entering the physics classroom. Students observe motion on a daily basis. They are constantly interpreting and making sense of their observations, unintentionally building their own understanding of kinematics before receiving any formal instruction. Unfortunately, when students…

This simple experiment uses an unusual graph straightening exercise to confirm the parallel axis theorem for an irregular object. Along the way, it estimates experimental values for g and the moment of inertia of a tennis racket. We use Excel to find a 95% confidence interval for the true values.

A vehicle starts from rest at constant acceleration, then cruises at constant speed for a time. Next, it decelerates at a constant rate.… This and similar statements are common in elementary physics courses. Students are asked to graph the motion of the vehicle or find the velocity, acceleration, and distance traveled by the vehicle from a given…

In a Montessori preschool classroom, students work independently on tasks that absorb their attention in part because the apparatus are carefully designed to make mistakes directly observable and limit exploration to one aspect or dimension. Control of error inheres in the apparatus itself, so that teacher intervention can be minimal. Inspired by…

Those who come to Cambridge soon learn that the fastest route between Harvard and MIT is by the subway. For many students, this short ride is a quick and easy way to link physics and calculus. A simple, homemade accelerometer provides all the instrumentation necessary to produce accurate graphs of acceleration, velocity, and displacement position…

We have recently developed a Graphs and Tracks model based on an earlier program by David Trowbridge, as shown in Fig. 1. Our model can show position, velocity, acceleration, and energy graphs and can be used for motion-to-graphs exercises. Users set the heights of the track segments, and the model displays the motion of the ball on the track…

In our science for non-science majors course "21st Century Physics," we investigate modern "Hubble plots" (plots of velocity versus distance for deep space objects) in order to discuss the Big Bang, dark matter, and dark energy. There are two potential challenges that our students face when encountering these topics for the…

Mobile device accelerometers are a simple and easy way for students to collect accurate and detailed data on an amusement park ride. The resulting data can be graphed to assist in the creation of force diagrams to help students explain their physical sensations while on the ride. This type of activity can help students overcome some of the…

In this age of the microcomputer-based lab (MBL), students are quite accustomed to looking at graphs of position, velocity, and acceleration versus time. A number of textbooks argue convincingly that the slope of the velocity graph gives the acceleration, the area under the velocity graph yields the displacement, and the area under the…

The Doppler shift is a frequency change of a repetitive effect, as measured by a receiver, due to the motion of the wave emitter, to the motion of the wave receiver, or both. A demonstration of the effect is provided by the sound of a car's horn that changes from a higher pitch to a lower pitch when a car drives past. Most derivations of the…

Reviews the equations of motion and the use of a sine-sine coordinate grid for convenient plotting or analyzing of Lissajous figures. (Author/GS)