Ciphers based on rotor machines were the state-of-the-art in the mid-1900s, with arguably the most famous being the German Enigma. We have found that students have great interest in the Enigma and its cryptanalysis, so we created our own rotor cipher that is simulated with shifting tables and can be cryptanalyzed. Ours and the historic rotor…

We show that the total distance traveled by an under-damped oscillating spring mass system with sinusoidal displacement results in a nice, closed-form expression.

The wave equation with variable tension is a classic partial differential equation that can be used to describe the horizontal displacements of a vertical hanging chain with one end fixed and the other end free to move. Using a web camera and TRACKER software to record displacement data from a vibrating hanging chain, we verify a modified version…

This article presents an application of standard undergraduate ODE techniques to a modern engineering problem, that of using a tuned mass damper to control the vibration of a skyscraper. This material can be used in any ODE course in which the students have been familiarized with basic spring-mass models, resonance, and linear systems of ODEs.…

When thinking of models for sinusoidal waves, examples such as tides of the ocean, daily temperatures for one year in your town, light and sound waves, and certain types of motion are used. Many textbooks [1, p. 222] also present a "Ferris wheel description problem" for students to work. This activity takes the Ferris wheel problem out of the…

Making connections is one of the most important foundations involved in learning mathematics. Two projects are presented in this article: one involving Newton's Second Law of Motion and the other involving the determination of star numbers, a type of figurate number. The two invoke seemingly different modalities for students at different levels of…

A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…

For a given orbital period and eccentricity, we determine the maximum time lapse between the winter solstice and the spring equinox on a planet. In addition, given an axial precession path, we determine the effects on the seasons. This material can be used at various levels to illustrate ideas such as periodicity, eccentricity, polar coordinates,…

An aspect of the long history of the struggle to quantify motion is briefly traced through three extended problem sets suitable for use in first year calculus courses. The sketch culminates with an application of simple mathematics to one of the most current problems in the theory of the universe. (Contains 2 figures.)

We describe a project for calculus and differential equations students involving trajectories of a spacecraft whose propulsion system depends solely on muting gravitational effects of heavenly bodies. In particular, we consider the spacecraft imagined by H. G. Wells, and focus on getting his spacecraft from the Moon to the Earth. Cavorite, angular…

Brownian motion is an important stochastic process used in modelling the random evolution of stock prices. In their 1973 seminal paper--which led to the awarding of the 1997 Nobel prize in Economic Sciences--Fischer Black and Myron Scholes assumed that the random stock price process is described (i.e., generated) by Brownian motion. Despite its…

We present a project for a differential equations class: an analysis of an H. G. Wells story in which is given a submersible's descent time, its ascent time, and its eruption from the surface before splashdown. (Contains 5 figures.)