We describe an easy and fun project using water rockets to demonstrate applications of single variable calculus concepts. We provide procedures and a supplies list for launching and videotaping a water rocket flight to provide the experimental data. Because of factors such as fuel expulsion and wind effects, the water rocket does not follow the parabolic model of a textbook projectile, so instead we develop a one-variable height vs. time polynomial model by interpolating observed data points. We then analyze this model using methods suitable to a first semester calculus course. We include a list of questions and partial solutions for our project in which students use calculus techniques to find quantities not apparent from direct observation. We also include a list of websites and other resources to complement and extend this project.

Water rockets provide an easily implemented, engaging activity that allows students to experience first-hand how scientists use mathematical modeling and the tools of calculus to determine properties not apparent from the raw data alone. In this activity, students develop a model for the rocket height of a single water rocket launch after videotaping it in front of a building of known dimensions and then use calculus concepts to analyze the rocket performance. Water rockets (see Figure 1) are cheap, re-usable, easy to launch, and have a very high fun-to-nuisance ratio. They also provide an example of some of the fundamental aspects of projectile behavior, while factors such as the propulsion mechanism and wind effects give them more complex flight paths than those of simple projectiles. Although there is a lot of available information about combustible-fuel model rockets (see section V), they move too fast for easy measurements and require too much space. Water rockets are not as expensive nor logistically difficult as combustiblefuel model rockets, yet they have just enough complexity in their flight paths to provide an opportunity to put the skills and concepts learned in calculus to work in a substantial way.

One of the main goals of the activity is to provide a physical context incorporating many central calculus concepts. Another objective is for students to create and interpret a model and realize that it provides information not available from the experimental data alone. In particular, once the rocket rises above the backdrop of the building, there is no longer a frame of reference to estimate its height experimentally, thus necessitating a mathematical model. Also, questions such as finding the impact velocity and maximum height can only be addressed with a model; these quantities cannot be determined from the raw data or even from the videotape. Students also explore some of the limitations of such modeling efforts.

Most standard calculus (or physics) textbooks include a parabolic model describing the flight of a projectile. In the one-dimensional case, the function s (t) = v^sub 0^t - ½ gt^sup 2^ models the vertical position of a projectile with respect to time t, where g = 32.2 ft/sec^sup 2^ is Earth's gravitational constant and v^sub 0^ is the initial velocity. Other than the initial acceleration, this model considers only gravity acting on the rocket, and ignores air resistance, for example.

Experiments demonstrating this parabolic model often require access to a large windless space (such as a hangar) and a mechanism to measure the initial velocity, neither of which was readily available to us. Rather than trying to design an experiment to demonstrate the parabolic model, or to fit a parabolic model to the experimental data, we decided to explore the vagaries introduced by wind (irregular gusts, not just air resistance) and varying propulsion by modeling individual water rocket launches.

To collect data, students first launch the rockets against a backdrop of known height, such as a building, and videotape the experiment. Then, they use the dimensions of the backdrop, the position of the camcorder, and an analysis of the videotape to determine the rocket height at several times. With these data points and a graphing calculator or CAS, students can use curve fitting tools to construct a higher order polynomial model of the height of the rocket as a function of time. We use polynomial interpolation since it is easier to motivate within our curriculum than, for example, a least squares fit, and a simple interpolation command is available in Maple. However, any curve fitting technique available on a graphing calculator or CAS would work as well.

This project is intended for students taking single variable differential calculus. Students must be able to determine and interpret first and second derivatives. Substantial trigonometry and three-dimensional geometry are required for estimating the heights from the videotape of the rocket against the building, which provides an excellent review and valuable context for these mathematical foundations. Students will need to solve several equations in several variables for the polynomial interpolation, or use a graphing calculator or CAS with polynomial interpolation capability. Advanced students familiar with calculus in three dimensions may do more advanced modeling and analysis, such as developing a parameterized curve for the rocket's flight path and using it to consider such aspects as arc length and tangential velocities