High school college-prep physics students will use the iterative engineering design process to build a model rocket capable of carrying a specific payload. During the test flights, the students will work both to ensure their designs fly with a minimal amount of horizontal velocity, and that they fly as high as possible. The students will measure and record the maximum height attained by their rockets.
The students will then describe the forces acting on the rockets when they are in flight, including a drag force model that varies with the square of velocity. Using Newton’s laws of motion, the students will develop the equations describing the motion of the rockets. The entire flight of each rocket is simulated by numerically integrating the equations of motion using Euler’s method of integration in an Excel spreadsheet-based simulation. This fairly detailed simulation models flight parameters that change over time, such as the overall mass, drag force, force of the rocket engine, etc. Such a simulation can be used to analyze a complex real life problem such as this and to gain insight about its motion. The goal for students goes beyond having them simply plug numbers into their numerical simulators as instructed — this lesson aims to give students a deep understanding of exactly how each cell of the simulator was calculated and why it makes sense to do it that way.
Determining the drag coefficient of the rocket is a challenging problem. The students will be able to use the numerical simulator to "back out" the value of the drag coefficient for their model rocket from the maximum height it achieved. Students will then be presented with the final challenge of predicting their rocket’s maximum height when it is launched with a different powered engine. This is accomplished by using the previously determined drag coefficient and swapping the rocket engine’s thrust model in the simulation. The rocket’s maximum height will be measured independently and compared with the maximum height from the simulation. This exercise will provide students with both a measure of the accuracy of their prediction and a platform from which to discuss the inherent errors associated with solving complex real life problems.
An advanced physics extension to the lesson prompts students to create their own numerical simulator, similar to the one provided.
The lesson also invites interdisciplinary collaboration between math and science teachers by providing extensions that guide students through deriving and/or using more advanced mathematical approaches to measuring maximum altitude, measuring the center of pressure of the rocket, analyzing the effect of improving the accuracy of estimated values, and predicting the instantaneous force of an engine through the use of piecewise functions. Several technology extensions are briefly described as well, should the necessary equipment be available.
This project is an ideal platform for students to see the practical application of mathematical concepts (such as Riemann sums) to which they have been exposed in their higher-level mathematics courses. Additionally, this project takes advantage of the cognitive maturity of calculus-based physics students, motivating them to use high-level mathematical paradigms (such as the Euler method of integration) to bring relevance to the theory they have learned in their math and science courses.