On the approximation of D.I.Y. water rocket dynamics including air drag

TL;DR

This paper presents a simple approximation method for modeling water rocket dynamics with air drag, enabling accurate predictions with minimal computational effort suitable for educational settings.

Contribution

It introduces an approximation technique based on definite integrals for analyzing water rocket motion including air resistance, validated against numerical solutions and experiments.

Findings

Approximation closely matches Runge-Kutta numerical results.

Analytic solutions for different flight phases are effective.

Drag coefficient estimated and confirmed experimentally.

Abstract

If you want to get accurate predictions for the motion of water and air propelled D.I.Y rockets, neglecting air resistance is not an option. But the theoretical analysis including air drag leads to a system of differential equations which can only be solved numerically. We propose an approximation which simply works by the estimate of a definite integral and which is even feasible for undergraduate physics courses. The results only slightly deviate from the reference data (received by the Runge-Kutta method). The motion is divided into several flight phases that are discussed separately and the resulting equations are solved by analytic and numeric methods. The different results from the flight phases are collected and are compared to data that has been achieved by well explained and documented experiments. Furthermore, we theoretically estimate the rocket's drag coefficient. The result is confirmed by a wind tunnel experiment.