Teaching Taylor series with rocket landing: kinematics, approximations, and engineering insights

Taylor series are fundamental mathematical tools in physics education, essential for approximating functions and understanding physical dynamics. While introductory curricula often emphasize their mathematical derivation, a nuanced discussion of practical implications and the inherent trade-offs in using different orders of approximation. This paper presents a pedagogical case study, designed for undergraduate physics and engineering students and their instructors, that uses a realistic rocket landing scenario to bridge this gap. We comparatively analyze zero-order (constant position), first-order (constant velocity), second-order (constant acceleration), and third-order (constant jerk) Taylor series approximations against a more realistic numerical trajectory, which incorporates a decaying thrust profile. The analysis effectively demonstrates a compelling progression in predictive capability—from ‘utterly flawed’ to providing a ‘safety margin,’ achieving a ‘successful landing,’ and finally approximating a ‘precision landing’ We explicitly highlight the computational trade-offs involved, defined here in terms of floating-point operations (FLOPs) per time step. For instance, the third-order model reduces landing velocity error to less than 20% compared to the true trajectory, but increases the FLOP count relative to the second-order model. This study also aims to foster a deeper understanding of the nature of science itself, revealing that scientific laws and models are fundamentally approximate descriptions, constantly refined but never absolute. It uniquely illuminates the balance between accuracy, computational efficiency, and practical utility in real-world systems, promoting a more critically engaged understanding of fundamental mechanics and the scientific enterprise.