The Hyperbolic Tangent as an Educational Tool for Teaching Variable Acceleration

TL;DR

This paper introduces the hyperbolic tangent as an educational tool to approximate solutions for variable acceleration problems, enhancing teaching methods and student understanding of non-constant motion in physics.

Contribution

It develops an approximate analytical solution to the Falling Astronaut Problem using hyperbolic tangent functions, demonstrating its pedagogical value for teaching variable acceleration.

Findings

Hyperbolic tangent provides a useful approximation for variable acceleration motion.

The technique enhances pedagogical approaches in physics education.

Potential applications extend beyond classroom teaching.

Abstract

This paper presents an approximate analytical solution to the Falling Astronaut Problem by means of the hyperbolic tangent, and it explores the educational opportunities presented by this technique. The author's previous paper presented a function for time in terms of position t(x) that modeled the motion of an astronaut as she falls from an arbitrary height to the surface of a spherical planet with no air resistance, but an exact analytical function for position in terms of time x(t) was not found. This paper derives an approximate function for x(t) using kinematic equations with constant acceleration and "switch functions," specifically the hyperbolic tangent. The paper concludes with a discussion of the pedagogical implications of the technique, its potential for deepening student understanding of non-constant motion, and applications beyond the classroom.