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A computational and pedagogical framework for projectile motion using Python visualizations

Leonardi Hernández Sánchez, Francisco Soto Eguibar, Irán Ramos Prieto, Héctor Manuel Moya Cessa

TL;DR

The paper addresses how to teach projectile motion by bridging analytic equations with conceptual understanding using Python-based visualizations. It integrates analytical expressions for key observables, such as $x_h = \frac{v_0^2 \sin(2\theta)}{g}$ and $y_v = \frac{v_0^2 \sin^2(\theta)}{2g}$, with reproducible simulations that produce trajectory plots, parameter-space maps, and iso-curves. Its main contributions are a cohesive computational pedagogical framework, demonstration of angular symmetry in range, and a parameter-space approach that links individual trajectories to global outcomes, all within open-source Python code. This framework facilitates active learning in undergraduate mechanics and is easily adaptable to related contexts like sports, engineering, and computer graphics, while offering avenues to incorporate more complex effects such as air resistance.

Abstract

Projectile motion is one of the most fundamental problems in introductory physics, offering a clear context to connect algebraic reasoning with conceptual understanding. This work presents a computational and pedagogical framework that combines the analytical formulation of projectile motion with interactive visualizations developed in Python. Using reproducible simulations, the dependence of the maximum height and horizontal range on the launch parameters $(v_0,θ)$ is examined through trajectory plots, parameter-space maps, and iso-curves. These visual representations reveal non-trivial combinations of initial conditions that yield equivalent outcomes, reinforcing physical intuition and providing an accessible open-source tool for teaching and learning classical mechanics.

A computational and pedagogical framework for projectile motion using Python visualizations

TL;DR

The paper addresses how to teach projectile motion by bridging analytic equations with conceptual understanding using Python-based visualizations. It integrates analytical expressions for key observables, such as and , with reproducible simulations that produce trajectory plots, parameter-space maps, and iso-curves. Its main contributions are a cohesive computational pedagogical framework, demonstration of angular symmetry in range, and a parameter-space approach that links individual trajectories to global outcomes, all within open-source Python code. This framework facilitates active learning in undergraduate mechanics and is easily adaptable to related contexts like sports, engineering, and computer graphics, while offering avenues to incorporate more complex effects such as air resistance.

Abstract

Projectile motion is one of the most fundamental problems in introductory physics, offering a clear context to connect algebraic reasoning with conceptual understanding. This work presents a computational and pedagogical framework that combines the analytical formulation of projectile motion with interactive visualizations developed in Python. Using reproducible simulations, the dependence of the maximum height and horizontal range on the launch parameters is examined through trajectory plots, parameter-space maps, and iso-curves. These visual representations reveal non-trivial combinations of initial conditions that yield equivalent outcomes, reinforcing physical intuition and providing an accessible open-source tool for teaching and learning classical mechanics.
Paper Structure (6 sections, 13 equations, 3 figures)

This paper contains 6 sections, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theoretical model and fundamentals of projectile motion
  3. Computational visualization
  4. Parameter-space analysis
  5. Discussion and didactic implications
  6. Conclusions

Figures (3)

  • Figure 1: Projectile motion for $v_0=20\,\mathrm{m/s}$ and $\theta=45^\circ$. The plot shows $x(t)$ and $y(t)$ as functions of time, illustrating the parabolic trajectory under uniform gravity.
  • Figure 2: Projectile trajectories obtained from the analytical equations implemented in Python, illustrating how variations in the launch parameters modify the parabolic path.
  • Figure 3: Parameter-space analysis of projectile motion. (a) Horizontal range and (b) maximum height as functions of $(\theta,v_0)$; in both panels, the red curve highlights the iso-level corresponding to the reference configuration ($v_0=20\,\mathrm{m/s}$, $\theta=45^\circ$). (c) The corresponding iso-curves $R=R^*$ and $H=H^*$ shown together, emphasizing their intersection at the reference point.