The Brachistochrone & Optimal Paths: Trajectory Optimization Under Gravity

Investigating time-minimizing pathways using differential calculus, computational Python modeling, and physical track trials.

A comprehensive study examining how acceleration distributes along candidate paths under gravity to identify the true path of quickest descent. Combines classical mechanics, numerical simulation, and custom-built experimental apparatus.

Key Project Metrics & Highlights:

Academic Mentorship

Directed by Dr. Omer Avci (University of Ottawa, former IMO Medalist).

Methodology

Analytical cycloid derivation, acceleration distribution modeling, and timed experimental validation using weighted bead tracks.

Current Status

Manuscript in development for submission to high school research journals and national STEM competitions.

The Brachistochrone & Optimal Paths: Trajectory Optimization Under Gravity

Theoretical Framework & Mathematical Derivation

Euler-Lagrange equations applied to path optimization.
Theoretical derivation of the cycloid as the true solution to the Brachistochrone problem.
Visual vector diagrams showcasing gravity components along varied curve profiles.

Computational Modeling & Python Simulations

Interactive/Visual Python simulation plots comparing linear, parabolic, and cycloidal trajectories.
Code snippets showcasing numerical time-of-flight integrations under friction and idealized conditions.

Experimental Apparatus & Physical Testing

High-resolution photography of custom-fabricated physical tracks.
Methodology for timed trials using weighted beads released across differing curve geometry.
Comparative data table showcasing predicted theoretical transit times vs. empirical experimental results.

Results, Conclusions & Manuscript

Error analysis discussing friction losses, rotational kinetic energy of beads, and real-world deviation from ideal cycloids.

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