Dolphin Kick Biomechanics
Mathematical Modelling · Biomechanics
Executed a biomechanical motion-tracking analysis to extract 72 discrete kinematic data points from underwater footage. Constructed a 39-equation piecewise model to map ankle trajectory and optimized the thrust-to-drag ratio via calculus to derive a theoretical maximum velocity of 2.64 m/s.
At a glance (one-pager)
Theoretical maximum velocity validated at 2.64 m/s
Optimization of the derived thrust-drag differential equation yielded a critical point at 2.64 m/s. This mathematically proves that the current kick frequency () is near optimal, isolating physical amplitude as the primary biomechanical bottleneck for performance gains.
Tracking Duration
GoPro at 29.97 fps; tracking ran until foot left frame.
Avg. Amplitude
Mean crest-to-trough across 4 peaks. Calibrated via foot length.
Avg. Frequency
Fitted by Desmos slider; gives ~4 kicks over 2.40 s.
Measured Velocity
Derived from x-displacement over ~30 frames (≈ 1 s).
Optimal Velocity
From setting on the thrust–drag objective.
Piecewise Segments
Required to accurately model the full kick shape.
The underwater dolphin kick is heavily restricted in elite competition due to its unparalleled propulsive efficiency. The objective of this research was to extract empirical kinematic data from human athletic performance and construct a mathematical model precise enough to isolate the biomechanical variables governing optimal swimming velocity.
Methodology & reflection
Modeling human biomechanics exposed the severe limitations of idealized mathematical assumptions. The primary limiting factor of the optimization was assuming a direct proportionality between the amplitude-frequency product and actual velocity without an experimentally determined constant . Future kinematic modeling must incorporate Computational Fluid Dynamics (CFD) to measure true volumetric drag and utilize 3D motion tracking to account for lateral force vectors.
Extracted 72 ankle coordinate pairs across a 2.40-second sample at 29.97 fps using Adobe After Effects motion tracking. Initial hypothesis testing proved a singular sinusoidal function insufficient due to multi-directional concavity reversals and near-vertical data intervals. Constructed a highly precise 39-equation piecewise continuous function using seven distinct algebraic families to map the physical geometry. This geometry was then generalized to extract mean amplitude and frequency parameters for calculus-based optimization.
Formulated an objective function to balance forward thrust against quadratic fluid drag. Thrust was derived using mass flow rate principles yielding a linear term of . Drag was modeled using literature-standard streamlined coefficients () and measured cross-sectional area. Setting the derivative equal to zero isolated the theoretical maximum velocity threshold.