Mathematical Modeling of Water Jet Trajectories in Amusement Parks: Application of Lagrange Interpolation and Projectile Mechanics

Abstract

Introduction: This study explores the physics of water jets in Disney attractions, applying a projectile motion model to design city fountains. The primary goal is to determine the optimal nozzle angle and initial water velocity to maximize jet height and span a tunnel path safely, ensuring water does not fall on passersby. The core hypothesis is that a specific angle-velocity combination exists for achieving this, using classical physics while neglecting air resistance.                            

Aim: The key tasks were to derive the jet's trajectory equation via Lagrange interpolation based on coordinates from a tunnel photograph, identify the optimal parameters for height and range, and examine the model's practical limits under realistic water velocities.

Materials and Methods: A geometric analysis of a tunnel photo was performed using GeoGebra to identify key trajectory points. Mathematical modeling employed Lagrange interpolation to fit a quadratic function. The standard projectile equations for range (l = V₀²sin(2θ)/g) and maximum height (  = v₀²sin²θ/(2g)) were used for parameter optimization. Model validity was assessed using R² values.

Results: The trajectory was best described by quadratic functions, with g(x) = -0.04521x² + 2.23158x - 4.10237 providing a better fit (R² = 0.849). The optimal parameters were an angle θ ≈ 65.89° and an initial velocity v₀ ≈ 25.53 units/s. Slight errors at extreme points indicated the influence of unmodeled factors like air resistance.

Conclusion: The research confirms that water jet paths can be accurately modeled with quadratic projectile equations. The identified parameters are valuable for designing effective and safe water features. Future work should incorporate air resistance for greater realism.