Michal Nowicki Edwin Perezic Zachary Conrad

Outline Basic Understanding of Effect 1. Physical Examples 1. Projectile Motion 2. Roberto Carlos 1997, Occidental Soccer Physics of effect 1. Fluid Dynamics and Aerodynamics 3. Mathematics of Movement 1. ODEs of projectile motion 2. Numerical Method for Solution: Runge-Kutta Method 3. Mathematica Modeling of solution

Introduction to Projectile Motion x = x o +v xo t y = 0 z = z o + v zo t – ½gt 2 x -> displacement in x direction y -> displacement in y direction z -> displacement in z direction g -> gravity constant v -> initial velocity

Physical Examples  Roberto Carlos 1997 Roberto Carlos 1997  Basketball Basketball

Fluid Dynamics Real world is not a vacuum Drag force and lift force F l =-.5p|v| 2 C l Lift coefficient Depends on spin Additional curvature of trajectory

ODEs of Projectile Motion x -> displacement in x direction y -> displacement in y direction z -> displacement in z direction k d -> drag coeffecient g -> gravity constant k l -> lift coeffecient γ -> angle between spin axis and ground plane (x,y plane) v -> initial velocity

Runge-Kutta Numerical Method  Essentially a modified Eulers Method Weighted Averages  Method used by Mathematica to solve ODEs  For all three equations in the ODE x(t 0 )=0 y(t 0 )=0 z(t 0 )=0

Runge-Kutta Cont. y’= f(t,y) y(t 0 )= n 0 t 0 =0 Timestep= h k 1 =hf(t 0,n 0 ) k 2 =hf(t 0 +h/2,n 0 +k 1 /2) k 3 =hf(t 0 +h/2,n 0 +k 2 /2) k 4 =hf(t 0 +h,n 0 +k 3 ) n 1 =n 0 +(k 1 +2k 2 +2k 3 +k 4 ) 6

Mathematica  Due to complexity of the system of ODEs we used Mathematica to solve and model this system

Conclusion  Gained understanding of the effect of the Magnus Effect on the flight of a soccer ball  Found ODEs for motion of a soccer ball  Used Mathematica model to solve the ODEs of flight of a soccer ball  Modeled famous free kicks such as Roberto Carlos 1997 using Mathematica