The Magnus Effect in Soccer: A Comprehensive Analysis
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Michal Nowicki Edwin Perezic Zachary Conrad
Outline
Basic Understanding of Effect
1.
Physical Examples
1.
Projectile Motion
2.
Roberto Carlos 1997, Occidental Soccer 2015
2.
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
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
Explore the Magnus Effect in soccer through a detailed examination of its physical examples, the physics behind it, mathematics of movement, and numerical methods for solution. Delve into real-world applications, such as Roberto Carlos' famous 1997 goal, to understand the impact of fluid dynamics and aerodynamics on projectile motion in sports.
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Presentation Transcript
BEND IT LIKE MAGNUS: AN BEND IT LIKE MAGNUS: AN INSPECTION OF THE INSPECTION OF THE MAGNUS EFFECT IN SOCCER MAGNUS EFFECT IN SOCCER Michal Nowicki Edwin Perezic Zachary Conrad
Outline Basic Understanding of Effect 1. Physical Examples 1. Projectile Motion 2. Roberto Carlos 1997, Occidental Soccer 2015 2. 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 = xo+vxot y = 0 z = zo+ vzot gt2 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 Basketball
Fluid Dynamics Real world is not a vacuum Drag force and lift force Fl=-.5p|v|2Cl 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 kd -> drag coeffecient g -> gravity constant kl -> lift coeffecient -> 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(t0)=0 y(t0)=0 z(t0)=0
Runge-Kutta Cont. y = f(t,y) y(t0)= n0 t0=0 Timestep= h k1=hf(t0,n0) k2=hf(t0+h/2,n0+k1/2) k3=hf(t0+h/2,n0+k2/2) k4=hf(t0+h,n0+k3) n1=n0+(k1+2k2+2k3+k4) 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
