A presentation describing how to simulate a soccer ball in flight in a simple way. A useful idea for a lab or short project.

1. 1
The trajectory of a soccer ball
How to simulate it: A Short Workshop
www.physicsandsport.com/en
Vassilios M Spathopoulos
Lecturer
Department of Aircraft Technology
Technological Educational Institute of Central Greece

2. 2
Motivation for workshop
 The trajectory of a soccer ball in flight is governed by
complex aerodynamic mechanisms similar to those
determining the motion of an aircraft

3. 3
Personal interest
 As an aeronautical engineer and a keen football fan,
have become interested in soccer ball flight mechanics
 This has lead to the development of a flight model in
the Matlab® environment (see figure above)
 One can also simulate the trajectory quite easily in the
Excel® environment
 This is what this presentation will describe!

4. 4
Airplane vs soccer ball
 The motion of both airplane and soccer ball is driven by
aerodynamic and gravitational forces
v

mg
Fdrag
Fmag
ω

5. 5
Workshop aims
 Present and familiarise with the basic aerodynamic
mechanisms involved
 Apply basic flight mechanics principles in order to
calculate the trajectory of a soccer ball in flight
 Use a trajectory simulation in the Excel® environment in
order to obtain important parameters relating to typical
free kicks

6. 6
Force diagram
 Gravitational force
Downward direction
 Drag
Direction opposite to velocity
 Magnus force
Direction normal to the plane determined by
the velocity and the spin vector
Note: This force diagram ignores side forces
due to seam orientation!
v

mg
Fdrag
FMagnus
ω

7. 7
Gravitational force
 Constant direction and magnitude
 m = 0.43 kg
 g = 9.81 m/s2
gW m
v

mg
ω

8. 8
Drag
 Varying direction and magnitude
 Direction opposite to velocity
 ρ = 1.2 kg/m3
 Α =0.038 m2
 Cd = Function, primarily, of the Reynolds number,
 D = 0.22 m
 μ = 18.2 μPa.s
u
u
uD dCA
2
2
1
μ
ρUD
Re

9. 9
Generation of drag (pressure drag)
•The flow separates from the surface producing a low pressure wake
behind the ball.
•The difference in pressure produces the drag force opposing the motion
of the ball.

10. 10
Drag crisis
•Once a critical Reynolds number, and therefore speed, is
exceeded, the flow changes from laminar to turbulent thus
delaying the separation and therefore producing a smaller
wake and smaller drag coefficient.
•The critical Reynolds number (and therefore the speed) at
which the drag is drastically reduced, is lower for rough
spheres.
•It is for this reason that golf balls are designed with dimples,
thus increasing their range.

11. 11
Magnus force
 Varying direction and magnitude
 Direction normal to the plane determined by the
velocity and spin vectors
 Cmag = function, primarily, of the spin parameter
U
ωR
s
uω
uω
uFmag magCA
2
2
1

12. 12
Generation of Magnus force
•Due to the rotation, at the upper surface we have a later flow
separation than at the lower one and as a result the wake is
deflected downwards.
•As a reaction to this deflection (Newton’s 3rd Law), an upwards
force is exerted on the ball.

13. 13
Effect of Magnus force
By determining the tilt of the rotation axis of the ball, the
player can produce the desired ball trajectory!

14. 14
Simulation methodology
 We divide time into small steps, dt = 0.01s
 If we know the values of x,y,z, vx,vy,vz at time t
 At time t+dt we have,
x(t+dt)=x(t)+ux(t)*dt
y(t+dt)=y(t)+uy(t)*dt
z(t+dt)=z(t)+uz(t)*dt
ux(t+dt)=ux(t)+ax(t)*dt
uy(t+dt)=uy(t)+ay(t)*dt
uz(t+dt)=uz(t)+az(t)*dt
 Numerical method: of Euler
 We need to know ax(t), ay(t), az(t)
 It is reminded that,
...,, 2
2
xx a
dt
xd
u
dt
dx

15. 15
Equations of motion of a soccer ball in flight
222
dt
dz
dt
dy
dt
dx
U
,
2
magmag C
m
A
k ,
2
dd C
m
A
k
Where,
sin2
2
dt
dy
k
dt
dx
kU
dt
xd
magd
dt
dy
k
dt
dz
dt
dx
kU
dt
yd
dmag cossin2
2
g
dt
dy
k
dt
dz
kU
dt
zd
magd cos2
2
angleaxisspin

16. 16
Drag and Magnus coefficients
For speeds greater than 15 m/s we can assume
that the drag coefficient has a constant value of
0.15.
For spin parameters greater than 0.2 we can
assume that the Magnus coefficient is equal to
the spin parameter (great simplification!)

17. 17
The simulation..
Has several simplifying assumptions:
o The drag coefficient is assumed constant throughout a
flight
o The Magnus coefficient is assumed to be equal to the
spin parameter
o Side forces due to seam orientation are ignored
o The spin rate is assumed constant
o A very simple numerical integration technique is
employed in order to solve the equations of motion
It can portray the basic features of soccer
ball flight!

18. 18
Simple hand calculation
Estimate the lateral deflection of the ball for
a free kick 20 yards from goal for 7 rev/s
sidespin given an initial velocity of 25 m/s
Assume that:
o Motion only occurs in the x-y plane (2D)
o The drag force is neglected
o The Magnus coefficient is constant at 0.2
Use Newton’s 2nd Law to solve!
2.0
25
0.117π2
U
Rfπ2
U
ωR
sCmag

19. 19
Solution
98.22.025038.0256.1
2
1
2
1 22
magCAuFmag
2
m/s93.6
43.0
98.2
m
F
a
mag
x
m
U
y
atax xx 85.1
25
28.18
93.6
2
1
2
1
2
1
22
2
Keep a note of the answer and compare to that obtained from the simulation!

20. 20
For more info on the physics of sports..
Please refer to my website:
www.physicsandsport.com/en
Thank you!