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Lat. Am. J. Phys. Educ. Vol. 8, No. 4, Dec. 2014
4505-1
http://www.lajpe.org
Modelling the flight characteristics of a soccer ball
Jeffrey Leela, Donna M. G. Comissiong, Karim Rahaman
Department of Mathematics and Statistics, Faculty of Science and Technology,
The University of the West Indies, St Augustine Campus, Trinidad and Tobago.
E-mail: Donna.Comissiong@sta.uwi.edu
(Received 3 June 2014, accepted 30 November 2014)
Abstract
During a football game the ball is airborne a large percentage of the time. For this reason it is important to understand
to principles that govern the motion of the ball in flight. This paper explores the various equations associated with the
movement of the ball in flight. Numeric techniques are utilized to solve some of the equations and represent them
graphically. The parameters in the model are manipulated to show how the graphs vary and agree with real life results.
Keywords: Mathematical modelling, drag coefficient, flight of ball, trajectory, velocity of ball.
Resumen
En un partido de fútbol la pelota está en el aire durante un gran porcentaje del tiempo. Por esta razón, es importante
entender los principios que gobiernan el movimiento de la bola en vuelo. Este artículo explora las diversas ecuaciones
asociadas con el movimiento de la pelota en vuelo. Se utilizan técnicas numéricas para resolver algunas de las
ecuaciones y representarlas gráficamente. Los parámetros en el modelo son manipulados para mostrar cómo los
gráficos varían y están de acuerdo con los resultados de la vida real.
Palabras clave: Modelos matemáticos, coeficiente de resistencia, vuelo de la bola, trayectoria, velocidad de la bola.
PACS: 01.40.Fk, 01.40.G, 89.90.+n ISSN 1870-
9095
I. INTRODUCCIÓN
Newton’s second law of motion F=ma where F is a force,
m is a mass, and a is the acceleration is used to determine
the flight of a ball. In the general case there are three forces
acting on the ball, the force of gravity and two forces
arising from the interaction with the air. For the latter of the
two forces due to the air the drag acts in a direction
opposite to the ball’s velocity. The other force is the
Magnus force which acts in the presence of spin and
deflects the ball perpendicular to the velocity and to the
axis of spin. If the spin axis of the ball is parallel to the
ground, the Magnus force can provide lift. However, if the
spin axis of the ball is perpendicular to the ground, the ball
is made to bend flight.
We begin by considering the case where the effects of
air are negligible. There is no horizontal force acting on the
ball therefore, the equation for the horizontal velocity u is:
0=
dt
du
m
.
Since there is no horizontal force acting on the ball the
horizontal component of velocity is constant, and u is equal
to the initial horizontal velocity u0. The horizontal
displacement, x is therefore:
tux o
=
. (1)
The equation for the vertical component of velocity v, is:
mg
dt
dv
m−=
,
where g is the acceleration due to gravity. This equation has
the solution:
gtvv o−=
,
where is the initial vertical velocity. The velocity v at time t
is obtained from:
dt
dy
v=
.
Therefore, the vertical displacement is obtained by
integrating:
gtv
dt
dy
o−=
,
to get:
Jeffrey Leela, Donna M. G. Comissiong and Karim Rahaman
Lat. Am. J. Phys. Educ. Vol. 8, No. 4, Dec. 2014
4505-2
http://www.lajpe.org
2
1
2
o
y v t gt=−
. (2)
Using equation (1) to eliminate t in equation (2) gives the
equation for the trajectory:
2
2
1
2
o
oo
vg
yx x
uu
=−
. (3)
This is the equation of a parabola.
The range R, of the flight is obtained by putting
0y=
in equation (3) since the ball starts off at y = 0 and returns
to the ground. On doing this gives for the range,
g
uv
Roo
2
=
. (4)
The time of flight is given by the time t=T, at which the
displacement y returns to zero. From equation (2) this is
given by:
g
v
To
2
=
.
Suppose that the initial angle between the initial trajectory
and the ground is θ0, then:
00 0 00 0
sin and cosvV uV
θθ
==
, (5)
where the initial velocity V0 is given by:
222
ooo uvV +=
.
In terms of V0 and θ0, the range given by equation (4) is:
2
000
2sin cosV
Rg
θθ
=
,
which is:
g
V
R0
2
02sin
θ
=
.
It is known that sin2θ0. has a maximum value at θ0 = 45o,
this angle gives the maximum range for a given V0,
2
0
max
V
Rg
=
Our attention now shifts to the drag. The equation for such
is given by:
2
0
1
2
d
FCpAV=
. (6)
Where the drag coefficient CD depends on the velocity, is ρ
the density of air, V is the velocity of the ball and A is its
cross–sectional area, in our case πa2, which is the maximum
cross–section, where is the ball’s radius.
Equation (6) as simple as it appears gives rise to
associated equations of motion that are rather involved.
This is due to the fact that the velocity depends on CD
and also the drag force couples the equation for the
horizontal and vertical components of the velocity.
Newton’s equations now become:
θ
cos
d
F
dt
du
m−=
. (7)
And:
mgF
dt
dv
md−−=
θ
sin
. (8)
where θ is the angle between the trajectory and the ground
at time t ,given by:
u
v
=
θ
tan
. (9)
II. METHODOLOGY
It is important to note that equation (7) to (9) do not have an
algebraic solution, but can be solved numerically. For
simplicity, CD is taken to be constant during the flight. For
our case we use CD = 0.2 since it is highly dependent on the
Reynolds number. Wind tunnel testing has determined
various drag coefficients for different shapes. For a sphere,
it ranges from 0.07 to 0.5. A smooth sphere is around 0.5
and a rough sphere is around. A football’s surface can be
described somewhere in between these values.
(www.grc.nasa.gov/www/k-12/airplane/shaped.html).
Using
and sin
1
θ
Vv =
θ
cos
1Vu =
equations (7) and
(8) can now be written as:
Vu
dt
du
1
1
α
−=
, (10)
gVv
dt
dv −−=1
1
α
. (11)
Where:
2
1
2
1
2uvV +=
. (12)
And:
m
A
CD
ρα
2
1
=
.
Equation (10) to (12) will be solved with CD = 0.2.
The density of air taken to be 1.2 kg m-3 at a
temperature of 20°C the mass of the ball approximately
0.3969 kgm-3 and its cross sectional area is 0.39m2, giving
the value α = 0.015m-1. (The Science of Soccer, John
Wesson). By solving for u and v we can integrate u and
v
dt
dy =
to obtain x and y respectively.
Modelling the flight characteristics of a soccer ball
Lat. Am. J. Phys. Educ. Vol. 8, No. 4, Dec. 2014
4505-3
http://www.lajpe.org
By manipulating V0 and θ0 we obtain numerical
simulations for the flight path of the ball. However, our
values for v0 and θ0 must be carefully chosen. Most medium
to long range aerial passes in football are kicked with
velocities ranging from 15m/s to 30m/s. Usually a 25m to
30m free kick is struck with a velocity of about 25m/s. This
enables about 1 second to elapse between contact with the
ball entering the goal. (www.soccerball
world.com/Physics.htm). Since our analysis is primarily
concerned with aerial passing during play at medium to
long range distances 20m to 70m, velocities of 15m/s to
30m/s is a good standard. One objective of the player is to
flight the ball with just the right velocity to allow a
teammate to collect and control. If too much velocity is
applied this may be difficult.
Other forces can play a pivotal role in the movement of
the football in flight. Let us consider the effect of wind
moving parallel to the field’s plane having speed w along a
parabolic path. The equations of motion would take the
form:
( )
du uwV,
dt
α
=− −
(13)
dv vV g ,
dt
α
=−
(14)
with V given by:
( )
2
2
2vwuV +−=
. (15)
Notice that the wind only affects the equation that relates to
the horizontal velocity component since it can act in such a
way to either speed up the motion or slow it down. We will
assume that +w corresponds to a trailing wind and –w
corresponds to a headwind.
Again we rely on numerical methods to solve these
equations. It is interesting to note if we make the
transformation
anduw u v v
′′
−→ →
, equations (13) to
(15) take the form of equation (10) to (12) which we are
already familiar with. We can see that u and v are replaced
by
anduv
′′
. If the equations are solved for
anduv
′′
and
anduv
′′
are calculated from
dx u
dt
′′
=
and
dy v
dt
′′
=
, then the
required solution can be achieved using the inverse
transformations:
,,
,.
uu w vv
xxwt yy
′′
=+ =
′′
=+ =
We now present numerical simulations for both the
horizontal and vertical displacement of a football when the
angle to the horizontal is fixed and the initial velocity of the
ball is varied between 15m/s and 30m/s. This is done in the
case of a trailing wind and a headwind.
Furthermore, we fix the velocity and vary the initial
angle of the ball between 10° and 45°. In each instance, we
used wind speeds of 4m/s and 7m/s. Wikipedia has
classified these wind speeds as ranging from light breeze to
moderate breeze. An Olympic sprinter running with a
trailing wind of 4m/s is said to be wind aided and the time
clocked is not recorded as legal. (International Athletics
Federation). This is to say that a football moving through
the air with winds speeds in this range will strongly
influence its behaviour.
Consequently in addition to the wind having a velocity
w moving parallel to the field’s plane, the wind can act
along the sides of the ball. The velocity is still taken as w
but the direction is denoted by z. The motion of the velocity
is this direction is denoted by V. The equation for the
change in velocity in this direction with respect to time is:
( )
Vwv
dt
dv
z
z−−=
α
, (16)
with:
( )
2
222 wvvuV z−++=
.
Solving for α yields 0.015m-1 the above equation can be
solved numerically. However, a simple procedure gives a
formula for the sideways deflection of the ball’s trajectory
which is accurate for most cases. The equation for the
forward motion is:
Vu
dt
du
α
−=
. (17)
Dividing equation (16) by equation (17) gives:
u
wv
du
dv zz −
=
. (18)
Integrating equation (18) gives the solution:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−=
0
1
u
u
wvz
, (19)
where u0 is the initial value of u and vz = 0 initially.
The deflection z is obtained by solving:
z
v
dt
dz =
.
Hence, by using equation (19) for:
0
0
tudt
zwt
u
⎛⎞
∫
=−
⎜⎟
⎝⎠
.
The deflection d, over the full trajectory is therefore:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−=
o
u
R
Twd
,
Jeffrey Leela, Donna M. G. Comissiong and Karim Rahaman
Lat. Am. J. Phys. Educ. Vol. 8, No. 4, Dec. 2014
4505-4
http://www.lajpe.org
where T is the time of flight and R is the range. It is
important to observe that T and R are slightly affected by
the side wind. A good approximation for d is obtained using
their values with no wind.
So far, we haven’t considered what transpires when spin
is applied. A force known as the Magnus effect describes
such motion. When a football is spinning the deflection
caused by the Magnus effect displaces the ball in a
perpendicular direction to the spin axis. This is to say that,
if the spin axis is horizontal, the Magnus effect provides
back lift and it the spin axis is vertical, the Magnus effect
causes the ball’s trajectory to bend sideways.
With medium and long range kicking we are primarily
concerned with getting the ball to lift up high enough to
clear opposing players and travel the required distance.
Back lift produces this type of motion. Conventionally this
force FL is given by:
2
2
1VACF LL
ρ
=
. (20)
by analogy with the drag force given in equation (6) This
formula has its origin in aeronautics and the subscript L
stands for the lift which would occur, for example, on a
wing. For our purpose this expression is somewhat
misleading because
L
C
depends on both the spin and the
velocity.
For a spinning ball
L
C
is proportional to
Vaq /
provided
that
Vaq /
is not too large and it is, therefore, convenient to
write:
sL C
V
aq
C=
,
where p is the angular frequency of the spin and is the
radius of the ball, then:
VqaACF sL
ρ
2
1
=
. (21)
Substituting for the air density
3
21 −
⋅=mkg
ρ
, the radius
a=0.11m and the cross-sectional area
2
0390mA ⋅=
equation
(21) becomes.
3
26 10
Ls
FCqV
−
=⋅×
. (22)
This sideways force produces a curved trajectory and the
force is balanced by the centrifugal force
2wheremV / R, R
is the radius of curvature of the
trajectory. Using equation (22) with the mass of 0.3969 kg,
the resulting radius of curvature is:
qC
V
R
s
153=
. (23)
If we measure the rotation by the number of revolutions per
second, f, then since
π
2/qf =
equation (23) becomes:
fC
V
R
s
24=
. (24)
It’s is more natural to think in terms of sideways
displacement of the ball as illustrated in figure 1 below. If
we approximate by taking the trajectory to have a constant
curvature then using Pythagoras’s equation:
( )
2
2
2RDRL =−+
,
and taking
RD <<
so that
2
D
is negligible:
R
L
D
2
2
=
.
Using equation (24) this becomes:
2
1
metres in
48
s
CLf
DVms
V
−
=
. (25)
FIGURE 1. Matematical construction for the trajectory.
The time of flight is L/V and so the number of revolutions
of the ball during its flights is n=Lf/V. Substitution of this
relation into equation (25) gives:
48
n
C
L
D
s
=
.
We have no direct measurement of
s
C
for footballs but
experiments with other spheres have given values in the
range 0.25 to 1 depending on the nature of the surface.
Taking
0.5
s
C=
we obtain the approximate relation:
100
n
L
D=
(The Science of Soccer John Wesson).
For example, a deviation of 1 m over a length of 30 m
would required the ball to undergo about 3 revolutions.
Modelling the flight characteristics of a soccer ball
Lat. Am. J. Phys. Educ. Vol. 8, No. 4, Dec. 2014
4505-5
http://www.lajpe.org
III. RESULTS
FIGURE 2. Ball trajectories for varying initial velocities (initial angle 10 degrees).
FIGURE 3. Ball trajectories for varying initial velocities with trailing wind 4 m/s (initial angle 10 degrees).
FIGURE 4. Ball trajectories for varying initial velocities with trailing wind 7 m/s (initial angle 10 degrees).
Jeffrey Leela, Donna M. G. Comissiong and Karim Rahaman
Lat. Am. J. Phys. Educ. Vol. 8, No. 4, Dec. 2014
4505-6
http://www.lajpe.org
FIGURE 5. Ball trajectories for varying initial angles with trailing wind 7 m/s (initial velocity 30m/s).
FIGURE 6. Ball trajectories for varying initial velocities with head wind 4 m/s (initial angle 10 degrees).
Modelling the flight characteristics of a soccer ball
Lat. Am. J. Phys. Educ. Vol. 8, No. 4, Dec. 2014
4505-7
http://www.lajpe.org
FIGURE 7. Ball trajectories for varying initial angles with head wind 4 m/s (initial velocity30m/s).
FIGURE 8. Ball trajectories for varying initial velocities with head wind 7 m/s (initial angle 10 degrees).
FIGURE 9. Ball trajectories for varying initial angles with head wind 7 m/s (initial velocity30m/s).
Jeffrey Leela, Donna M. G. Comissiong and Karim Rahaman
Lat. Am. J. Phys. Educ. Vol. 8, No. 4, Dec. 2014
4505-8
http://www.lajpe.org
The following graphs show the vertical and horizontal
displacements of the ball when back spin is applied. The
amount of spin determines to amount of lift. First, we have
the code.
FIGURE 10. Initial velocity vs sideways displacement.
FIGURE 11. Initial velocity vs number of revolutions during
flight.
FIGURE 12. Vertical displacement vs sidedways displacement.
FIGURE 13. Vertical displacement vs number of revolutions
during flight.
IV. DISCUSSION AND CONCLUSIONS
It is also important to note that the ratio of f/V appearing in
equation (25) is related to the ratio of the rotational energy
to the kinetic energy. This ratio is
2
2
2
1
2
1
Vm
wI
E
E
K
R=
and since:
2
3
2amI =
2
320⎟
⎠
⎞
⎜
⎝
⎛
⋅=
v
f
E
E
K
R
,
1−
sminV
.
For the example, a ball travelling at
( )
1
41330 −
⋅smmph
with a spin of 3 revolutions per second has a rotational
energy of
%61 ⋅
of its kinetic energy. Our results clearly
demonstrate the close link between our numerical
simulation and reality. The trajectories with drag force
acting against the motion of the ball will definitely affect its
range as seen in figure 2 and 3. It clearly demonstrates that
the maximum projectile range will not be achieved with a
launch angle of 45°. The results show that a trailing wind
will increase the range of the trajectory for the same
velocity and launch angle with just drag whereas, a
headwind will decrease the trajectory. This is quite
consistent with the laws of Physics.
The results obtained when spin is applied shows that the
ball bends perpendicular to the spin axis. If the spin axis is
perpendicular to the ground the ball deviates left or right
depending on if the rotation is clockwise or anti-clockwise.
If the spin axis is parallel to the ground, the spin has two
forms. These are known as topspin where the ball is rotated
towards the ground or lift where the rotation is towards the
Modelling the flight characteristics of a soccer ball
Lat. Am. J. Phys. Educ. Vol. 8, No. 4, Dec. 2014
4505-9
http://www.lajpe.org
air. The graphs show the more intense the upward rotation,
the more lift is generated. This in turn has the effect of
increasing the range of the trajectory. Soccer players utilize
this technique in long range passing. These results can
inform coaches and other personnel in the sport how to
implement certain approaches in their training regime to
develop greater accuracy and efficiency in medium and
long range passing. A possible follow up paper can explore
the how the materials, design, size and shape used in
various soccer balls can influence their behaviour in flight.
REFERENCES
[1] Dunlop, J., Free Flight Aerodynamics of Sports Balls
(2003). Consulted in: uniaxe.com.au/.../Free%20Flight%20
Aerodynamics%20of%20Sports%2.Accessed in: May 10,
2011.
[2] Drag force on a sphere. Consulted in;
www.scribd.com/doc/5174368/Drag-Force-on-a-Sphere.
Accessed April 30, 2011).
[3] Erichson, H., Maximum projectile range with drag and
lift, with particular application to golf, American Journal of
Physics 51,357-362 (1982).
[4] International Athletics Federation. Consulted in:
http://www.iaaf.org/home. Accessed in: September 16,
2011.
[5] Shape effects on drag. Consulted in: www.grc.nasa.gov/
www/k-12/airplanes/shaped.html. Accessed in: August 31,
2011.
[6] Soccer Ball Physics. Consulted in: www.soccerball
world.com/Physics.htm. Accessed in: April 30, 2011.
[7] Wesson J., The Science of Soccer, (Institute of Physics
Publishing, Bristol and Philadelphia, 2002).
[8] Magnus_effect. Consulted in: en.wikipedia.org/wiki/
Magnus_effect. Accessed in: April 30, 2011.
