Mathematical Modeling of Badminton Shuttlecock Trajectory and Flight

This report provides a comprehensive analysis of the factors influencing the trajectory of a badminton shuttlecock. It begins by highlighting the importance of aerodynamics and the unique bluff body characteristics of the shuttlecock, which contribute to its high drag and steep flight path. The report then explores the mathematical modeling of the shuttlecock's trajectory, emphasizing the roles of gravitational force and air resistance. It delves into the impact of stroke force and stroke angle on flight direction, referencing research on smash, jump-smash, clear, and drop shots. The analysis incorporates Newton's second law of motion, examining the forces at play, including gravitational force, aerodynamic drag, and buoyancy. The report then develops equations to model the shuttlecock's flight, considering the impact of air resistance, Reynolds number, and terminal velocity. It differentiates between the effects of feather and synthetic shuttlecocks. The paper provides the equations for vertical and horizontal velocities, and the horizontal distance travelled. The report concludes by emphasizing the terminal velocity's significance in determining the flight trajectory and highlights the importance of stroke angle and force in influencing the shuttlecock's path and landing position. The report also references experimental data on the drag coefficient of feather and synthetic shuttlecocks in relation to the Reynolds number.

Modeling the trajectory of the shuttlecock in badminton 1
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Modeling the trajectory of the shuttlecock in badminton 2
Modeling the trajectory of the shuttlecock in badminton
Aerodynamic plays a key role in the speed at which the shuttlecock of badminton is able
to exhibit. Finding the relationship between air resistance and the shuttlecock speed is important
to understand the trajectory path of the badminton shuttlecock. First, it is important to understand
that badminton shuttlecock has a bluff body (Alam, Chowdhury, Theppadungporn and Subic,
2010). This helps to develop high aerodynamic drag and steep flight trajectory and therefore
making it to behave different from other sport balls. The flight trajectory is affected by the
gravitational force and air resistance. Developing a proper modeling of shuttlecock trajectory
will result in determining the final velocity of the badminton shuttlecock (Mehta, Alam and
Subic, 2008). Although the sport is much common, there is no much information on its flight
trajectory. This paper will be able explore mathematical exploration in modeling the flight
trajectory for the badminton shuttlecock. The following equation has been used to define the
flight trajectory of shuttlecock.
Equation 1
In the past, different researchers have been able to research on different sections on the
movement and aspect of badminton shuttlecock movement. Under their studies, the researchers
were able to look at the speed of the shuttlecock (Smits & Ogg, 2013). The major analysis found
that the speed of the shuttlecock is related to the jump-smash, smash, clear and drop. The speed
of the shuttlecock is widely affected by the initial stroke force. In addition, the stroke angle
affects the flight direction of the shuttlecock and this affect the trajectory. Therefore the two

Modeling the trajectory of the shuttlecock in badminton 3
factors which affect the flight trajectory of the badminton shuttlecock are the stroke force and
stroke angle. According to Chang (2010), execution stroke actions of smash and jump-smash is
able to show a wide extension of upper arm, a sharp elbow joint angle and accelerated wring
angular velocity. From the analysis of player, Chang found that the clear and drop paths were
slower than the smash and jump-smash stroke paths.
In addition to the stroke force and stroke angle, the flight trajectory of badminton
shuttlecock is also affected by the air resistance force. It is clear that this resistance will affect the
path which the shuttlecock has to follow and the speed which it will move with (Seo, Kobayashi
& Murakami, 2004). Reynolds number, R is mostly used to determine if the laws of linearity or
quadratic are applied in air resistance. The analysis of the air resistance will therefore help in the
analysis of the trajectory flight of the shuttlecock. This paper will be able to develop
mathematical equation based on the different factors which are able to affect the flight path of
the shuttlecock (Chen, Pan and Chen, 2009). This will be achieved by analyzing the velocities of
the shuttlecock and the different factors which affect the velocity.
In addition, it is important to understand the shuttlecock in order to be able to properly
formulate its flight trajectory. The shuttlecocks are made of different materials and dimensions.
The most common types of shuttlecock include the feather shuttlecock and the synthetic
shuttlecock (Alam et al., 2009). The feather shuttlecock is made of 16 goose feathers. It has a
skirt diameter of 65mm, a mass of around 5.2 grams and total length of 85mm.

Modeling the trajectory of the shuttlecock in badminton 4
Figure 1: different types of shuttlecocks (Julien et al., 2011)
First, Newton laws are important in analyzing the trajectory of the shuttlecock. According
to Newton second law of motion, the force is equal to the mass of the body multiplied by the
acceleration (Chang, 2010). For a moving shuttlecock, different forces will be analyzed to
determine forces which play at each angle. The following equation represents the Newton second
law;
W + FV + B = ma
Equation 2
From the above equation, W is the gravitational force, Fv is the aerodynamic drag which
the shuttlecock experiences ad B is the buoyancy. The shuttlecock moves in air and it buoyancy
factor helps it to float on air. This factor as well is able to affect the flight trajectory. The
buoyancy factor will be related to the air density. Since air density will be able to change from
time to time, the flight trajectory of the shuttlecock will similarly change (Chen, Pan and Chen,
2009). Nevertheless, the effect of the buoyancy in flight trajectory of the shuttlecock is minimal
compared to the contribution from the gravitational forces and aerodynamic drag. All these
forces are affected by different factors which contribute to the flight trajectory (Tong. 2012). For

Modeling the trajectory of the shuttlecock in badminton 5
instance, the aerodynamic drag force will be depended on relative speed of the shuttlecock
through the air. The aerodynamic drag is usually in the opposite direction of the shuttlecock and
therefore it contributed to slower velocity. The aerodynamic drag is therefore a resistance force
to the shuttlecock. In general, the amount of resistance is represented by;
Fv = byn
Equation 3
From the above equation, v is the speed of the shuttlecock relative to air, while n is a real
number and b is a constant which depends on air properties and shape of the shuttlecock. b and n
are generally determined through experiments (The Engineering Tool Box, 2009). As the
shuttlecock falls down through a vertical line, the above equation shows that the speed and
resistance force will increase. When the resistance force is in balance with the shuttlecock
weight, the overall rate of acceleration turns out to be zero. At that point, the shuttlecock will
reach its terminal velocity, VT and then continues to move at zero acceleration. Then the
shuttlecock will achieve a constant velocity rate which it will be moving with. Equation 1 and 2
can be used to get the terminal velocity. At this point, the buoyancy force will be neglected since
its contribution is minimal (Tong. 2012). In addition, the acceleration a will be set at 0. Since the
aerodynamic is a resistance force, it will be a negative factor thus showing it will contribute
negative force reducing the speed. This will therefore give;
mg – bvnT = 0 therefore vT = (mg/b)1/n
Equation 4

Modeling the trajectory of the shuttlecock in badminton 6
Therefore at this point, the parameter of b and n can be replaced with the terminal
velocity VT. The terminal velocity is therefore an important factor which will help in finding the
flight trajectory of the shuttlecock. In addition, the resistance force which the shuttlecock faces
can be formulated in two different ways (Alam et al., 2009). One is that is proportional to the
speed of the shuttlecock and second to the square of the speed. When the shuttlecock is hit, it
usually has an horizontal factor as well as an vertical factor of the velocity. When the shuttlecock
is hit with an initial velocity vi, the horizontal and vertical velocities factors are expressed as
follows;
Vxi = vi cosθi and vyi = vi cosθi
Equation 5
The angle θi is the initial angle which the shuttlecock is able to make with the horizontal
line. The amount of resistance will therefore be composed of the horizontal and vertical drag
forces (The Engineering Tool Box, 2009). Setting n = 1, the vertical and horizontal air drag
forces will be;
Fv=Fvxi + Fvyj
Equation 6
Fvx = bvx and Fvy = bvy , where the x is the horizontal air drag and y is the vertical air drag
forces.
In order to formulate a proper trajectory, let now consider each direction alone. Looking
at the vertical component of the air resistance force,
-mg – bvy = m(dvy/dt)

Modeling the trajectory of the shuttlecock in badminton 7
Equation 7
The value of vy will move in two directs of up and down. The upward movement will be
considered positive (+) and the downward direction negative (-). Intregrating the above equation
leads to;
Vy (t) = (vt + vyi) e-gt/vi - vt
Equation 8
And height of;
Y (t) = vt/g (vt + vyi)(1 – e-gt/vt) – vtt
Equation 9
m will be the mass of the shuttlecock, vyi is the initial velocity which is taken at y = 0 and
t = 0, g is the gravitational acceleration and vt is the terminal velocity represented as mg/b.
equation 7 is able to provide the top of the trajectory, which will be the maximum height which
the shuttlecock will be able to attain (Alam et al., 2008). At that point, the vy will be zero and this
leads to the flight time of the shuttlecock as;
t = vt/g ln (vt + vyi)/ vt
Equation 10
And the horizontal component at vy = 0 will be;
-bvx= m (dvx/dt)
Equation 11

Modeling the trajectory of the shuttlecock in badminton 8
Therefore integrating the above equation to get the horizontal velocity;
Vx = xxi e-gt/vt
Equation 12
While the horizontal distance will be
X = {(vtvxi)g}()1 – e-gt/vt)
Equation 13
In order to get the equation of the trajectory, we combine the equations 8 and 12, which leads to;
Y = x{(vt + vyi)/ vxi} – v2t/g ln{(vtvxi)/ (vtvxi - gx)}
Equation 14
In addition, changing the value of n to be 2 can help to derive another equation of
trajectory. The n and b were factors which were related to air properties. At the end, it is clear
that since the terminal velocity can be measured; there is no need to carrying out experiment to
find these two factors (Alam et al., 2008). These equations are able to prove that the terminal
velocity is the important factor which is required to come up with the shuttlecock flight
trajectory. Measuring the terminal velocity of the shuttlecock is able to ensure that the flight
trajectory can be found at different time and horizontal distances.
Moreover, it is clear that the structure and shape of the shuttlecock makes it to have
unsymmetrical motion and form of trajectory. Using the terminal velocity, it is clear that the air
resistance force is proportional to the square of the shuttlecock speed. This is seen when the
terminal velocity of the shuttlecock is formulated using n = 2. The best model to find the

Modeling the trajectory of the shuttlecock in badminton 9
relationship between the vertical fall and terminal velocity was found to be the quadratic air
resistance force (Tong, 2012). In addition, the angle of stroke and the strength of the stroke are
other key factors which influence the shuttlecock trajectory. These two factors will be able to
work on both the linear and quadratic air resistance force laws. As per the above formulation, it
is therefore not important to find parameters such as air drag forces. This is because the terminal
velocity factor is able to factor the factors and able to find the trajectory with them. Therefore the
most important factors to consider while playing badminton are the angle and forces of stroke.
These two will be able to define the path and trajectory of the shuttlecock and the position which
it is able to land (Alam, Subic, Watkins, Naser, Rasul, 2008). The two factors are able to give the
motion equations which define the flight trajectory path. In addition, fitting experimental data on
the equations helps to understand the shuttlecock speed of the smash-jump, smash, clear and
drop. This is explained by the equations showing the trajectory of the shuttlecock.
In addition, according to experiment by Julien et al., (2011), the trajectory of the
shuttlecock is able to depend on the type of shuttlecock used and also the drag coefficient. In the
experiment, the drag coefficient was compared with the Reynolds number. The experiment
aimed at determining the effects of these two factors on the trajectory path for the feather and
synthetic shuttlecocks (Chang, 2010). From the experiment, the authors were able to show that
the mostly the drag coefficient was able to increase with increase in the Reynolds number.

Modeling the trajectory of the shuttlecock in badminton 10
Figure 2: Drag Coefficient Vs Reynolds number for feather and synthetic shuttlecocks
(Julien et al., 2011)
In addition, according to the different drag coefficients, the authors were able to analyze
the flight trajectory path for the feather and synthetic shuttlecocks. The shuttlecocks were
analyzed at different speeds of 40, 70, 100 and 130 km/h. in addition, it has to be noted that the
authors in this section were able to rely on the earlier developed equations of trajectory (Alam et
al., 2008). After the analysis, the authors were able to agree that their trajectory paths are able to
agree with the developed equations. Most importantly, the authors were able to show that the
feather shuttlecock is able to process a ore steep curve at the end o the flight than the synthetic
shuttlecock.

Modeling the trajectory of the shuttlecock in badminton 11
Figure 3: flight Trajectory for feather shuttlecock (Julien et al., 2011)
Figure 4: Flight trajectory for synthetic shuttlecock (Julien et al., 2011)
In addition, according to the above diagrams, Julien et al., (2011), showed that the
trajectory path will also depend on the type of material used to make the shuttlecock. The
trajectory paths show that the horizontal distance in the feather shuttlecock case is much shorter
than the synthetic shuttlecock. The altitude or the maximum height also is able to vary. The
synthetic shuttlecock is able to have a high attitude than the feather shuttlecock. The authors

Modeling the trajectory of the shuttlecock in badminton 12
noted that the initial speed was similar as well as the launch angle for both shuttlecocks. Even
with the differences in the types of shuttlecocks used, Julien et al (2011) was able to agree that
the results for the formulas developed were correct. The results achieved from these experiments
were able to agree to the trajectories equations developed.
The flight trajectory is in form of a parabola. There are different angles which the
shuttlecock can be launched on as seen above. These will be able to define the flight trajectory.
Defining two points, the following equation can be used to calculate the height at different
instances.
y = a(x-h)2+ k
Equation 15
The stroke angle as it is seen is able to define the position where the shuttlecock will land
and therefore defining the horizontal distance.
In conclusion, the modeling equations of the flight trajectories were able to confirm that
the terminal velocity is important in defining the shuttlecock trajectories. From the equations, it