JOURNAL
OF
SPORTS SCIENCE &
MEDICINE

Research
article

A STUDY OF SHUTTLECOCK'S TRAJECTORY IN BADMINTON 

LungMing Chen^{1}, YiHsiang Pan^{2} and YungJen Chen^{3} 

^{1}Department of Computer and Communication, SueTe University, Taiwan ROC, ^{2}Graduate Institute of Physical Education, National Taiwan Sport University, Taiwan ROC, ^{3}Taipei County EllChorng Elementary School, Taipei County , Taiwan ROC. 



© Journal of Sports Science and Medicine (2009) 8, 657  662 


ABSTRACT  
The main purpose of this study was to construct and validate a
motion equation for the flight of the badminton and to find the relationship
between the air resistance force and a shuttlecock's speed. This research
method was based on motion laws of aerodynamics. It applied aerodynamic
theories to construct motion equation of a shuttlecock's flying trajectory
under the effects of gravitational force and air resistance force. The result
showed that the motion equation of a shuttlecock's flight trajectory could
be constructed by determining the terminal velocity. The predicted shuttlecock
trajectory fitted the measured data fairly well. The results also revealed
that the drag force was proportional to the square of a shuttlecock velocity.
Furthermore, the angle and strength of a stroke could influence trajectory.
Finally, this study suggested that we could use a scientific approach to
measure a shuttlecock's velocity objectively when testing the quality of
shuttlecocks. And could be used to replace the traditional subjective method
of the Badminton World Federation based on players' striking shuttlecocks,
as well as applying research findings to improve professional knowledge
of badminton player training.
Key words: Projectile motion, aerodynamics, badminton. 

INTRODUCTION  
Most studies in the past explored the speed of shuttlecocks. Virtually
no research has been done on the flight trajectory of the badminton shuttlecock.
By understanding the flight trajectory of the badminton shuttlecock we
can predict the speed, time, direction and path, and this would provide
helpful information for training of players. 

METHODS  
When a shuttlecock is in flight, according to Newton's Second Law:
In equation (1): is gravitational force, is aerodynamic drag force, and is buoyancy. As a matter of fact, the measured air buoyancy of standard shuttlecock, for example which has a volume of 19 cm^{3} in the air (density~1.205kg·m^{3}), to be about 0.02 gw. We may neglect its very slight influence, when compared with the force of gravity and aerodynamic drag. The magnitude of aerodynamic drag force depends on the relative speed of the shuttlecock through the air, and its direction is always opposite to the direction of the shuttlecock. Generally, the magnitude of resistance force can be expressed as:
Where v is the speed of the shuttlecock relative to air, parameter n is a real number and b is a constant that depends on the properties of the air and the shape and dimension of the shuttlecock. The two parameters b and n in general were determined by experiments. As a shuttlecock is falling down vertically, the speed and resistance force will increase (Eq. 2). The rate of acceleration becomes zero when the increased resistance force eventually balances the weight. At this point, the shuttlecock reaches its terminal velocity V_{T} and from then on it continues to move with zero acceleration. After this point, the motion of a shuttlecock is under constant velocity. The terminal velocity can be obtained from Eq. (1) and (2) by neglecting buoyancy and setting α= dv/dt = 0. This gives
Here, measuring the parameters b and n could be replaced measuring its terminal velocity V_{T}.This helps us to find the eligible trajectory of the shuttlecock. Resistance force could be modeled in two ways, either proportional to object speed or to the speed squared. Assume the shuttlecock is hit with initial velocity , then, the horizontal and vertical velocities are expressed by
where angle sinӨ_{i} is the initial angle. n = 1, we have
F_{vx} = bv_{x} and F_{vy} = bv_{y} are the x  component and y  component of air resistance. We consider vertical and horizontal directions
respectively.
The value of vy is "+" for upward and "" for downward. Then we may integrate Eq. (6) to obtain vertical velocity.
and height
Where m is the mass of shuttlecock, v_{yi}
is the initial vertical velocity and we have taken y = 0 at t = 0, g is
the gravitational acceleration, and
is the terminal velocity.
(b) Horizontal component:
We may integrate Eq. (10) to obtain horizontal velocity
and horizontal distance
Combine Eq. (8) and (12), then we have the equations of the trajectory as
n = 2, consider the air resistance force of the
shuttlecock as ,
here and are
the x  component and y  component of air resistance force, respectively.
Obviously, the factor v in each of the force component expressions is
the essential coupling between the x and y equations that prevents any
analytic solutions from being found. These equations can only be solved
satisfactorily accurately using analytical method. But in order to describe
the role of terminal velocity and approach the practical trajectory of
a shuttlecock, let us consider motion in the vertical and horizontal directions
separately.
After integration, we have the solution of vertical velocity as
Where _{} is the terminal velocity. When the shuttlecock reaches the highest point where we have v_{y} = 0, , the time of flight 't' is expressed as follows:
At this moment, the height of apex is
B. Horizontal direction: here we have
and could find the horizontal speed as
and horizontal distance as
Combine Eq.(17) and (20), we have the equation of the trajectory as
Theoretically, the relationship between air resistance and speed was revealed in n and b. At first sight, it seems that to find the trajectory of the shuttlecock, we need to measure n and b in the beginning. However, from the derived results of Eq. (13) or Eq. (21), we have found that the coefficients n and b could be determined by the terminal velocity v_{t}. In other words, if the terminal velocity was measured, the shuttlecock's trajectory might be found. 

RESULTS  
Figure 1 is the experimental schematic diagram, where X is the video camera, Q_{A} is the initial shuttlecock position, Q_{Adj} is the position after parallax correction, and d is the distance between the shuttlecock and scale. We found that the terminal velocity of some shuttlecocks ranged from 6.51 to 6.87 m·s^{1}. One of the experimental shuttlecocks has the mass of 5.19 g, a head diameter of 2.70 cm and neck diameter of 4.20 cm, a tail diameter of 6.50 cm and a headtotail length of 9.00 cm. The shuttlecock was released from a height of 18 m, and experimental y(t) was measured. In Figure 2a and 2b, it was found that the best equation for measuring and calculating a shuttlecock trajectory was that of the quadratic air resistance force. From the measured data, we could plot the v(t)t diagram in Figure 2c and the terminal velocity 6.86 m·s^{1} was found. This is very close to the value 6.80 m·s^{1} that Peastrel, Lynch and Angelo (1980) had measured. Figure 3 and 4 are the theoretical trajectories from Eq. (13) and (21). Where Y (n = 1) and Y1(n = 2) in Figure 3 both have same initial velocity v_{i}= 30 m·s^{1} Ө = 30^{o} but consider the different condition tha F_{v} = bv and F_{v} = bv^{2} respectively. We found the shuttlecock landed with a horizontal distance of 17.48 m and 9.65 m respectively. Therefore, the shuttlecock in the first case will land out side of the standard court, which has a length of 13.41 m (44 ft). Generally, in practice, the initial speed of a shuttlecock in overhead stroke hit is below 56 m·s^{1}. In other words, at Ө = 30^{o} the initial speed of most players is often greater than v_{i} = 30 m·s^{1}. Very often, the flying shuttlecock lands inside the court. This also helps us to judge intuitionally. What is the relationship between air drag resistance force and the speed of a shuttlecock that have been described above. Apparently, the possibility of the first condition, which states that air drag resistance force is proportional to the speed of a shuttlecock (i.e. F_{v} = bv), is very low. Nevertheless, we must point out, in particular, the relationship between air drag resistance force and the speed of a shuttlecock. The trajectory of a shuttlecock could be described by using terminal velocity. However, we can test again with fast overhead stroke that has an initial velocity of v_{i} = 56 m·s^{1} Ө = 30° and 60°. Their trajectories Y2 (n = 2) and Y3 (n = 2) are also shown in Figure 3. From Eq. (16) and (17), we also found that the shuttlecock needs 0.88s and 0.94s to reach the top of 6.81m and 9.31m high respectively, and totally 2.33s and 2.63s to alight with a horizontal distance of 12.52m and 10.65m. From Eq. (15) and (19), we could also found that the vertical and horizontal velocities. For example, if the initial angle is adjusted to 60 degree, their vertical and horizontal landing velocities are 6.4 m/s and 1.5 m·s^{1}. Their ratio is about 4.3 : 1, which interplays the almost vertical landing phenomenon observed. Next in Fig. 4, the trajectories of clear, smash, and drop have their initial velocity of v_{i} = 45m/s, Ө = 0°, v_{i} = 70m/s, Ө = 10° and v_{i} = 25m/s, Ө = 20° were shown. Y4 (n=2), Y5 (n=2), and Y6 (n=2) respectively. Their reaction time for received player also could be found if necessary. 

DISCUSSION  
The special structure of
a shuttlecock makes its trajectory perform unsymmetrical motion when playing.
From the terminal velocity of a shuttlecock, we could find that the air
drag force is proportional to the square of the speed of the shuttlecock.
The result is consistent with Peastrel, Lynch and Angelo (1980), who performed an experiment
on vertical fall to measure the terminal velocity of a shuttlecock. They
found that the best model was the quadratic air resistance force. Tong,
2004
performed an identical experiment that also came up with the same result.
Furthermore, the angle and strength of a stroke could influence its trajectory.
And the equation we found can predict the trajectory of the shuttlecock.
Some known values were substituted to check the experimental data. This
equation also works adequately on the motion of a shuttlecock under smashing.
In this paper, we have considered both the linear and the quadratic air
resistance force laws respectively. We found that the trajectory of a
shuttlecock could be expressed in terms of its terminal velocity, which
means it is unnecessary to find the other parameters, like air drag force.
The results revealed that the motion equation of a shuttlecock's flying
trajectory could fit experimental data. It also revealed that the angle
and force of a stroke could influence trajectory, therefore playing an
important part in making strategic plans. This should be helpful for badminton
player training. These findings were the same as what Tsai et al. (1995,
1997) had come up with. From
the motion equations, it is easy to understand why the order of a shuttlecock
speed is: jumpsmash, smash, clear, and drop among badminton forehand
overhead strokes  because stroke force and angle can affect the trajectory
of a shuttlecock significantly. 

AUTHORS BIOGRAPHY  
LungMing CHEN Employment: Professor, Department of Computer and Communication, SueTe University, Taiwan ROC. Degree: PhD. Research interests: Sport biomechanics. Email: chenlm@mail.stu.edu.tw 

YiHsiang PAN Employment: Assistant Professor, Graduate Institute of Physical Education, National Taiwan Sport University, Taiwan ROC. Degree: PhD. Research interests: Physical Education teaching and training. Email: poterpan@seed.net.tw 

YungJen CHEN Employment: Taipei County EllChorng Elementary School, Taiwan ROC. Degree: BSc. Research interests: Physical education teaching and training. Email: chicken1008@yahoo.com.tw 
