Discuss about the MGT 209 Modeling the Trajectory of the Shuttlecock in Badminton. 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.

## Factors Affecting Flight Trajectory

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 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.

## Analysis of Shuttlecock

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.

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 + F_{V} + B = ma

**Equation ****2**

From the above equation, W is the gravitational force, F_{v} 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 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;

F_{v} = by^{n}

**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, V_{T} 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;

## Newton Laws and Shuttlecock Trajectory

mg – bv^{n}_{T} = 0 therefore v_{T} = (mg/b)^{1/n}

**Equation ****4**

Therefore at this point, the parameter of b and n can be replaced with the terminal velocity V_{T}. 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 v_{i}, the horizontal and vertical velocities factors are expressed as follows;

V_{xi} = v_{i} cosθ_{i} and v_{yi} = v_{i} cosθ_{i}

**Equation ****5**

The angle θ_{i} 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=F_{vx}i + F_{vy}j

**Equation ****6**

F_{vx} = bv_{x} and F_{vy} = bv_{y} , 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 – bv_{y} = m(dv_{y}/dt)

**Equation ****7**

The value of v_{y} 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;

V_{y} (t) = (v_{t} + v_{yi}) e^{-gt/vi }- v_{t}

**Equation ****8**

And height of;

Y (t) = v_{t}/g (v_{t} + v_{yi})(1 – e^{-gt/vt}) – v_{t}t

**Equation ****9**

m will be the mass of the shuttlecock, v_{yi} is the initial velocity which is taken at y = 0 and t = 0, g is the gravitational acceleration and v_{t} 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 v_{y} will be zero and this leads to the flight time of the shuttlecock as;

t = v_{t}/g ln (v_{t} + v_{yi})/ v_{t}

**Equation ****10**

And the horizontal component at v_{y} = 0 will be;

-bv_{x}= m (dv_{x}/dt)

**Equation ****11**

Therefore integrating the above equation to get the horizontal velocity;

## Formulating the Trajectory

V_{x} = x_{xi} e^{-gt/vt}

**Equation ****12**

While the horizontal distance will be

X = {(v_{t}v_{xi})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{(v_{t} + v_{yi})/ v_{xi}} – v^{2}_{t/g }ln{(v_{t}v_{xi})/ (v_{t}v_{xi} - 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 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.

## Conclusion

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.

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.

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 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 is found that the air resistance drag is proportional to the square of the shuttlecock velocity. In addition, the key factors which are found to impact the trajectory of the shuttlecock include the stroke force and angle. In addition, Julien et al., (2011) was able to demonstrate that the material and type of shuttlecock is also a key factor in influencing the flight trajectory. The mathematical modeling was able to show that the motion equations are most important in developing the shuttlecock flight trajectory. The only this required in the formulation of the flight trajectory is the terminal velocity which is done in aerodynamics. In addition, the mathematical modeling also showed that the air resistance force is proportional to the square of the shuttlecock velocity. Most importantly, the model was able to show that the trajectory angle also has key influence on the trajectory as well as the stroke strength. In addition, from Julien et al experiment, it was found that the different shuttlecocks are able to have different trajectories. This showed that although their trajectory paths are able to follow the mathematical models, the materials play important roles in influencing the shape of the trajectory paths. The authors in these experiments were able to show that the Reynolds number and the coefficients of drag also affect the kind of trajectories achieved. All in all, the mathematical formulation is important in explaining the shuttlecock trajectories. The terminal velocity is important to be able to explain and take care of the air factors. Therefore it can be concluded that the trajectories can be influenced by the materials of the shuttlecock, the stroke force and the stroke angle.

References

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Julien Le Personnic, Firoz Alam, Laurent Le Gendre, Harun Chowdhury, and Aleksandar Subic (2011). Flight trajectory simulation of badminton shuttlecocks (PDF Download Available). Retrieved from: https://www.researchgate.net/publication/251716847_Flight_trajectory_simulation_of_badminton_shuttlecocks [accessed Mar 14 2018].

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