This value allows us to evaluate the projectile terminal velocity in free fall, {{U}_{\infty }}=\sqrt{g\mathcal{L}}=6.7\;{\rm m}\;{{{\rm s}}^{-1}}, which corresponds to the steady state d{\bf U}/dt={\bf 0}. The exposed section S=\pi {{(D/2)}^{2}} is equal to 28\;{\rm c}{{{\rm m}}^{2}} for both shuttlecocks. In order to answer this question, shuttlecock prototypes have been constructed. We conduct an experimental study of shuttlecock mass as a function of humidity conditions at T = 20{{\;}^{\circ}}C. Corresponding results are gathered in table B2 The authors acknowledge C and P Mace and J Careil for bringing our attention to badminton specificities. Parameters influencing \mathcal{L} are determined, such as drag coefficients measured in a wind tunnel, and results are plotted in figure 12. (a) The time interval between each position is 5 ms, the shuttlecock initial velocity is {{U}_{0}}\approx 18.6\;{\rm m}\;{{{\rm s}}^{-1}} and its initial angular velocity is \dot{{{\varphi }_{0}}}=206\;{\rm rad}\;{{{\rm s}}^{-1}}. The different characteristic times arising from (1) can finally be compared to the data. A typical example of the time evolution of is plotted in figure 4. Measurements of the frequency of different strokes (first column) as referenced in figure 19 for all playing shots (second column) and for killing shots (third column). This phenomenon may also contribute to the players' preference for feathered shuttlecocks. Experienced players prefer shuttlecocks submitted to a slightly larger drag, such as feathered ones, in order to hit them violently without exiting the court. The only stroke whose range is short compared to the court size is the net shot. The first one is the flipping time {{\tau }_{f}} needed for to vary from 180° to 0°. In practice, badminton players try to delay stabilization with net drops, in order to prevent the opponent from hitting the projectile correctly.
The dynamics of a shuttlecock and its influence on the badminton game have been questioned.

Time evolution of the angle between the shuttlecock axis of symmetry and its velocity {\bf U}. The effect of rotation on the flight can also be discussed. Badminton is a very simple game but it also involves a lot of physics between the players and the ground and the racket with the shuttle cock and many more.
One may wonder whether rotation induces gyroscopic stabilization. For clear strokes, the trajectory ends with a nearly vertical fall. Because this light particle experiences a large drag, its trajectory is nearly triangular [8] and it highly depends on the projectile properties. Players usually counterbalance this effect by using lighter shuttlecocks when air is hotter. (b) The shuttlecock initial velocity is {{U}_{0}}\approx 10.4\;{\rm m}\;{{{\rm s}}^{-1}} and its initial angular velocity is \dot{{{\varphi }_{0}}}=28\;{\rm rad}\;{{{\rm s}}^{-1}}. When the hit intensity decreases, the dynamics of the shuttlecock slows down. Shuttlecocks have a top speed of up to 137 m s−1 [2]. Table B1. In this regime, x0 highly depends on shuttlecock and air properties through the aerodynamic length \mathcal{L}. For the feathered projectile, it is {{C}_{D\,f}}=0.65\pm 0.05 whereas we have for the plastic one {{C}_{D\,p}}=0.68\pm 0.05. If you are 13 years old when were you born? All Rights Reserved. Inserting the previous relation in equation (13) provides: In the limit S{{M}_{C}}\gg s{{M}_{B}}, we obtain: where M={{M}_{C}}+{{M}_{B}}. All the data (blue dots) are distributed around a line of slope 1.1. For the maximal initial speed ever recorded ({{U}_{{\rm max} }}=137\;{\rm m}\;{{{\rm s}}^{-1}}), the shuttlecock maximum range xmax is 13.8 m [14]. We now discuss how the shuttlecock geometry influences its flipping dynamics, which ideally might explain why a shuttlecock opening angle Λ close to 45 °was selected (figure 2(c)). Equation (15) corresponds to (1) studied in this paper. Figure 1. In this limit, the equation of motion can be integrated with the initial conditions \varphi (t=0)=\pi and \dot{\varphi }(t=0)={{\dot{\varphi }}_{0}}, which yields: Hence, the flipping dynamics of the shuttlecock depends on the pulsation {{\omega }_{0}} and initial angular velocity {{\dot{\varphi }}_{0}}. This remark leads one to neglect the damping term \dot{\varphi }/{{\tau }_{s}} in equation (1). Volume 17, We propose classifying badminton strokes in the diagram drawn in figure 19(b). 17 063001, https://doi.org/10.1088/1367-2630/17/6/063001. Figure 6(a) compares the experimental flipping time {{\tau }_{f\,{\rm exp} }} with the theoretical one {{\tau }_{f\,th}} predicted by solving equation (2). Badminton is played either by two opposing players (singles) or two opposing pairs (doubles). Its location depends on the pressure profile around the projectile. This explains that shuttlecocks generally perform less than a complete turn after an impact with a racket.


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