Full code

Clear["Global`*"]
(*References:​​NEWJOURNALOFPHYSICS.​​​​"The physics of badminton"​​​​CarolineCohen,BaptisteDarboisTexier,DavidQuéréandChristopheClanet.​​​​LadHyX,UMR7646duCNRS,EcolePolytechnique,91128PalaiseauCedex,France​​PMMH,UMR7636duCNRS,ESPCI,75005Paris,France​​*)
TIME=2;​​mC=0.004;(*Shuttlecockcorkmass(Csphere)[kg]*)​​mB=0.001;(*Shuttlecockskirtmass(Bsphere)[kg]*)​​d=0.06;(*Shuttlecockskirtdiameter(Bsphere)[m]*)​​Lgc=0.03;(*Distancebetweenthecenterofmassandcenterofpressure[m]*)​​ρ=1.204;(*Densityofair[kg/m^3]*)​​Cd=0.65;(*CoefficientsofDrag[-]*)​​V=13.4;(*ShuttlecockinitialVelocity[m/s]*)​​γ=58Degree;(*ShuttlecockinitialAngle[∘]*)​​g=9.81;(*Gravitationalacceleration[m/s^2]*)​​S=π*
2
d
4
;(*Crosssection(Bsphere)[m^2]*)​​m=mC+mB;(*Shuttlecocktotalmass[kg]*)
(*FlipModel-------------------------------------*)
In[]:=
Phi=NDSolveValue​​ϕ''[t]+
ρ*S*Cd*V
2*mB*1+
mB
mC
*ϕ'[t]+
ρ*S*Cd*
2
V
2*(mC+mB)*Lgc
*Sin[ϕ[t]]0,​​ϕ[0]π,​​ϕ'[0]20​​,​​ϕ,{t,0,TIME};
In[]:=
Plot[{Phi[t],π,2π},{t,0,1},PlotLegends"Expressions",PlotRangeAll]
Out[]=
0.2
0.4
0.6
0.8
1.0
3
4
5
6
7
Phi(t)
π
2π
(*Trajectories------------------------------------------------------------------------------------------------------------*)
In[]:=
{X,Y}=NDSolveValuem*x''[t]-
1
2
*ρ*S*Cd*
2
x'[t]
,m*y''[t]==-m*g-
1
2
*ρ*S*Cd*
2
y'[t]
,x[0]==0,y[0]==0,WhenEvent[y[t]0,end=t;"StopIntegration"],​​x'[0]==V*Cos[γ],y'[0]==V*Sin[γ],{x,y},{t,0,Infinity};
In[]:=
ParametricPlot[{X[t],Y[t]},{t,0,end}]
Out[]=
1
2
3
4
5
0.5
1.0
1.5
2.0
2.5
3.0
In[]:=
{sss=FindRoot[Y[t]0,{t,1}],{"X max =",X[t]/.sss}}//Quiet
Out[]=
{{t1.41284},{X max =,5.28464}}
In[]:=
Clear["Global`*"]
In[]:=
T=20;(*Temperature[∘Celsius]*)​​Δ=0.6;(*Relativehumaidity[%/100]*)​​p=101325;(*Absolutepressure[Pa]*)​​Md=0.028964;(*Molarmassofdryair[kg/mol]*)​​Mv=0.018016;(*Molarmassofwatervapor[kg/mol]*)​​R=8.314;(*Universalgasconstant[J/(K*mol)]*)​​mC=0.0035;(*Shuttlecockcorkmass(Csphere)[kg]*)​​mB=0.0025;(*Shuttlecockskirtmass(Bsphere)[kg]*)​​d=0.06;(*Shuttlecockskirtdiameter(Bsphere)[m]*)​​dc=0.03;(*Shuttlecockcorkdiameter(Csphere)[m]*)​​Lgc=0.03;(*DistancebetweenthecenterofmassandCpoint(centerofmassCsphere)[m]*)​​Lgb=
mC*Lgc
mB
;(*DistancebetweenthecenterofmassandBpoint(centerofmassBsphere)[m]*)​​Cd=0.65;(*CoefficientsofDrag[1]*)​​V=10;(*ShuttlecockinitialVelocity[m/s]*)​​γ=30Degree;(*ShuttlecockinitialAngle[∘]*)​​g=9.81;(*Gravitationalacceleration[m/s^2]*)​​S=π*
2
d
4
;(*Crosssection(Bsphere)[m^2]*)​​s=π*
2
dc
4
;(*Crosssection(Csphere)[m^2]*)​​m=mC+mB;(*Shuttlecocktotalmass[kg]*)​​psat=100*6.1078*
7.5*T
T+237.3
10
;(*Saturationvapourpressureofwater[Pa]*)​​pv=Δ*psat;(*Pressureofwatervapor[Pa]*)​​pd=p-pv;(*Partialpressure[Pa]*)​​ρ=
pd*Md+pv*Mv
R*(T+273.15)
;(*Densityofair[kg/m^3]*)​​
In[]:=
{Phi,X,Z}=NDSolveValue​​ϕ''[t]*(mB*
2
Lgb
+mC*
2
Lgc
)-ρ*S*Cd*V*(V*Sin[ϕ[t]]+Lgb*ϕ'[t])*Lgb-ρ*s*Cd*V*(-V*Sin[ϕ[t]]+Lgc*ϕ'[t])*Lgc,​​m*x''[t]-
1
2
*ρ*S*Cd*
2
x'[t]
,​​m*z''[t]==-m*g-
1
2
*ρ*S*Cd*
2
z'[t]
,​​WhenEvent[z[t]0,TIME=t;"StopIntegration"],​​x[0]==0,​​z[0]1,​​x'[0]==V*Cos[γ],​​z'[0]==V*Sin[γ],​​ϕ[0]Pi,​​ϕ'[0]20​​,​​{ϕ,x,z},{t,0,Infinity};

Spinning net shot