Merry-go-round

The equation of motion in of a ball a stationary x,y plane is defined by:

MatrixForm@{(x[t]==(Subscript[v, x])*t-R/2),(y[t]==(Subscript[v, y])*t)}//TraditionalForm

(* Naming the rotational coordinate system x'، y' and is obtained by multiplying the stationary coordinate system by the rotation matrix λ*)

(MatrixForm[{x',y'}]==MatrixForm@RotationMatrix[ω*t].MatrixForm[{x[t],y[t]}])//TraditionalForm

(* subsituting x,y by their equations and compute the matrix multiplication yields*)

x[t_]:=Vx*t-R/2

y[t_]:=Vy*t

MatrixForm@(RotationMatrix[ω*t].{x[t],y[t]})//TraditionalForm

The trajectory of this equation of motion in a stationary coordinate system is shown below:

(* Taking R=5, ω=5, Vx=6, Vy=4*)

R=5; ω=5; Vx=6; Vy=4;

Manipulate[Show[ParametricPlot[{R*Cos[t],R*Sin[t]
},{t,0,2Pi},PlotStyle->{Black,Thickness[0.005]}],ListPlot[{Table[{R*Cos[ω*l+w],R*Sin[ω*l+w]},{w,0,10,Pi/5}],Table[{R*Cos[ω*l+w],R*Sin[ω*l+w]},{w,0,10,Pi/5}]},PlotStyle->{{Yellow,PointSize[0.05]},{Red,PointSize[0.025]}}],ParametricPlot[{{x[t],y[t]}},{t,0.0000001,l},PlotStyle->Blue],ListPlot[{{x[l],y[l]}
}, PlotStyle->{PointSize[0.02],{Black,Green}}]],{l,0,1.02}]

The trajectory of ball when we observe it in a rotating non-inertial frame:

Manipulate[Show[ParametricPlot[{R*Cos[t],R*Sin[t]
},{t,0,2Pi},PlotStyle->{Black,Thickness[0.005]}],ListPlot[{Table[{R*Cos[w],R*Sin[w]},{w,0,10,Pi/5}],Table[{R*Cos[w],R*Sin[w]},{w,0,10,Pi/5}]},PlotStyle->{{Yellow,PointSize[0.05]},{Red,PointSize[0.025]}}],ParametricPlot[{RotationMatrix[ω*t].{x[t],y[t]}},{t,0.0000001,l},PlotStyle->Blue],ListPlot[{RotationMatrix[ω*l].{x[l],y[l]}}, PlotStyle->{PointSize[0.02],Gray}]],{l,0,1.02}]

It is important to show this observation, both of the trajectories will exit the merry-go-round at the same time, which gives us an intuition that the ball have physically exited the merry-go-round no matter what frame you are observing it from:

Manipulate[Show[ParametricPlot[{R*Cos[T],R*Sin[T]
},{T,0,2Pi},PlotStyle->{Black,Thickness[0.005]}],ListPlot[{Table[{R*Cos[-ω*t+w],R*Sin[-ω*t+w]},{w,0,10,Pi/5}],Table[{R*Cos[-ω*t+w],R*Sin[-ω*t+w]},{w,0,10,Pi/5}]},PlotStyle->{{Yellow,PointSize[0.05]},{Red,PointSize[0.025]}}],ParametricPlot[{RotationMatrix[ω*T].{x[T],y[T]},{x[T],y[T]}},{T,0.0000001,t},PlotStyle->Blue],ListPlot[{RotationMatrix[ω*t].{x[t],y[t]}}, PlotStyle->{PointSize[0.02],Gray}],ListPlot[{{x[t],y[t]}}, PlotStyle->{PointSize[0.02],Green}]],{t,0,1.02}]

Another interesting observation is when we set ω = 0, the trajectories will follow the same path - as expected:

Manipulate[Show[ParametricPlot[{R*Cos[T],R*Sin[T]
},{T,0,2Pi},PlotStyle->{Black,Thickness[0.005]}],ListPlot[{Table[{R*Cos[-ω*t+w],R*Sin[-ω*t+w]},{w,0,10,Pi/5}],Table[{R*Cos[-ω*t+w],R*Sin[-ω*t+w]},{w,0,10,Pi/5}]},PlotStyle->{{Yellow,PointSize[0.05]},{Red,PointSize[0.025]}}],ParametricPlot[{RotationMatrix[ω*T].{Vx*T-R/2,Vy*T},{Vx*T-R/2,Vy*T}},{T,0.0000001,t},PlotStyle->Blue],ListPlot[{RotationMatrix[ω*t].{Vx*t-R/2,Vy*t}}, PlotStyle->{PointSize[0.03],Gray}],ListPlot[{{Vx*t-R/2,Vy*t}}, PlotStyle->{PointSize[0.02],Green}]],{{t,0,"t"},0,1.02},{{ω,0},1}]