Rotating Launch

td = 0.02;
a = 1 - td 4 Pi;
Rsoln[ti_] :=
  NDSolve[{x''[t] == -x[t]/(x[t]^2 + y[t]^2)^(3/2),
    y''[t] == -y[t]/(x[t]^2 + y[t]^2)^(3/2),
    x[0] == Cos[\[Theta][ti]], y[0] == Sin[\[Theta][ti]],
    x'[0] == (a - td \[Theta]'[ti]) Sin[\[Theta][ti]],
    y'[0] == -(a - td \[Theta]'[ti]) Cos[\[Theta][ti]]}, {x[t],
    y[t]}, {t, 0, 10}];
Lsoln[ti_] :=
  NDSolve[{x''[t] == -x[t]/(x[t]^2 + y[t]^2)^(3/2),
    y''[t] == -y[t]/(x[t]^2 + y[t]^2)^(3/2),
    x[0] == Cos[\[Theta][ti]], y[0] == Sin[\[Theta][ti]],
    x'[0] == -(a + td \[Theta]'[ti]) Sin[\[Theta][ti]],
    y'[0] == (a + td \[Theta]'[ti]) Cos[\[Theta][ti]]}, {x[t],
    y[t]}, {t, 0, 10}];

\[Theta][t_] :=
  If[t < 0, Pi/2,
   If[t <= 1, Pi/2 + 2 Pi t^2, 4.5 Pi - 2 Pi (t - 2)^2]];
Manipulate[
 lsoln = Rsoln[\[Tau]];
 rsoln = Lsoln[\[Tau]];
 Show[
  ParametricPlot[
   First[{x[t], y[t]} /. lsoln /. {t -> tt}], {tt, 0, 0.7},
   PlotStyle -> Directive[Thick, Red],
   PlotRange -> {{-1.1, 1.1}, {-1.1, 1.1}}, Axes -> None],
  ParametricPlot[
   First[{x[t], y[t]} /. rsoln /. {t -> tt}], {tt, 0,
    0.7 + 2.8 Max[Min[\[Tau], 2 - \[Tau]], 0]},
   PlotStyle -> Directive[Thick, Blue]],
  Graphics[{
    Point[{Cos[\[Theta][\[Tau]]], Sin[\[Theta][\[Tau]]]}],
    FaceForm[RGBColor[0.3, 0.5, 0.8]],
    Disk[{0, 0}, 0.9], White,
    Table[
     Line[{{0, 0},
       0.9 {Cos[\[Theta][\[Tau]] - i],
         Sin[\[Theta][\[Tau]] - i]}}], {i, 0, 2 Pi - 0.01, Pi/4}]
    }]
  ], {\[Tau], -0.2, 2}]
(*it's that easy*)