x[s_] := s; y[s_] := -Sin[s]

track[s_] := {x[s], y[s]};

ParametricPlot[track[s], {s, 0, π}, AspectRatio -> Automatic, 
      Compiled -> False]

(Subscript[v, 0]) = 0; g = 9.8;

Plot[(Sqrt[Abs])[((x'[s]^2 + y'[s]^2)/(((Subscript[v, 0])^2) - 2  g*y[s]))], {s, 0, 2  π}]

t[σ_] := 
      NIntegrate[(Sqrt[Abs])[((x'[s]^2 + y'[s]^2)/(((Subscript[v, 0])^2) - 2  g*y[s]))], {s, 0, σ}]

g0 = Plot[t[σ], {σ, 0, 3.14158}]

g1 = 
      Plot[(ArcSin[((2  σ)/π) - 1]/π +  .5)*
          t[π], {σ, 0, π}, 
        PlotStyle -> Hue[ .4]]

g2 = 
      Plot[(ArcSin[((Sin[2  σ]/a + 2  σ)/π) - 1]/π +  .5)*t[π], {σ, 0, π}, 
        PlotStyle -> Hue[ .2]]

g3 = 
      Plot[(ArcSin[((Sin[4  σ]/(a^3) + Sin[2  σ]/a + 2  σ)/π) - 1]/π +  .5)*t[π], {σ, 0, π}, 
        PlotStyle -> Hue[ .6]]

Show[g0, g1, g2, g3]

(Sqrt[NIntegrate])[(((t[σ] - (ArcSin[((Sin[2  σ]/a + 2  σ)/π) - 1]/π +  .5)*t[π]))^2), {σ, 0, 3.14158}]

This is the (R^2) for g2

(Sqrt[NIntegrate])[(((t[σ] - (ArcSin[((Sin[4  σ]/(a^3) + Sin[2  σ]/a + 2  σ)/π) - 1]/π +  .5)*t[π]))^2), {σ, 0, 3.14158}]

Solve[(ArcSin[((2  σ)/π) - 1]/π +  .5)*
          t[π] == τ, σ]

Solve[(ArcSin[((2  σ)/k2) - 1]/k2 + (1/2))*
          k1 == τ, σ]

pos[τ_] := -(π/2) (-1 + 
            Sin[((π (t[π] - 2 τ))/(2  t[π]))])

Plot[pos[τ], {τ, 0, 6}]

pos[k_] := 
      If[EvenQ[IntegerPart[(k/t)[π]]], 
        FindRoot[
          t[σ] == k, {σ, 
            Round[(k/t)[π]] 3.1415 +  .00008, 0, π}, 
          Compiled -> False, MaxIterations -> 100], 
        FindRoot[
          t[σ] == 
            2  t[π] - 
              k, {σ, (2 - Round[(k/t)[π]]) 3.1415 +  .00008, 
            0, π}, Compiled -> False, 
          MaxIterations -> 100]]

test[k_] := If[EvenQ[IntegerPart[(k/t)[π]]], even, odd]

k = 6; dk =  .1;

Off[Solve::"nlim"]; 
    Off[NIntegrate::"nlim"]; << Graphics`Animation`

MatrixForm[Table[pos[a], {a, dk, k, dk}]]

A = Join[{0}, %];

B = 
        MatrixForm[
          Table[{n, A[[IntegerPart[((n + dk)/dk)]]]}, {n, 0, k, 
              dk}]];

g1 = 
      ParametricPlot[{x[s], y[s]}, {s, 0, π}, 
        AspectRatio -> Automatic]

Table[
      Show[{g1, 
          Graphics[{PointSize[ .05], 
              Point[{x[B[[1, IntegerPart[((t + dk)/dk)], 2]]], 
                  y[B[[1, IntegerPart[((t + dk)/dk)], 2]]]}]}]}, 
        AspectRatio -> 1, 
        PlotRange -> {{-0.1, π +  .1}, {-1.1, 1}}], {t, 0, 
        k, dk}]

Export["Sine.GIF", %, AnimationDisplayTime -> dk]