x[s_] := 
      If[s < (π/2), (3/(2  π)) (s^2), 
        Sin[s - π/2] + ((s - π/2)/2) + 3  π/8]; 
    y[s_] := If[
        s < π/2, -5 Sin[s], -2.45 Sin[s] - 2.55]

ParametricPlot[{x[s], y[s]}, {s, 0, 16}, AspectRatio -> Automatic, 
      PlotRange -> {{0, 3  π}, {-5, 0}}]

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

Plot[{x'[s], 1.5}, {s, 0, 16}]

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

k = 3.1; dk =  .1;

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

MatrixForm[
      Table[FindRoot[t[σ] == a, {σ, 0.1}, 
          Compiled -> False], {a, dk, k, dk}]]

A = Join[{0}, σ /. %];

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

g1 = 
      ParametricPlot[{x[s], y[s]}, {s, 0, 
          B[[1, ((k + dk)/dk), 2]]}]

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 -> {{- .1, 
              x[B[[1, ((k + dk)/dk), 2]] +  .1]}, {-5.2, 0.5}}], {t, 
        0, k, dk}]

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