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

ParametricPlot[{x[s], y[s]}, {s, 0, 2}, 
      AspectRatio -> Automatic]

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

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

Plot[t[σ], {σ, 0, 2  π}]

k = 5; 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, x[A[[((k + dk)/dk)]]]}, 
        PlotRange -> {{- .1, 
              x[B[[1, ((k + dk)/dk), 2]]]}, {y[
                B[[1, ((k + dk)/dk), 2]]], 0.5}}, 
        AspectRatio -> -y[B[[1, ((k + dk)/dk), 1]]]/
            x[B[[1, ((k + dk)/dk), 1]]]]

Table[
      Show[{g1, 
          Graphics[{PointSize[ .05], 
              Point[{x[B[[1, IntegerPart[((t + dk)/dk)], 2]]], 
                  y[B[[1, IntegerPart[((t + dk)/dk)], 2]]]}]}]}, 
        AspectRatio -> -y[B[[1, ((k + dk)/dk), 1]]]/
            x[B[[1, ((k + dk)/dk), 1]]], 
        PlotRange -> {{- .1, 
              x[B[[1, ((k + dk)/dk), 2]] + 5]}, {y[
                B[[1, ((k + dk)/dk), 2]]], 0.5}}], {t, 0, k, 
        dk}]

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