Ball bouncing, off walls too

(* much faster than yesterday's *)

g = 0.1; k = 0.01; TMax = 100; num = 40; vxmax = 0.99; boxwidth = 15; \
tail = TMax;
path[vx_] := NDSolve[{x'[0] == vx, y'[0] == Sqrt[1 - vx^2],
    x''[t] == -20 Sign[x[t]] ( Tanh[x[t]^2 - boxwidth^2] + 1),
    y''[t] == -0.1 - 20 (Tanh[20 y[t]] - 1)/2 - k y'[t], x[0] == 0,
    y[0] == 0.1}, {x[t], y[t]}, {t, 0, TMax}];

paths = Table[path[v], {v, -vxmax, vxmax, (2 vxmax)/num}];
frame[T_] :=
  Show[Table[
    ParametricPlot[
     First[{x[t], y[t]} /. paths[[i]]] /. {t -> tt}, {tt,
      Max[0, T - tail], T},
     PlotRange -> {{-boxwidth, boxwidth}, {-1, 10}}, Axes -> None,
     PlotStyle ->
      Directive[Opacity[0.8],
       ColorData["RedBlueTones"][i/Length[paths]]]]
    , {i, Length@paths}],
   Graphics[{
     PointSize[Medium], RGBColor[0.3, 0.4, 0.9],
     Table[{Darker@Darker@ColorData["RedBlueTones"][i/Length[paths]],
       Point[First[{x[t], y[t]} /. paths[[i]] /. {t -> T}]]}
      , {i, Length@paths}]

     }
    ]
   ];

Manipulate[frame[T], {T, 0.01, TMax}]