x[s_, t_] := 
      t*(E^s)

    y[s_, t_] := 0

    u[s_, t_] := 
      (t^2)*(E^(2  s))


    p1 = {RGBColor[1, 0, 0], Thickness[0.01]};

    p2 = {RGBColor[0.5, 0.5, 0], Thickness[0.01]};

    p3 = {RGBColor[0, 1, 0], Thickness[0.01]};

    p4 = {RGBColor[0, 0.5, 0.5], Thickness[0.01]};

    p5 = {RGBColor[0, 0, 1], Thickness[0.01]};

    s1 = {x[0, t], y[0, t], u[0, t], p1};

    s2 = {x[0.2, t], y[0.2, t], u[0.2, t], p2};

    s3 = {x[0.4, t], y[0.4, t], u[0.4, t], p3};

    s4 = {x[0.6, t], y[0.6, t], u[0.6, t], p4};

    s5 = {x[0.8, t], y[0.8, t], u[0.8, t], p5};

    t1 = {x[s, 0], y[s, 0], u[s, 0], p1};

    t2 = {x[s, 0.5], y[s, 0.5], u[s, 0.5], p2};

    t3 = {x[s, 1], y[s, 1], u[s, 1], p3};

    t4 = {x[s, 1.5], y[s, 1.5], u[s, 1.5], p4};

    t5 = {x[s, 2], y[s, 2], u[s, 2], 
          p5};


    At1[T_] := 
        ParametricPlot3D[t1, {s, -5, T - 6}, AspectRatio -> 1, 
          DisplayFunction -> Identity];

    At2[T_] := 
        ParametricPlot3D[t2, {s, -5, T - 9}, AspectRatio -> 1, 
          DisplayFunction -> Identity];

    At3[T_] := 
        ParametricPlot3D[t3, {s, -5, T - 12}, AspectRatio -> 1, 
          DisplayFunction -> Identity];

    At4[T_] := 
        ParametricPlot3D[t4, {s, -5, T - 15}, AspectRatio -> 1, 
          DisplayFunction -> Identity];

    At5[T_] := 
        ParametricPlot3D[t5, {s, -5, T - 18}, AspectRatio -> 1, 
          DisplayFunction -> Identity];

    As1[T_] := 
        ParametricPlot3D[s1, {t, -5, T - 18}, AspectRatio -> 1, 
          DisplayFunction -> Identity];

    As2[T_] := 
        ParametricPlot3D[s2, {t, -5, T - 24}, AspectRatio -> 1, 
          DisplayFunction -> Identity];

    As3[T_] := 
        ParametricPlot3D[s3, {t, -5, T - 30}, AspectRatio -> 1, 
          DisplayFunction -> Identity];

    As4[T_] := 
        ParametricPlot3D[s4, {t, -5, T - 36}, AspectRatio -> 1, 
          DisplayFunction -> Identity];

    As5[T_] := 
        ParametricPlot3D[s5, {t, -5, T - 42}, AspectRatio -> 1, 
          DisplayFunction -> 
            Identity];


    D1 := Show[As1[T], AspectRatio -> 1, 
        DisplayFunction -> Identity]

    D2 := Show[As1[T], At1[T], AspectRatio -> 1, 
        DisplayFunction -> Identity]

    D3 := Show[As1[T], At1[T], As2[T], AspectRatio -> 1, 
        DisplayFunction -> Identity]

    D4 := Show[As1[T], At1[T], As2[T], At2[T], AspectRatio -> 1, 
        DisplayFunction -> Identity]

    D5 := Show[As1[T], At1[T], As2[T], At2[T], As3[T], AspectRatio -> 1, 
        DisplayFunction -> Identity]

    D6 := Show[As1[T], At1[T], As2[T], At2[T], As3[T], At3[T], 
        AspectRatio -> 1, 
        DisplayFunction -> Identity]

    D7 := Show[As1[T], At1[T], As2[T], At2[T], As3[T], At3[T], As4[T], 
        AspectRatio -> 1, 
        DisplayFunction -> Identity]

    D8 := Show[As1[T], At1[T], As2[T], At2[T], As3[T], At3[T], As4[T], 
        At4[T], AspectRatio -> 1, 
        DisplayFunction -> Identity]

    D9 := Show[As1[T], At1[T], As2[T], At2[T], As3[T], At3[T], As4[T], 
        At4[T], As5[T], DisplayFunction -> Identity]

    D10 := Show[As1[T], At1[T], As2[T], At2[T], As3[T], At3[T], As4[T], 
        At4[T], As5[T], At5[T], AspectRatio -> 1, 
        DisplayFunction -> 
          Identity]



    Q := Which[T ≤ 5, D1, 5 < T ≤ 10, D2, 
        10 < T ≤ 15, D3, 15 < T ≤ 20, D4, 
        20 < T ≤ 25, D5, 25 < T ≤ 30, D6, 
        30 < T ≤ 35, D7, 35 < T ≤ 40, D8, 
        40 < T ≤ 45, D9, 45 < T ≤ 50, 
        D10]

    W := Table[
          Show[Q, DisplayFunction -> $DisplayFunction, 
            PlotRange -> {{-3, 2.5}, {0, 2.5}, {-5, 10}}], {T, 
            0.0001, 50, 0.5}];

    R = Plot3D[(x^2), {x, -5, 5}, {y, -5, 5}, 
        PlotRange -> {{-3, 2.5}, {-5, 2.5}, {-5, 
              10}}]

    Show[W, R, DisplayFunction -> $DisplayFunction]