<< Graphics`Arrow`

o = {0, 0};

x[t_] = Cos[t]/((t^2) + 10) +  .01  t;
y[t_] = t Sin[t]  + t - 2;
r[t_] = {x[t], y[t]};

tmin = -50;
tmax = 120;
p1 = 
        Show[Graphics[Table[Arrow[o, r[ .05 t]], {t, tmin, tmax}]], 
          DisplayFunction -> Identity];
p2 = 
        ParametricPlot[r[ .05 t], {t, tmin, tmax}, 
          PlotStyle -> {Thickness[0.01], Hue[0.895]}, 
          PlotRange -> All, 
          DisplayFunction -> Identity];
Show[GraphicsArray[{{p1, p2}}]];
Show[{p1, p2}, 
        DisplayFunction -> $DisplayFunction];

Table[
        Show[{p2, Graphics[Arrow[o, r[ .05 t]]]}, 
          PlotRange -> {{-0.08, 0.11}, {-3, 2.5}}, 
          DisplayFunction -> $DisplayFunction], {t, tmin, 
          tmax}];