Velocity Vector

Author

Eric W. Weisstein
May 3, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialGeometry/VelocityVector.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/VelocityVector.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Plot

<<Graphics`Arrow`
<<Graphics`Animation`

gifs="Phosphine:Troves:MathOztex:gifs:";
SetOptions[Export,ConversionOptions->{"AnimationDisplayTime"->0.1}]

p=Point[N[{(1/2)^(1/2),(1/2)^(1/2)}]];
v=Arrow[N[{(1/2)^(1/2),(1/2)^(1/2)}],N[{0,2^(1/2)}]];

Show[Graphics[{Circle[{0,0},1],Hue[0],PointSize[.03],p,v},PlotRange->{{-2,2},{-2,2}},AspectRatio->Automatic,Axes->Automatic,Ticks->{Range[-2,2],Range[-2,2]}]]

hvelovect[s_]:={Point[{Sinh[s],Cosh[s]}],Arrow[{Sinh[s],Cosh[s]},{Exp[s],Exp[s]}]};

hyper[s_]:=Show[{ParametricPlot[{Sinh[t],Cosh[t]},{t,-2,2.7},DisplayFunction->Identity],Graphics[{PointSize[.05],RGBColor[1,0,0],hvelovect[s]}]},AxesOrigin->{0,0},PlotRange->{{-4,8},{0,8}},Ticks->{Range[-4,8],Range[0,8]},AspectRatio->Automatic,DisplayFunction->$DisplayFunction];

<<partition`

hvelovect[s_]:={Point[{Sinh[s],Cosh[s]}],{RGBColor[1,0,0],Thickness[.02],Arrow[{Sinh[s],Cosh[s]},{Exp[s],Exp[s]},HeadWidth->1.3]}};

Show[GraphicsArray[partition[Table[hyper[s],{s,-2,2,4/5}],3]]]

Export[gifs<>"velvec.gif",Table[hyper[s],{s,-2,2,.1}]]

color[speed_]:=Hue[(speed/10)+.5];
speed1[s_]:=N[((Exp[2 s]+Exp[-2 s])/2)^(1/2)];
speed2[s_]:=N[((1+2 s^2)/(1+s^2))^(1/2)];	hvelovect1[s_]:={Point[{Sinh[s],Cosh[s]}],Arrow[{Sinh[s],Cosh[s]},{Exp[s],Exp[s]}]};
hvelovect2[s_]:={Point[{s,(1+s^2)^(1/2)}],Arrow[{s,(1+s^2)^(1/2)},{s+1,(1+s+s^2)/((1+s^2)^(1/2))}]};
twoh[s_]:=Show[{ParametricPlot[{Sinh[t],Cosh[t]},{t,-2,2.7},DisplayFunction->Identity],Graphics[{color[speed1[s]],PointSize[.03],hvelovect1[s]}],Graphics[{color[speed2[s]],PointSize[.03],hvelovect2[s]}]},AxesOrigin->{0,0},PlotRange->{{-4,8},{0,8}},Ticks->{Range[-4,8],Range[0,8]},AspectRatio->Automatic,DisplayFunction->$DisplayFunction];
hmovie[n_]:=Table[twoh[s],{s,-2,2,4/n}];

Export[gifs<>"velvec2.gif",hmovie[16]]

Show[GraphicsArray[partition[hmovie[5],3]]]