Manipulate[
Animate[
Plot[{Piecewise[{{F[z-v1 t]+R F[-(z+v1 t)],z<0}},(1+R) F[(v1/v2)(z-v2 t)]]}/.{v1->1,v2->1.5},{z,-5,6},PlotRange->{-1.05,2.05},Ticks->None,Axes->{False,True},PlotStyle->{{Thick,Black}}],
{t,-4,4}],{{R,1/2,"R"},-1,1} ,Initialization:>{F[x_]:=(E^(Piecewise[{{-2/3(x^2),x<0}},-4(x^3)]))} ]

F[x_]:=(E^(Piecewise[{{-2/3(x^2),x<0}},-4(x^3)]))

tabulka=Table[
Show[
Plot[{Piecewise[{{F[z-v1 t]+R F[-(z+v1 t)],z<0}},(1+R) F[(v1/v2)(z-v2 t)]]}/.{v1->1,v2->1.5,R->-1},{z,-5,5},PlotRange->{-1.05,2.05},Ticks->None,Axes->{False,True},PlotStyle->{{Thick,Black}}],
Graphics[Text[Style["R = -1, \!\(\*SubscriptBox[\(v\), \(\(1\)\(\\ \)\)]\) < (Subscript[v, 2])",FontSize->12],{-3.5,1.7}]]
],
{t,-4,4,0.1}];

Export["odrazy.gif",tabulka,"GIF"]

tab=Table[0,{i,1,5}]

tab[[5]]=tabulka;

tab2=Flatten[tab,1];

Export["odraz.gif",tab2,"GIF"]