{NotebookFileName[],DateString[]}

Earth’s orbit

PlanetData["Earth","OrbitPeriod"]

((PlanetData["Earth","OrbitPeriod"])/(2π))

Quantity[1,"AstronomicalUnit"]

style[
t_
]:=
Style[t,Bold,FontFamily->"Arial",FontSize->14]

Block[{is=400,arcr=0.2, θ=(π/6),ω=1,t0=(π/6),p1,p2,v,t},
v=D[{Cos[ω t],Sin[ω t]},t]/.t->t0;
p1={1,0};
p2={Cos[θ],Sin[θ]};
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2}],
Arrow[{p2,p2+v}],
Text[style["v"],p2+(v/2),{2,0}],
Arrow[{p2,p2+v{0,1}}],
Text[style["(Subscript[v, y])"],p2+((v{0,1})/2),{-1.5,0}],
Arrow[{p2+v{0,1},p2+v}],
Text[style["(Subscript[v, x])"],p2+v{(1/2),1},{0,-1}],

Line[{p1,{0,0},p2}],
Circle[{0,0},arcr,{0,θ}],
Text[style["θ=π/6"],arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.3,0}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,ImageSize->is
]
]

Block[{is=400,arcr=0.2, θ=(π/6),ω=1,t0=(π/6),p1,p2,v,t},
v=D[{Cos[ω t],Sin[ω t]},t]/.t->t0;
p1={1,0};
p2={Cos[θ],Sin[θ]};
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2}],
Arrow[{p2,p2+v}],
Text[style["v"],p2+(v/2),{2,0}],
Arrow[{p2,p2+v{0,1}}],
Text[style["(Subscript[v, y])"],p2+((v{0,1})/2),{-1.5,0}],
Arrow[{p2+v{0,1},p2+v}],
Text[style["(Subscript[v, x])"],p2+v{(1/2),1},{0,-1}],

Line[{p1,{0,0},p2}],
Line[{p2,{1,0}p2}],
Circle[{0,0},arcr,{0,θ}],
Text[style["θ=π/6"],arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.3,0}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}],
Text[style["cos(θ)"],p2{(1/2),0},{0,1}],
Text[style["sin(θ)"],p2{1,(1/2)},{-1.2,0}]
}
,ImageSize->is
]
]

Δ t=1

dt=dt1=1

Block[{is=400,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δt="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = (_______, _______)",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = (_______, _______)",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = (_______, _______)",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = ("<>ToString[N[Cos[θ3]-Cos[θ2]]]<>", "<>ToString[N[Sin[θ3]-Sin[θ2]]]<>")",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = ("<>ToString[N[Cos[θ3]-Cos[θ2]]]<>", "<>ToString[N[Sin[θ3]-Sin[θ2]]]<>")",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) =  ("<>ToString[N[((Cos[θ3]-Cos[θ2])/(dθ))]]<>", "<>ToString[N[((Sin[θ3]-Sin[θ2])/(dθ))]]<>")",
}
]//Column

Δ t=0.5

dt=dt1=0.5

Block[{is=400,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = (_______, _______)",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = (_______, _______)",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = (_______, _______)",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = ("<>ToString[N[Cos[θ3]-Cos[θ2]]]<>", "<>ToString[N[Sin[θ3]-Sin[θ2]]]<>")",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = ("<>ToString[N[Cos[θ3]-Cos[θ2]]]<>", "<>ToString[N[Sin[θ3]-Sin[θ2]]]<>")",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) =  ("<>ToString[N[((Cos[θ3]-Cos[θ2])/(dθ))]]<>", "<>ToString[N[((Sin[θ3]-Sin[θ2])/(dθ))]]<>")",
}
]//Column

Δ t=0.1

dt=dt1=0.1

Block[{is=400,as=GoldenRatio,exp=1.3,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v,xmin,xmax,dx,ymin,ymax,dy},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{ymin,ymax}=MinMax[{p2,p3}[[;;,2]]];
dy=exp(ymax-ymin);
ymin=((ymin+ymax-dy)/2);
ymax=((ymin+ymax+dy)/2);
{xmin,xmax}=MinMax[{p2,p3}[[;;,1]]];
dx=Max[xmax-xmin,as dy];
xmin=((xmin+xmax-dx)/2);
xmax=((xmin+xmax+dx)/2);
Print[{{xmin,xmax},{ymin,ymax}}];
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,PlotRange->{{xmin,xmax},{ymin,ymax}}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = (_______, _______)",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = (_______, _______)",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,as=GoldenRatio,exp=1.3,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v,xmin,xmax,dx,ymin,ymax,dy},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{ymin,ymax}=MinMax[{p2,p3}[[;;,2]]];
dy=exp(ymax-ymin);
ymin=((ymin+ymax-dy)/2);
ymax=((ymin+ymax+dy)/2);
{xmin,xmax}=MinMax[{p2,p3}[[;;,1]]];
dx=Max[xmax-xmin,as dy];
xmin=((xmin+xmax-dx)/2);
xmax=((xmin+xmax+dx)/2);
Print[{{xmin,xmax},{ymin,ymax}}];
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,PlotRange->{{xmin,xmax},{ymin,ymax}}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = (_______, _______)",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,as=GoldenRatio,exp=1.3,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v,xmin,xmax,dx,ymin,ymax,dy},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{ymin,ymax}=MinMax[{p2,p3}[[;;,2]]];
dy=exp(ymax-ymin);
ymin=((ymin+ymax-dy)/2);
ymax=((ymin+ymax+dy)/2);
{xmin,xmax}=MinMax[{p2,p3}[[;;,1]]];
dx=Max[xmax-xmin,as dy];
xmin=((xmin+xmax-dx)/2);
xmax=((xmin+xmax+dx)/2);
Print[{{xmin,xmax},{ymin,ymax}}];
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,PlotRange->{{xmin,xmax},{ymin,ymax}}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = ("<>ToString[N[Cos[θ3]-Cos[θ2]]]<>", "<>ToString[N[Sin[θ3]-Sin[θ2]]]<>")",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,as=GoldenRatio,exp=1.3,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v,xmin,xmax,dx,ymin,ymax,dy},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{ymin,ymax}=MinMax[{p2,p3}[[;;,2]]];
dy=exp(ymax-ymin);
ymin=((ymin+ymax-dy)/2);
ymax=((ymin+ymax+dy)/2);
{xmin,xmax}=MinMax[{p2,p3}[[;;,1]]];
dx=Max[xmax-xmin,as dy];
xmin=((xmin+xmax-dx)/2);
xmax=((xmin+xmax+dx)/2);
Print[{{xmin,xmax},{ymin,ymax}}];
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,PlotRange->{{xmin,xmax},{ymin,ymax}}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = ("<>ToString[N[Cos[θ3]-Cos[θ2]]]<>", "<>ToString[N[Sin[θ3]-Sin[θ2]]]<>")",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) =  ("<>ToString[N[((Cos[θ3]-Cos[θ2])/(dθ))]]<>", "<>ToString[N[((Sin[θ3]-Sin[θ2])/(dθ))]]<>")"
}
]//Column

Δ t=0.01

dt=dt1=0.01

Block[{is=400,as=GoldenRatio,exp=1.3,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v,xmin,xmax,dx,ymin,ymax,dy},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{ymin,ymax}=MinMax[{p2,p3}[[;;,2]]];
dy=exp(ymax-ymin);
ymin=((ymin+ymax-dy)/2);
ymax=((ymin+ymax+dy)/2);
{xmin,xmax}=MinMax[{p2,p3}[[;;,1]]];
dx=Max[xmax-xmin,as dy];
xmin=((xmin+xmax-dx)/2);
xmax=((xmin+xmax+dx)/2);
Print[{{xmin,xmax},{ymin,ymax}}];
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,PlotRange->{{xmin,xmax},{ymin,ymax}}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = (_______, _______)",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = (_______, _______)",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,as=GoldenRatio,exp=1.3,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v,xmin,xmax,dx,ymin,ymax,dy},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{ymin,ymax}=MinMax[{p2,p3}[[;;,2]]];
dy=exp(ymax-ymin);
ymin=((ymin+ymax-dy)/2);
ymax=((ymin+ymax+dy)/2);
{xmin,xmax}=MinMax[{p2,p3}[[;;,1]]];
dx=Max[xmax-xmin,as dy];
xmin=((xmin+xmax-dx)/2);
xmax=((xmin+xmax+dx)/2);
Print[{{xmin,xmax},{ymin,ymax}}];
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,PlotRange->{{xmin,xmax},{ymin,ymax}}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = (_______, _______)",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,as=GoldenRatio,exp=1.3,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v,xmin,xmax,dx,ymin,ymax,dy},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{ymin,ymax}=MinMax[{p2,p3}[[;;,2]]];
dy=exp(ymax-ymin);
ymin=((ymin+ymax-dy)/2);
ymax=((ymin+ymax+dy)/2);
{xmin,xmax}=MinMax[{p2,p3}[[;;,1]]];
dx=Max[xmax-xmin,as dy];
xmin=((xmin+xmax-dx)/2);
xmax=((xmin+xmax+dx)/2);
Print[{{xmin,xmax},{ymin,ymax}}];
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,PlotRange->{{xmin,xmax},{ymin,ymax}}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = ("<>ToString[N[Cos[θ3]-Cos[θ2]]]<>", "<>ToString[N[Sin[θ3]-Sin[θ2]]]<>")",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) = (_______, _______)"
}
]//Column

Block[{is=400,as=GoldenRatio,exp=1.3,arcr=0.2,ω=1,t0=(π/6),dθ,θ1,θ2,θ3,p1,p2,p3,v,xmin,xmax,dx,ymin,ymax,dy},
dθ=ω dt;
{θ1,θ2,θ3}={0,ω t0,ω(t0+dt)};
{p1,p2,p3}=({Cos[#],Sin[#]}&)/@{θ1,θ2,θ3};
{ymin,ymax}=MinMax[{p2,p3}[[;;,2]]];
dy=exp(ymax-ymin);
ymin=((ymin+ymax-dy)/2);
ymax=((ymin+ymax+dy)/2);
{xmin,xmax}=MinMax[{p2,p3}[[;;,1]]];
dx=Max[xmax-xmin,as dy];
xmin=((xmin+xmax-dx)/2);
xmax=((xmin+xmax+dx)/2);
Print[{{xmin,xmax},{ymin,ymax}}];
{
Graphics[{
Thick,PointSize[Large],
Circle[],
Point[{p1,p2,p3}],
{Red,
Arrow[{p2,p3}],
Text[style["(Subscript[v, avg])Δt"],((p2+p3)/2),{-1,0}],
Arrow[{p2,p2{1,0}+p3{0,1}}],
Text[style["(Subscript[v, avg, y])Δt"],p2{1,(1/2)}+p3{0,(1/2)},{-1.1,0}],
Arrow[{p2{1,0}+p3{0,1},p3}],
Text[style["(Subscript[v, avg, x])Δt"],p2{(1/2),0}+p3{(1/2),1},{0,-1}]
},

Line[{p1,{0,0},p2}],
Line[{p3,{0,0}}],
Circle[{0,0},arcr,{θ1,θ2}],
Circle[{0,0},arcr,{θ2,θ3}],
Text[style["θ=π/6"],arcr{Cos[((θ2)/2)],Sin[((θ2)/2)]},{-1.3,0}],
Text[style["Δθ="<>ToString[N[dθ]]],arcr{Cos[((θ2+θ3)/2)],Sin[((θ2+θ3)/2)]},{-1,-1}],
Text[style["(cos(θ), sin(θ))"],p2,{-1,0}]
}
,PlotRange->{{xmin,xmax},{ymin,ymax}}
,ImageSize->is
],
"θ = "<>ToString[N[θ2]]<>"     θ+Δθ = "<>ToString[N[θ3]],
"(cos(θ), sin(θ)) = ("<>ToString[N[Cos[θ2]]]<>", "<>ToString[N[Sin[θ2]]]<>")",
"(cos(θ+Δθ), sin(θ+Δθ)) = ("<>ToString[N[Cos[θ3]]]<>", "<>ToString[N[Sin[θ3]]]<>")",
"(cos(θ+Δθ)-cos(θ), sin(θ+Δθ)-sin(θ)) = ("<>ToString[N[Cos[θ3]-Cos[θ2]]]<>", "<>ToString[N[Sin[θ3]-Sin[θ2]]]<>")",
"(((cos (θ + Δθ) - cos (θ))/(Δt)), ((sin (θ + Δθ) - sin (θ))/(Δt))) =  ("<>ToString[N[((Cos[θ3]-Cos[θ2])/(dθ))]]<>", "<>ToString[N[((Sin[θ3]-Sin[θ2])/(dθ))]]<>")"
}
]//Column

Forward Euler

Now, I want to do something sort of like Forward Euler. To do this, I need a way to find velocity as a function of position. Of course, that’s not the way Newtonian mechanics works: it gives you acceleration as a function of position, and then you get velocity by integrating acceleration. So I’m just going to fudge this. I will have a velocity whose magnitude is always one, with components (-sin θ,cos θ), with θ=arctan(x,y). This obviously has the unit circle as a solution. (Of course, in the Forward Euler approximation, it will deviate somewhat, but I want to demonstrate that.)

dt1=0.2;
t1=(π/6.0);
ω1=1.0;

vel1[
{x_,y_},
ω_:ω1,
r_:1
]:=Module[{θ,v},
θ=ArcTan[x,y];
v=ω r{-Sin[θ],Cos[θ]};
v
]

Block[{is=400,arcr=0.2, θ,ω=1,t0=(π/6),p1,p2,v,t},
θ=ω t0;
p1={1,0};
p2={Cos[θ],Sin[θ]};
v=vel1[p2];
Graphics[{
Thick,PointSize[Large],
Circle[],
Line[{p2,{0,0},p1}],
Point[p2],
Circle[{0,0},arcr,{0,θ}],
Text[style["θ"],arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.5,0}],
{Red,
Arrow[{p2,p2+v}],
Text[style["v"],p2+(v/2),{-2,0}]
}
}
,ImageSize->is
]
]

Block[{is=400,arcr=0.2, θ,ω=1,t0=(π/6),p1,p2,p3,v,t,dt=dt1,dxy},
θ=ω t0;
p1={1,0};
p2={Cos[θ],Sin[θ]};
v=vel1[p2];
dxy=dt v;
p3=p2+dxy;
Graphics[{
Thick,PointSize[Large],
Circle[],
Line[{p2,{0,0},p1}],
Point[p2],
Text[style["(x((Subscript[t, 0])), y((Subscript[t, 0])))"],p2,{-1.2,0}],
Circle[{0,0},arcr,{0,θ}],
Text[style["θ"],arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.5,0}],
{Red,
Arrow[{p2,p3}],
Text[style["v Δt"],p2+(dxy/2),{-1.5,0}],
Point[p3],
Text[style["(x((Subscript[t, 0])+ Δt), y((Subscript[t, 0])+Δt))"],p3,{-1.1,-1}]
}
}
,ImageSize->is
]
]

Block[{is=400,arcr=0.2, θ,ω=1,t0=(π/6),p1,p2,p3,p4,v,t,dt=dt1,dxy},
θ=ω t0;
p1={1,0};
p2={Cos[θ],Sin[θ]};
v=vel1[p2];
dxy=dt v;
p3=p2+dxy;
v=vel1[p3];
dxy=dt v;
p4=p3+dxy;
Graphics[{
Thick,PointSize[Large],
Circle[],
Line[{p2,{0,0},p1}],
Point[p2],
Text[style["(x((Subscript[t, 0])), y((Subscript[t, 0])))"],p2,{-1.2,0}],
Circle[{0,0},arcr,{0,θ}],
Text[style["θ"],arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.5,0}],
{Red,
Arrow[{p2,p3}],
Point[p3],
Text[style["(x((Subscript[t, 1])), y((Subscript[t, 1])))"],p3,{-1.2,0}]
},
{Red,
Arrow[{p3,p4}],
Text[style["(Subscript[v, 1]) Δt"],p3+(dxy/2),{-1.5,0}],
Point[p4],
Text[style["(x((Subscript[t, 2])), y((Subscript[t, 2])))"],p4,{-1.2,0}]
}
}
,ImageSize->is
]
]

Block[{is=400,arcr=0.2, θ,ω=1,t0=(π/6),p1,p2,p3,v,t,dt=dt1,dxy,nsteps=10,points,vecs},
θ=ω t0;
p1={1,0};
p2={Cos[θ],Sin[θ]};
p3=p2;
points=Table[
v=vel1[p3];
dxy=dt v;
p3+=dxy;
p3,
{nsteps}
];
vecs=Arrow/@Partition[Prepend[points,p2],2,1];
Graphics[{
Thick,PointSize[Large],
Circle[],
Line[{p2,{0,0},p1}],
Point[p2],
Text[style["(x((Subscript[t, 0])), y((Subscript[t, 0])))"],p2,{-1.2,0}],
Circle[{0,0},arcr,{0,θ}],
Text[style["θ"],arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.5,0}],
{Red,
Point[points],
vecs
}
}
,ImageSize->is
]
]

Block[{is=400,arcr=0.2, θ,ω=1,t0=(π/6),p1,p2,p3,v,t,dt=dt1,dxy,nsteps=45,points,vecs},
θ=ω t0;
p1={1,0};
p2={Cos[θ],Sin[θ]};
p3=p2;
points=Table[
v=vel1[p3];
dxy=dt v;
p3+=dxy;
p3,
{nsteps}
];
vecs=Arrow/@Partition[Prepend[points,p2],2,1];
Graphics[{
Thick,PointSize[Large],
Circle[],
Line[{p2,{0,0},p1}],
Point[p2],
Text[style["(x((Subscript[t, 0])), y((Subscript[t, 0])))"],p2,{-1.2,0}],
Circle[{0,0},arcr,{0,θ}],
Text[style["θ"],arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.5,0}],
{Red,
Point[points],
vecs
}
}
,ImageSize->is
]
]

dt1/2

Block[{is=400,arcr=0.2, θ,ω=1,t0=(π/6),p1,p2,p3,v,t,dt=(dt1/2),dxy,nsteps=40*2,points,vecs},
θ=ω t0;
p1={1,0};
p2={Cos[θ],Sin[θ]};
p3=p2;
points=Table[
v=vel1[p3];
dxy=dt v;
p3+=dxy;
p3,
{nsteps}
];
vecs=Arrow/@Partition[Prepend[points,p2],2,1];
Graphics[{
Thick,PointSize[Large],
Circle[],
Line[{p2,{0,0},p1}],
Point[p2],
Text[style["(x((Subscript[t, 0])), y((Subscript[t, 0])))"],p2,{-1.2,0}],
Circle[{0,0},arcr,{0,θ}],
Text[style["θ"],arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.5,0}],
{Red,
Point[points],
vecs
}
}
,ImageSize->is
]
]

Block[{is=400,arcr=0.2, θ,ω=1,t0=(π/6),p1,p2,p3,v,t,dt=(dt1/10),dxy,nsteps=45*10,points,vecs},
θ=ω t0;
p1={1,0};
p2={Cos[θ],Sin[θ]};
p3=p2;
points=Table[
v=vel1[p3];
dxy=dt v;
p3+=dxy;
p3,
{nsteps}
];
vecs=Line/@Partition[Prepend[points,p2],2,1];
Graphics[{
Thick,PointSize[Large],
Circle[],
Line[{p2,{0,0},p1}],
Point[p2],
Text[style["(x((Subscript[t, 0])), y((Subscript[t, 0])))"],p2,{-1.2,0}],
Circle[{0,0},arcr,{0,θ}],
Text[style["θ"],arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.5,0}],
{Red,
(*Point[points],*)
vecs
}
}
,ImageSize->is
]
]

Block[{is=400,arcr=0.2, θ,ω=1,t0=(π/6),p1,p2,p3,v,t,dt=(dt1/100),dxy,nsteps=45*100,points,vecs},
θ=ω t0;
p1={1,0};
p2={Cos[θ],Sin[θ]};
p3=p2;
points=Table[
v=vel1[p3];
dxy=dt v;
p3+=dxy;
p3,
{nsteps}
];
vecs=Line/@Partition[Prepend[points,p2],2,1];
Graphics[{
Thick,PointSize[Large],
Circle[],
Line[{p2,{0,0},p1}],
Point[p2],
Text[style["(x((Subscript[t, 0])), y((Subscript[t, 0])))"],p2,{-1.2,0}],
Circle[{0,0},arcr,{0,θ}],
Text[style["θ"],arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.5,0}],
{Red,
(*Point[points],*)
vecs
}
}
,ImageSize->is
]
]

VectorPlot[vel1[{x,y}],{x,-1.2,1.3},{y,-1.2,1.3}
,VectorStyle->Red
,VectorScale->Small
,Epilog->{
Thick,Circle[]
}
]