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Relativistic Acceleration
Verfasst: Mi 23. Jun 2021, 03:51
von Yukterez
This is the english version.
Eine deutschspachige Version findet sich
hierIndex:
constant force |
constant coordinate acceleration |
constant power, discussion (autotranslate from german):
click here
Relativistic Acceleration
Verfasst: Mi 23. Jun 2021, 03:53
von Yukterez
CONSTANT FORCE
The velocity v is derived by knowing the gammafactor γ
and the relativistic momentum p
where the force F, rest mass m and proper acceleration a
are connected with the differential of momentum p over coordinate time t. With that and F=constant we get
Plot with coordinate time t as x-axis (SI-units, momentum and energy as specific quantities per kg rest mass):
Analytical solutions for proper time τ and distance x:
Code:
Code: Alles auswählen
(* Eigenbeschleunigung, Methode 1 *)
a=1000000m/sek^2; (* Beschleunigung *)
v0=0m/sek; (* Anfangsgeschwindigkeit *)
c=3*^8m/sek; (* Lichtgeschwindigkeit *)
m=sek=1; (* Einheiten *)
γ[v_]:=1/Sqrt[1-v^2/c^2]; γ0=γ[v0]; (* Gamma *)
v[t_]:=(a t+v0 γ0)/Sqrt[1+(a t+v0 γ0)^2/c^2]; (* Geschwindigkeit *)
e[t_]:=c^2 γ[v[t]]-c^2; (* spezifische kinetische Energie *)
p[t_]:=v[t]γ[v[t]]; (* spezifischer Impuls *)
sol=NDSolve[{τ'[t]==Sqrt[1-v[t]^2/c^2], τ[0]==0, X'[t]==v[t], X[0]==0},
{τ, X}, {t, 0, 1000}, Method->{"ImplicitRungeKutta", "DifferenceOrder"->20},
MaxSteps->Infinity, InterpolationOrder->All, WorkingPrecision->32];
т[t_]:=Evaluate[τ[t]/.sol][[1]]; (* Eigenzeit *)
ť[t_]:=т'[t]; ṫ[t_]:=1/ ť[t]; (* Zeitdilatation *)
α[t_]:=Evaluate[X''[t]/.sol][[1]]; (* Koordinatenbeschleunigung *)
A[t_]:=α[t] γ[v[t]]^3; (* Eigenbeschleunigung *)
x[t_]:=Evaluate[X[t]/.sol][[1]]; (* Position *)
plot[f_]:=Plot[f[t], {t, 0, 1000}, Frame->True, ImagePadding->{{50, 8}, {16, 8}},
ImageSize->320, PlotRange->All, PlotStyle->Thick, AspectRatio->1/2]
Grid[{{plot[т], plot[ť]},
{" Eigenzeit τ", " Zeitdilatation dτ/dt"},
{}, {plot[v], plot[p]},
{" Geschwindigkeit v", " Impuls p"},
{}, {plot[x], plot[e]},
{" Strecke x", " kinetische Energie"},
{}, {plot[A], plot[α]},
{" Eigenbeschleunigung a", " Koordinatenbeschleunigung α"}},
Alignment->Center]Code: Alles auswählen
(* Eigenbeschleunigung, Methode 2 *)
a=1000000m/sek^2; (* Beschleunigung *)
v0=0m/sek; (* Anfangsgeschwindigkeit *)
c=3*^8m/sek; (* Lichtgeschwindigkeit *)
m=sek=1; (* Einheiten *)
γ[v_]:=1/Sqrt[1-v^2/c^2]; γ0=γ[v0]; (* Gamma *)
e[t_]:=c^2 γ[v[t]]-c^2; (* spezifische kinetische Energie *)
p[t_]:=v[t] γ[v[t]]; (* spezifischer Impuls *)
sol=NDSolve[{τ'[t]==Sqrt[1-V[t]^2/c^2], τ[0]==0,
(V[t]^2 V'[t])/(c^2 (1-V[t]^2/c^2)^(3/2))+V'[t]/Sqrt[1-V[t]^2/c^2]==a, V[0]==v0,
X'[t]==V[t], X[0]==0}, {τ, V, X}, {t, 0, 1000},
Method->{"ImplicitRungeKutta", "DifferenceOrder"->20}, MaxSteps->Infinity,
InterpolationOrder->All, WorkingPrecision->32];
т[t_]:=Evaluate[τ[t]/.sol][[1]]; (* Eigenzeit *)
v[t_]:=Evaluate[V[t]/.sol][[1]]; (* Geschwindigkeit *)
ť[t_]:=т'[t]; ṫ[t_]:=1/ť[t] (* Zeitdilatation *)
α[t_]:=Evaluate[X''[t]/.sol][[1]]; (* Koordinatenbeschleunigung *)
A[t_]:=α[t]γ[v[t]]^3; (* Eigenbeschleunigung *)
x[t_]:=Evaluate[X[t]/.sol][[1]]; (* Position *)
plot[f_]:=Plot[f[t], {t, 0, 1000}, Frame->True, ImagePadding->{{50, 8}, {16, 8}},
ImageSize->320, PlotRange->All, PlotStyle->Thick, AspectRatio->1/2]
Grid[{{plot[т], plot[ť]},
{" Eigenzeit τ", " Zeitdilatation dτ/dt"},
{}, {plot[v], plot[p]},
{" Geschwindigkeit v", " Impuls p"},
{}, {plot[x], plot[e]},
{" Strecke x", " kinetische Energie"},
{}, {plot[A], plot[α]},
{" Eigenbeschleunigung a", " Koordinatenbeschleunigung α"}},
Alignment->Center]Code: Alles auswählen
(* Eigenbeschleunigung, Methode 3 *)
a=1000000m/sek^2; (* Beschleunigung *)
v0=0m/sek; (* Anfangsgeschwindigkeit *)
c=3*^8m/sek; (* Lichtgeschwindigkeit *)
m=sek=1; (* Einheiten *)
γ[v_]:=1/Sqrt[1-v^2/c^2]; γ0=γ[v0]; (* Gamma *)
e[t_]:=c^2 γ[v[t]]-c^2; (* spezifische kinetische Energie *)
p[t_]:=v[t] γ[v[t]]; (* spezifischer Impuls *)
sol=NDSolve[{T'[τ]==1/Sqrt[1-V[τ]^2/c^2], T[0]==0, V'[τ]==a(1-V[τ]^2/c^2),
V[0]==v0, X'[τ]==V[τ]/Sqrt[1-V[τ]^2/c^2], X[0]==0}, {V, T, X}, {τ, 0, 1000},
Method->{"ImplicitRungeKutta", "DifferenceOrder"->20}, MaxSteps->Infinity,
InterpolationOrder->All, WorkingPrecision->32];
t[τ_]:=Evaluate[T[τ]/.sol][[1]]; (* Koordinatenzeit *)
т=Interpolation[Table[{t[j], j}, {j, 0, 1000}]]; (* Eigenzeit *)
v[t_]:=Evaluate[V[т[t]]/.sol]; (* Geschwindigkeit *)
ṫ[t_]:=Evaluate[T'[т[t]]/.sol]; ť[t_]:=1/ṫ[t]; (* Zeitdilatation *)
α[t_]:=x''[t]; (* Koordinatenbeschleunigung *)
A[t_]:=α[t]γ[v[t]]^3; (* Eigenbeschleunigung *)
x[t_]:=Evaluate[X[т[t]]/.sol]; (* Position *)
plot[f_]:=Plot[f[t], {t, 0, 1000}, Frame->True, ImagePadding->{{50, 8}, {16, 8}},
ImageSize->320, PlotRange->All, PlotStyle->Thick, AspectRatio->1/2]
Grid[{{plot[т], plot[ť]},
{" Eigenzeit τ", " Zeitdilatation dτ/dt"},
{}, {plot[v], plot[p]},
{" Geschwindigkeit v", " Impuls p"},
{}, {plot[x], plot[e]},
{" Strecke x", " kinetische Energie"},
{}, {plot[A], plot[α]},
{" Eigenbeschleunigung a", " Koordinatenbeschleunigung α"}},
Alignment->Center]For a side by side comparison of the 3 different ways to do it
click here
Relativistic Acceleration
Verfasst: Mi 23. Jun 2021, 03:57
von Yukterez
CONSTANT COORDINATE ACCELERATION
With α=dv/dt=d²x/dt²=a/γ³=constant the equation for v is simply
With constant coordinate acceleration the speed of light would be reached when
Plot with coordinate time t as x-axis (SI-units, momentum and energy as specific quantities per kg rest mass):
Analytical solutions for proper time τ and distance x:
Code:
Code: Alles auswählen
(* konstante Koordinatenbeschleunigung ẍ *)
ẍ=1000000m/sek^2; (* Koordinatenbeschleunigung *)
v0=0m/sek; (* Anfangsgeschwindigkeit *)
c=3*^8m/sek; (* Lichtgeschwindigkeit *)
m=sek=1; (* Einheiten *)
γ[v_]:=1/Sqrt[1-v^2/c^2]; γ0=γ[v0]; (* Gamma *)
v[t_]:=ẍ t+v0 ; (* Geschwindigkeit *)
e[t_]:=c^2 γ[v[t]]-c^2; (* spezifische kinetische Energie *)
p[t_]:=v[t]γ[v[t]]; (* spezifischer Impuls *)
sol=NDSolve[{τ'[t]==Sqrt[1-v[t]^2/c^2], τ[0]==0, X'[t]==v[t], X[0]==0},
{τ, X}, {t, 0, (c-v0)/ẍ-1},
Method->{"ImplicitRungeKutta", "DifferenceOrder"->20},
MaxSteps->Infinity, InterpolationOrder->All, WorkingPrecision->32];
т[t_]:=Evaluate[τ[t]/.sol]; (* Eigenzeit *)
ť[t_]:=т'[t]; ṫ[t_]:=1/ ť[t]; (* Zeitdilatation *)
α[t_]:=x''[t]; (* Koordinatenbeschleunigung *)
A[t_]:=α[t] γ[v[t]]^3; (* Eigenbeschleunigung *)
x[t_]:=Evaluate[X[t]/.sol][[1]]; (* Position *)
plot[f_]:=Plot[f[t], {t, 0, (c-v0)/ẍ-1},
Frame->True, ImagePadding->{{50, 8}, {16, 8}}, ImageSize->320,
PlotRange->All, PlotStyle->Thick, AspectRatio->1/2]
Grid[{{plot[т], plot[ť]},
{" Eigenzeit τ", " Zeitdilatation dτ/dt"},
{}, {plot[v], plot[p]},
{" Geschwindigkeit v", " Impuls p"},
{}, {plot[x], plot[e]},
{" Strecke x", " kinetische Energie"},
{}, {plot[A], plot[α]},
{" Eigenbeschleunigung a", " Koordinatenbeschleunigung α"}},
Alignment->Center]
Relativistic Acceleration
Verfasst: Mi 23. Jun 2021, 04:03
von Yukterez
CONSTANT POWER
With constant power L the energy mc²γ inreases linearly with t. The initial energy plus the applied work equals the new energy
where W is the applied work
If we solve for v we get
Plot with coordinate time t as x-axis (SI-units, momentum and energy as specific quantities per kg rest mass):
Analytical solutions for proper time τ and distance x:
Code:
Code: Alles auswählen
(* konstante Leistung *)
L=10^14 m^2/sek^3; (* spezifische Leistung *)
v0=0m/sek; (* Anfangsgeschwindigkeit *)
c=3*^8m/sek; (* Lichtgeschwindigkeit *)
m=sek=1; (* Einheiten *)
γ[v_]:=1/Sqrt[1-v^2/c^2]; γ0=γ[v0]; (* Gamma *)
v[t_]:=(c Sqrt[(-1+t L/c^2+γ0) (1+t L/c^2+γ0)])/(t L/c^2+γ0); (* Geschwindigkeit *)
e[t_]:=c^2 γ[v[t]]-c^2; (* spezifische kinetische Energie *)
p[t_]:=v[t] γ[v[t]]; (* spezifischer Impuls *)
sol=NDSolve[{τ'[t]==1/γ[v[t]], τ[0]==0, X'[t]==v[t], X[0]==0}, {τ, X},
{t, 0, 1000}, Method->{"ImplicitRungeKutta", "DifferenceOrder"->20},
MaxSteps->Infinity, InterpolationOrder->All, WorkingPrecision->32];
т[t_]:=Evaluate[τ[t]/.sol]; (* Eigenzeit *)
ť[t_]:=т'[t]; ṫ[t_]:=1/ť[t]; (* Zeitdilatation *)
x[t_]:=Evaluate[X[t]/.sol][[1]]; (* Position *)
α[t_]:=Evaluate[X''[t]/.sol][[1]]; (* Koordinatenbeschleunigung *)
A[t_]:=α[t] γ[v[t]]^3; (* Eigenbeschleunigung *)
plot[f_]:=Plot[f[t], {t, 0, 1000},
Frame->True, ImagePadding->{{50, 8}, {16, 8}}, ImageSize->320, PlotRange->All,
PlotStyle->Thick, AspectRatio->1/2]
Grid[{{plot[т], plot[ť]},
{" Eigenzeit τ", " Zeitdilatation dτ/dt"},
{}, {plot[v], plot[p]},
{" Geschwindigkeit v", " Impuls p"},
{}, {plot[x], plot[e]},
{" Strecke x", " kinetische Energie"},
{}, {plot[A], plot[α]},
{" Eigenbeschleunigung a", " Koordinatenbeschleunigung α"}},
Alignment->Center]