Cohete con aceleración uniforme
Variable
t τ d β γ μ
tiempo terrestre
t
=
{\displaystyle t=}
—
c
a
sinh
a
τ
c
{\displaystyle {\frac {c}{a}}\sinh {\frac {a\tau }{c}}}
[ formula 1]
d
c
1
+
2
c
2
a
d
{\displaystyle {\frac {d}{c}}{\sqrt {1+{\frac {2c^{2}}{ad}}}}}
[ formula 2]
c
a
β
1
−
β
{\displaystyle {\frac {c}{a}}{\sqrt {\frac {\beta }{1-\beta }}}}
[ formula 3]
c
a
γ
−
1
{\displaystyle {\frac {c}{a}}{\sqrt {\gamma -1}}}
[ formula 2]
c
a
μ
β
e
−
μ
−
β
e
2
{\displaystyle {\frac {c}{a}}\,{\frac {\mu ^{\beta _{\mathrm {e} }}-\mu ^{-\beta _{\mathrm {e} }}}{2}}}
tiempo propio
τ
=
{\displaystyle \tau =}
c
a
sinh
−
1
a
t
c
{\displaystyle {\frac {c}{a}}\sinh ^{-1}\ {\frac {at}{c}}}
[ formula 1]
—
c
a
cosh
−
1
(
1
+
a
d
c
2
)
{\displaystyle {\frac {c}{a}}\cosh ^{-1}\ {\Big (}1+{\frac {ad}{c^{2}}}{\Big )}}
[ formula 1]
c
a
tanh
−
1
β
{\displaystyle {\frac {c}{a}}\tanh ^{-1}\ \beta }
[ formula 2]
c
a
cosh
−
1
γ
{\displaystyle {\frac {c}{a}}\cosh ^{-1}\gamma }
c
a
β
e
log
μ
{\displaystyle {\frac {c}{a}}\,\beta _{\mathrm {e} }\log \mu }
distancia
d
=
{\displaystyle d=}
c
2
a
[
1
+
(
a
t
c
)
2
−
1
]
{\displaystyle {\frac {c^{2}}{a}}{\Bigg [}{\sqrt {1\!+\!{\Big (}{\frac {at}{c}}{\Big )}^{2}}}\!-\!1{\Bigg ]}}
[ formula 2]
c
2
a
[
cosh
a
τ
c
−
1
]
{\displaystyle {\frac {c^{2}}{a}}{\Big [}\cosh {\frac {a\tau }{c}}\!-\!1{\Big ]}}
[ formula 1]
—
c
2
a
[
1
1
−
β
2
−
1
]
{\displaystyle {\frac {c^{2}}{a}}{\Bigg [}{\frac {1}{\sqrt {1\!-\!\beta ^{2}}}}\!-\!1{\Bigg ]}}
c
2
a
(
γ
−
1
)
{\displaystyle {\frac {c^{2}}{a}}(\gamma -1)}
[ formula 2]
c
2
a
[
μ
β
e
+
μ
−
β
e
2
−
1
]
{\displaystyle {\frac {c^{2}}{a}}{\Bigg [}{\frac {\mu ^{\beta _{\mathrm {e} }}\!+\!\mu ^{-\beta _{\mathrm {e} }}}{2}}\!-\!1{\Bigg ]}}
velocidad relativa
β
=
{\displaystyle \beta =}
a
t
/
c
1
+
(
a
t
c
)
2
{\displaystyle {\frac {at/c}{\sqrt {1+{\big (}{\frac {at}{c}}{\big )}^{2}}}}}
[ formula 3]
tanh
a
τ
c
{\displaystyle \tanh {\frac {a\tau }{c}}}
[ formula 2]
1
−
(
1
+
a
d
c
2
)
−
2
{\displaystyle {\sqrt {1-{\Big (}1+{\frac {ad}{c^{2}}}{\Big )}^{-2}}}}
—
1
−
1
γ
2
{\displaystyle {\sqrt {1-{\frac {1}{\gamma ^{2}}}}}}
1
−
μ
−
2
β
e
1
+
μ
−
2
β
e
{\displaystyle {\frac {1-\mu ^{-2\beta _{\mathrm {e} }}}{1+\mu ^{-2\beta _{\mathrm {e} }}}}}
[ formula 4]
factor de Lorentz
γ
=
{\displaystyle \gamma =}
1
+
(
a
t
c
)
2
{\displaystyle {\sqrt {1+{\Big (}{\frac {at}{c}}{\Big )}^{2}}}}
[ formula 2]
cosh
a
τ
c
{\displaystyle \cosh {\frac {a\tau }{c}}}
1
+
a
d
c
2
{\displaystyle 1+{\frac {ad}{c^{2}}}}
1
1
−
β
2
{\displaystyle {\frac {1}{\sqrt {1-\beta ^{2}}}}}
—
μ
β
e
+
μ
−
β
e
2
{\displaystyle {\frac {\mu ^{\beta _{\mathrm {e} }}+\mu ^{-\beta _{\mathrm {e} }}}{2}}}
cociente de masa
μ
=
{\displaystyle \mu =}
[
a
t
c
+
1
+
(
a
t
c
)
2
]
1
β
e
{\displaystyle {\Bigg [}{\frac {at}{c}}\!+\!{\sqrt {1\!+\!{\Big (}{\frac {at}{c}}{\Big )}^{2}}}{\Bigg ]}^{\frac {1}{\beta _{\mathrm {e} }}}}
exp
a
τ
c
β
e
{\displaystyle \exp {\frac {a\tau }{c\beta _{\mathrm {e} }}}}
[
1
+
a
d
c
2
(
1
+
1
+
c
2
a
d
)
]
1
β
e
{\displaystyle {\Bigg [}1\!+\!{\frac {ad}{c^{2}}}{\Bigg (}\!1\!+\!{\sqrt {1\!+\!{\frac {c^{2}}{ad}}}}{\Bigg )}{\Bigg ]}^{\frac {1}{\beta _{\mathrm {e} }}}}
(
1
+
β
1
−
β
)
1
2
β
e
{\displaystyle {\Bigg (}{\frac {1+\beta }{1-\beta }}{\Bigg )}^{\frac {1}{2\beta _{\mathrm {e} }}}}
[ formula 4]
[
γ
+
γ
2
−
1
]
1
β
s
e
{\displaystyle {\Big [}\gamma +{\sqrt {\gamma ^{2}\!-\!1}}{\Big ]}^{\frac {1}{\beta _{\mathrm {se} }}}}
—
Viaje en cohete con aceleración y deceleración uniformes
Variable
t τ d β γ μ
tiempo terrestre
t
=
{\displaystyle t=}
—
2
c
a
sinh
a
τ
2
c
{\displaystyle {\frac {2c}{a}}\sinh {\frac {a\tau }{2c}}}
d
c
1
+
4
c
2
a
d
{\displaystyle {\frac {d}{c}}{\sqrt {1+{\frac {4c^{2}}{ad}}}}}
2
c
a
β
1
−
β
{\displaystyle {\frac {2c}{a}}{\sqrt {\frac {\beta }{1-\beta }}}}
2
c
a
γ
−
1
{\displaystyle {\frac {2c}{a}}{\sqrt {\gamma -1}}}
2
c
a
μ
β
e
2
−
μ
−
β
e
2
2
{\displaystyle {\frac {2c}{a}}\,{\frac {\mu ^{\frac {\beta _{\mathrm {e} }}{2}}-\mu ^{-{\frac {\beta _{\mathrm {e} }}{2}}}}{2}}}
tiempo propio
τ
=
{\displaystyle \tau =}
2
c
a
sinh
−
1
a
t
2
c
{\displaystyle {\frac {2c}{a}}\sinh ^{-1}\ {\frac {at}{2c}}}
—
2
c
a
cosh
−
1
(
1
+
a
d
2
c
2
)
{\displaystyle {\frac {2c}{a}}\cosh ^{-1}\ {\Big (}1+{\frac {ad}{2c^{2}}}{\Big )}}
2
c
a
tanh
−
1
β
{\displaystyle {\frac {2c}{a}}\tanh ^{-1}\ \beta }
2
c
a
cosh
−
1
γ
{\displaystyle {\frac {2c}{a}}\cosh ^{-1}\gamma }
2
c
a
β
e
2
log
μ
{\displaystyle {\frac {2c}{a}}\,{\frac {\beta _{\mathrm {e} }}{2}}\log \mu }
distancia
d
=
{\displaystyle d=}
2
c
2
a
[
1
+
(
a
t
2
c
)
2
−
1
]
{\displaystyle {\frac {2c^{2}}{a}}{\Bigg [}{\sqrt {1\!+\!{\Big (}{\frac {at}{2c}}{\Big )}^{2}}}\!-\!1{\Bigg ]}}
2
c
2
a
[
cosh
a
τ
2
c
−
1
]
{\displaystyle {\frac {2c^{2}}{a}}{\Big [}\cosh {\frac {a\tau }{2c}}\!-\!1{\Big ]}}
—
2
c
2
a
[
1
1
−
β
2
−
1
]
{\displaystyle {\frac {2c^{2}}{a}}{\Bigg [}{\frac {1}{\sqrt {1\!-\!\beta ^{2}}}}\!-\!1{\Bigg ]}}
2
c
2
a
(
γ
−
1
)
{\displaystyle {\frac {2c^{2}}{a}}(\gamma -1)}
2
c
2
a
[
μ
β
e
2
+
μ
−
β
e
2
2
−
1
]
{\displaystyle {\frac {2c^{2}}{a}}{\Bigg [}{\frac {\mu ^{\frac {\beta _{\mathrm {e} }}{2}}\!+\!\mu ^{-{\frac {\beta _{\mathrm {e} }}{2}}}}{2}}\!-\!1{\Bigg ]}}
velocidad relativa máxima
β
=
{\displaystyle \beta =}
a
t
/
(
2
c
)
1
+
(
a
t
2
c
)
2
{\displaystyle {\frac {at/(2c)}{\sqrt {1+{\big (}{\frac {at}{2c}}{\big )}^{2}}}}}
β
=
tanh
a
τ
2
c
{\displaystyle \beta =\tanh {\frac {a\tau }{2c}}}
1
−
(
1
+
a
d
2
c
2
)
−
2
{\displaystyle {\sqrt {1-{\Big (}1+{\frac {ad}{2c^{2}}}{\Big )}^{-2}}}}
—
1
−
1
γ
2
{\displaystyle {\sqrt {1-{\frac {1}{\gamma ^{2}}}}}}
1
−
μ
−
β
e
1
+
μ
−
β
e
{\displaystyle {\frac {1-\mu ^{-\beta _{\mathrm {e} }}}{1+\mu ^{-\beta _{\mathrm {e} }}}}}
factor de Lorentz máximo
γ
=
{\displaystyle \gamma =}
1
+
(
a
t
2
c
)
2
{\displaystyle {\sqrt {1+{\Big (}{\frac {at}{2c}}{\Big )}^{2}}}}
cosh
a
τ
2
c
{\displaystyle \cosh {\frac {a\tau }{2c}}}
1
+
a
d
2
c
2
{\displaystyle 1+{\frac {ad}{2c^{2}}}}
1
1
−
β
2
{\displaystyle {\frac {1}{\sqrt {1-\beta ^{2}}}}}
—
μ
β
e
2
+
μ
−
β
e
2
2
{\displaystyle {\frac {\mu ^{\frac {\beta _{\mathrm {e} }}{2}}+\mu ^{-{\frac {\beta _{\mathrm {e} }}{2}}}}{2}}}
cociente de masa
μ
=
{\displaystyle \mu =}
[
a
t
2
c
+
1
+
(
a
t
2
c
)
2
]
2
β
e
{\displaystyle {\Bigg [}{\frac {at}{2c}}\!+\!{\sqrt {1\!+\!{\Big (}{\frac {at}{2c}}{\Big )}^{2}}}{\Bigg ]}^{\frac {2}{\beta _{\mathrm {e} }}}}
exp
a
τ
c
β
e
{\displaystyle \exp {\frac {a\tau }{c\beta _{\mathrm {e} }}}}
[
1
+
a
d
2
c
2
(
1
+
1
+
2
c
2
a
d
)
]
2
β
e
{\displaystyle {\Bigg [}1\!+\!{\frac {ad}{2c^{2}}}{\Bigg (}\!1\!+\!{\sqrt {1\!+\!{\frac {2c^{2}}{ad}}}}{\Bigg )}{\Bigg ]}^{\frac {2}{\beta _{\mathrm {e} }}}}
(
1
+
β
1
−
β
)
1
β
e
{\displaystyle {\Bigg (}{\frac {1+\beta }{1-\beta }}{\Bigg )}^{\frac {1}{\beta _{\mathrm {e} }}}}
[
γ
+
γ
2
−
1
]
2
β
e
{\displaystyle {\Big [}\gamma +{\sqrt {\gamma ^{2}\!-\!1}}{\Big ]}^{\frac {2}{\beta _{\mathrm {e} }}}}
—
↑ a b c d Misner (1973) , ecuación 6.4 (estos autores usan c = 1)↑ a b c d e f g Semay (2006) , ecuaciones 3, 7, 10↑ a b Walter (2006) , ecuación 23c (este autor usa
w
/
c
=
β
e
{\displaystyle w/c=\beta _{\mathrm {e} }}
↑ a b Ackeret (1946) , ecuaciones 16 y 17 (este autor usa
U
/
c
=
β
{\displaystyle U/c=\beta }
![{\displaystyle {\frac {c^{2}}{a}}{\Bigg [}{\sqrt {1\!+\!{\Big (}{\frac {at}{c}}{\Big )}^{2}}}\!-\!1{\Bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75fd29d5c11bdce25a5916e83b5af8c5b1f425d8)
![{\displaystyle {\frac {c^{2}}{a}}{\Big [}\cosh {\frac {a\tau }{c}}\!-\!1{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/449ca35136e2ab5bf327be441bb4f63fbb4d5046)
![{\displaystyle {\frac {c^{2}}{a}}{\Bigg [}{\frac {1}{\sqrt {1\!-\!\beta ^{2}}}}\!-\!1{\Bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc964f7cf0bf42823b3ca7e6a006e6b0f0b8bfd1)
![{\displaystyle {\frac {c^{2}}{a}}{\Bigg [}{\frac {\mu ^{\beta _{\mathrm {e} }}\!+\!\mu ^{-\beta _{\mathrm {e} }}}{2}}\!-\!1{\Bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/321a4e981f816daa7b09e5fdd061e4e10986bcec)
![{\displaystyle {\Bigg [}{\frac {at}{c}}\!+\!{\sqrt {1\!+\!{\Big (}{\frac {at}{c}}{\Big )}^{2}}}{\Bigg ]}^{\frac {1}{\beta _{\mathrm {e} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f30e27e30a9fce659ea3006ef244b4112c1f12f)
![{\displaystyle {\Bigg [}1\!+\!{\frac {ad}{c^{2}}}{\Bigg (}\!1\!+\!{\sqrt {1\!+\!{\frac {c^{2}}{ad}}}}{\Bigg )}{\Bigg ]}^{\frac {1}{\beta _{\mathrm {e} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7f97a39b88f8bfee54d8b4596ff6cb23d693a3d)
![{\displaystyle {\Big [}\gamma +{\sqrt {\gamma ^{2}\!-\!1}}{\Big ]}^{\frac {1}{\beta _{\mathrm {se} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8182fe802b6f653f4ca8f4b15da1c1564d7a6ae6)
![{\displaystyle {\frac {2c^{2}}{a}}{\Bigg [}{\sqrt {1\!+\!{\Big (}{\frac {at}{2c}}{\Big )}^{2}}}\!-\!1{\Bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/675d02ea95f42e890b4d294c7370b8621217cf51)
![{\displaystyle {\frac {2c^{2}}{a}}{\Big [}\cosh {\frac {a\tau }{2c}}\!-\!1{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d247ea5b893b45b5bcc4e368285317afe27b1bd7)
![{\displaystyle {\frac {2c^{2}}{a}}{\Bigg [}{\frac {1}{\sqrt {1\!-\!\beta ^{2}}}}\!-\!1{\Bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/731d7fc4ce9194d5a86d43b239e40a79025c3fad)
![{\displaystyle {\frac {2c^{2}}{a}}{\Bigg [}{\frac {\mu ^{\frac {\beta _{\mathrm {e} }}{2}}\!+\!\mu ^{-{\frac {\beta _{\mathrm {e} }}{2}}}}{2}}\!-\!1{\Bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31ba66b672d8e713516fe603069c1d245a5c8c5e)
![{\displaystyle {\Bigg [}{\frac {at}{2c}}\!+\!{\sqrt {1\!+\!{\Big (}{\frac {at}{2c}}{\Big )}^{2}}}{\Bigg ]}^{\frac {2}{\beta _{\mathrm {e} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a99e7e84a6f78624504c190f34dd4e3a5c4d6e)
![{\displaystyle {\Bigg [}1\!+\!{\frac {ad}{2c^{2}}}{\Bigg (}\!1\!+\!{\sqrt {1\!+\!{\frac {2c^{2}}{ad}}}}{\Bigg )}{\Bigg ]}^{\frac {2}{\beta _{\mathrm {e} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1f8733c8ea03504c38d6805fabce361269c7ffd)
![{\displaystyle {\Big [}\gamma +{\sqrt {\gamma ^{2}\!-\!1}}{\Big ]}^{\frac {2}{\beta _{\mathrm {e} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20a6ef8941c548250770182542457744941c0107)
