Usuario:Corriendodelohpacoh/Cohete relativista

El cohete relativista es aquel cohete que alcanza una velocidad cercana a la de la luz de modo que se hacen notar losefectos de la relatividad especial, en particular la dilatación del tiempo.

Mecánica del cohete

Ecuación de cohete

A escribir.

Cinética del cohete

A escribir.

Cinética y uso de combustible del cohete relativista. Los parámetros son la aceleración uniforme a y la velocidad relativa de escape efectiva β_e, ligada al impulso específico I_esp por β_e = g₀I_esp/c. Se dan las fórmulas para la distancia recorrida d, el tiempo terrestre transcurrido t, el tiempo propio τ, la velocidad máxima alcanzada en el sistema de referencia terrestre (vía β y γ) y el cociente de masa μ (inverso de la fracción de carga útil).

Cohete con aceleración uniforme
Variable		t	τ	d	β	γ	μ
tiempo terrestre	$t=$	—	${\frac {c}{a}}\sinh {\frac {a\tau }{c}}$ ​	${\frac {d}{c}}{\sqrt {1+{\frac {2c^{2}}{ad}}}}$ ​	${\frac {c}{a}}{\sqrt {\frac {\beta }{1-\beta }}}$ ​	${\frac {c}{a}}{\sqrt {\gamma -1}}$ ​	${\frac {c}{a}}\,{\frac {\mu ^{\beta _{\mathrm {e} }}-\mu ^{-\beta _{\mathrm {e} }}}{2}}$
tiempo propio	$\tau =$	${\frac {c}{a}}\sinh ^{-1}\ {\frac {at}{c}}$ ​	—	${\frac {c}{a}}\cosh ^{-1}\ {\Big (}1+{\frac {ad}{c^{2}}}{\Big )}$ ​	${\frac {c}{a}}\tanh ^{-1}\ \beta$ ​	${\frac {c}{a}}\cosh ^{-1}\gamma$	${\frac {c}{a}}\,\beta _{\mathrm {e} }\log \mu$
distancia	$d=$	${\frac {c^{2}}{a}}{\Bigg [}{\sqrt {1\!+\!{\Big (}{\frac {at}{c}}{\Big )}^{2}}}\!-\!1{\Bigg ]}$ ​	${\frac {c^{2}}{a}}{\Big [}\cosh {\frac {a\tau }{c}}\!-\!1{\Big ]}$ ​	—	${\frac {c^{2}}{a}}{\Bigg [}{\frac {1}{\sqrt {1\!-\!\beta ^{2}}}}\!-\!1{\Bigg ]}$	${\frac {c^{2}}{a}}(\gamma -1)$ ​	${\frac {c^{2}}{a}}{\Bigg [}{\frac {\mu ^{\beta _{\mathrm {e} }}\!+\!\mu ^{-\beta _{\mathrm {e} }}}{2}}\!-\!1{\Bigg ]}$
velocidad relativa	$\beta =$	${\frac {at/c}{\sqrt {1+{\big (}{\frac {at}{c}}{\big )}^{2}}}}$ ​	$\tanh {\frac {a\tau }{c}}$ ​	${\sqrt {1-{\Big (}1+{\frac {ad}{c^{2}}}{\Big )}^{-2}}}$	—	${\sqrt {1-{\frac {1}{\gamma ^{2}}}}}$	${\frac {1-\mu ^{-2\beta _{\mathrm {e} }}}{1+\mu ^{-2\beta _{\mathrm {e} }}}}$ ​
factor de Lorentz	$\gamma =$	${\sqrt {1+{\Big (}{\frac {at}{c}}{\Big )}^{2}}}$ ​	$\cosh {\frac {a\tau }{c}}$	$1+{\frac {ad}{c^{2}}}$	${\frac {1}{\sqrt {1-\beta ^{2}}}}$	—	${\frac {\mu ^{\beta _{\mathrm {e} }}+\mu ^{-\beta _{\mathrm {e} }}}{2}}$
cociente de masa	$\mu =$	${\Bigg [}{\frac {at}{c}}\!+\!{\sqrt {1\!+\!{\Big (}{\frac {at}{c}}{\Big )}^{2}}}{\Bigg ]}^{\frac {1}{\beta _{\mathrm {e} }}}$	$\exp {\frac {a\tau }{c\beta _{\mathrm {e} }}}$	${\Bigg [}1\!+\!{\frac {ad}{c^{2}}}{\Bigg (}\!1\!+\!{\sqrt {1\!+\!{\frac {c^{2}}{ad}}}}{\Bigg )}{\Bigg ]}^{\frac {1}{\beta _{\mathrm {e} }}}$	${\Bigg (}{\frac {1+\beta }{1-\beta }}{\Bigg )}^{\frac {1}{2\beta _{\mathrm {e} }}}$ ​	${\Big [}\gamma +{\sqrt {\gamma ^{2}\!-\!1}}{\Big ]}^{\frac {1}{\beta _{\mathrm {se} }}}$	—
Viaje en cohete con aceleración y deceleración uniformes
Variable		t	τ	d	β	γ	μ
tiempo terrestre	$t=$	—	${\frac {2c}{a}}\sinh {\frac {a\tau }{2c}}$	${\frac {d}{c}}{\sqrt {1+{\frac {4c^{2}}{ad}}}}$	${\frac {2c}{a}}{\sqrt {\frac {\beta }{1-\beta }}}$	${\frac {2c}{a}}{\sqrt {\gamma -1}}$	${\frac {2c}{a}}\,{\frac {\mu ^{\frac {\beta _{\mathrm {e} }}{2}}-\mu ^{-{\frac {\beta _{\mathrm {e} }}{2}}}}{2}}$
tiempo propio	$\tau =$	${\frac {2c}{a}}\sinh ^{-1}\ {\frac {at}{2c}}$	—	${\frac {2c}{a}}\cosh ^{-1}\ {\Big (}1+{\frac {ad}{2c^{2}}}{\Big )}$	${\frac {2c}{a}}\tanh ^{-1}\ \beta$	${\frac {2c}{a}}\cosh ^{-1}\gamma$	${\frac {2c}{a}}\,{\frac {\beta _{\mathrm {e} }}{2}}\log \mu$
distancia	$d=$	${\frac {2c^{2}}{a}}{\Bigg [}{\sqrt {1\!+\!{\Big (}{\frac {at}{2c}}{\Big )}^{2}}}\!-\!1{\Bigg ]}$	${\frac {2c^{2}}{a}}{\Big [}\cosh {\frac {a\tau }{2c}}\!-\!1{\Big ]}$	—	${\frac {2c^{2}}{a}}{\Bigg [}{\frac {1}{\sqrt {1\!-\!\beta ^{2}}}}\!-\!1{\Bigg ]}$	${\frac {2c^{2}}{a}}(\gamma -1)$	${\frac {2c^{2}}{a}}{\Bigg [}{\frac {\mu ^{\frac {\beta _{\mathrm {e} }}{2}}\!+\!\mu ^{-{\frac {\beta _{\mathrm {e} }}{2}}}}{2}}\!-\!1{\Bigg ]}$
velocidad relativa máxima	$\beta =$	${\frac {at/(2c)}{\sqrt {1+{\big (}{\frac {at}{2c}}{\big )}^{2}}}}$	$\beta =\tanh {\frac {a\tau }{2c}}$	${\sqrt {1-{\Big (}1+{\frac {ad}{2c^{2}}}{\Big )}^{-2}}}$	—	${\sqrt {1-{\frac {1}{\gamma ^{2}}}}}$	${\frac {1-\mu ^{-\beta _{\mathrm {e} }}}{1+\mu ^{-\beta _{\mathrm {e} }}}}$
factor de Lorentz máximo	$\gamma =$	${\sqrt {1+{\Big (}{\frac {at}{2c}}{\Big )}^{2}}}$	$\cosh {\frac {a\tau }{2c}}$	$1+{\frac {ad}{2c^{2}}}$	${\frac {1}{\sqrt {1-\beta ^{2}}}}$	—	${\frac {\mu ^{\frac {\beta _{\mathrm {e} }}{2}}+\mu ^{-{\frac {\beta _{\mathrm {e} }}{2}}}}{2}}$
cociente de masa	$\mu =$	${\Bigg [}{\frac {at}{2c}}\!+\!{\sqrt {1\!+\!{\Big (}{\frac {at}{2c}}{\Big )}^{2}}}{\Bigg ]}^{\frac {2}{\beta _{\mathrm {e} }}}$	$\exp {\frac {a\tau }{c\beta _{\mathrm {e} }}}$	${\Bigg [}1\!+\!{\frac {ad}{2c^{2}}}{\Bigg (}\!1\!+\!{\sqrt {1\!+\!{\frac {2c^{2}}{ad}}}}{\Bigg )}{\Bigg ]}^{\frac {2}{\beta _{\mathrm {e} }}}$	${\Bigg (}{\frac {1+\beta }{1-\beta }}{\Bigg )}^{\frac {1}{\beta _{\mathrm {e} }}}$	${\Big [}\gamma +{\sqrt {\gamma ^{2}\!-\!1}}{\Big ]}^{\frac {2}{\beta _{\mathrm {e} }}}$	—

Referencias

Referencias

• J. Ackeret, Zur theorie der Racketen (1946), Helvetica Physica Acta, 19, 103
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation (1973), W. H. Freeman, San Francisco ISBN 0-7167-0344-0
• Ulrich Walter, Relativistic rocket and space flight (2006), Acta Astronautica 59, 453
• Claude Semay, Observer with a constant proper acceleration, doi:10.1088/0143-0807/27/5/015