Two spaceships
Question
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The following quantities appear in the problem:
Geschwindigkeit \(v\) /
The following formulas must be used to solve the exercise:
\(u' = \frac{u-v}{1-\frac{uv}{c^2}} \quad \)
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Exercise:
Two spaceships leave Earth in opposite directions each with a speed of .c with respect to Earth. abcliste abc What is the velocity of the first spaceship relative to the second spaceship? abc What is the velocity of spaceship two relative to spaceship one? abcliste
Solution:
center tikzpicture scoperotate scale. %flaps coordinate P at --; drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scoperotate around:Pshift-cm-cmyscalexscal drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scope %rocket drawfillyellow!!white -- tooutin - tooutin tooutin --; %peak drawfillblue!!white -. tooutin . tooutin tooutin -.; %central flap drawfillblue!!white - tooutin - - tooutin -; %porthole drawcolorgrayultra thickfillwhite!!blue -. circle .cm; %fuel foreach i in ... pgfmathsetmacroxrand*. + *- - . pgfmathsetmacroyrand*. + *.-. pgfmathsetmacroopacValrand*.+ drawcolorred fillred!!yellow xy circle .cm; scope pgftransformxshift filldrawcolorblue fillblue!!white circle .cm; drawcolorgreen!!black-latex --; drawcolorgreen!!black-latex --; nodecolorgreen!!black at .. mathcalS'; pgftransformxshift drawcolorgreen!!black-latex --; drawcolorgreen!!black-latex --; nodecolorgreen!!black at .. mathcalS; drawcolorblulatex .--. noderight u_r' .c; drawcolorblulatex -.---. nodeleft u_l' -.c; drawcolorred-latex -..---. nodeleft v -.c; drawcolorred-latex -.-.---.-. nodeleft u_l -.c; scoperotate scale. %flaps coordinate P at --; drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scoperotate around:Pshift-cm-cmyscalexscal drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scope %rocket drawfillyellow!!white -- tooutin - tooutin tooutin --; %peak drawfillblue!!white -. tooutin . tooutin tooutin -.; %central flap drawfillblue!!white - tooutin - - tooutin -; %porthole drawcolorgrayultra thickfillwhite!!blue -. circle .cm; %fuel foreach i in ... pgfmathsetmacroxrand*. + *- - . pgfmathsetmacroyrand*. + *.-. pgfmathsetmacroopacValrand*.+ drawcolorred fillred!!yellow xy circle .cm; scope tikzpicture center abcliste abc If we name the Earths frame mathcalS' and the frame of spaceship glqq rightgrqq mathcalS then the relative velocity of mathcalS' earth with respect to mathcalS is v-.c. The speed of spaceship glqq leftgrqq as seen from mathcalS' earth is then u_l'-.c. The speed of spaceship glqq leftgrqq in mathcalS is: u_l fracu_l'+v+fracu_l'vc^ frac-.c + -.c+frac-.c-.cc^ -.c abc Since the problem is symmetric we know that u_x''.c. abcliste
Two spaceships leave Earth in opposite directions each with a speed of .c with respect to Earth. abcliste abc What is the velocity of the first spaceship relative to the second spaceship? abc What is the velocity of spaceship two relative to spaceship one? abcliste
Solution:
center tikzpicture scoperotate scale. %flaps coordinate P at --; drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scoperotate around:Pshift-cm-cmyscalexscal drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scope %rocket drawfillyellow!!white -- tooutin - tooutin tooutin --; %peak drawfillblue!!white -. tooutin . tooutin tooutin -.; %central flap drawfillblue!!white - tooutin - - tooutin -; %porthole drawcolorgrayultra thickfillwhite!!blue -. circle .cm; %fuel foreach i in ... pgfmathsetmacroxrand*. + *- - . pgfmathsetmacroyrand*. + *.-. pgfmathsetmacroopacValrand*.+ drawcolorred fillred!!yellow xy circle .cm; scope pgftransformxshift filldrawcolorblue fillblue!!white circle .cm; drawcolorgreen!!black-latex --; drawcolorgreen!!black-latex --; nodecolorgreen!!black at .. mathcalS'; pgftransformxshift drawcolorgreen!!black-latex --; drawcolorgreen!!black-latex --; nodecolorgreen!!black at .. mathcalS; drawcolorblulatex .--. noderight u_r' .c; drawcolorblulatex -.---. nodeleft u_l' -.c; drawcolorred-latex -..---. nodeleft v -.c; drawcolorred-latex -.-.---.-. nodeleft u_l -.c; scoperotate scale. %flaps coordinate P at --; drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scoperotate around:Pshift-cm-cmyscalexscal drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scope %rocket drawfillyellow!!white -- tooutin - tooutin tooutin --; %peak drawfillblue!!white -. tooutin . tooutin tooutin -.; %central flap drawfillblue!!white - tooutin - - tooutin -; %porthole drawcolorgrayultra thickfillwhite!!blue -. circle .cm; %fuel foreach i in ... pgfmathsetmacroxrand*. + *- - . pgfmathsetmacroyrand*. + *.-. pgfmathsetmacroopacValrand*.+ drawcolorred fillred!!yellow xy circle .cm; scope tikzpicture center abcliste abc If we name the Earths frame mathcalS' and the frame of spaceship glqq rightgrqq mathcalS then the relative velocity of mathcalS' earth with respect to mathcalS is v-.c. The speed of spaceship glqq leftgrqq as seen from mathcalS' earth is then u_l'-.c. The speed of spaceship glqq leftgrqq in mathcalS is: u_l fracu_l'+v+fracu_l'vc^ frac-.c + -.c+frac-.c-.cc^ -.c abc Since the problem is symmetric we know that u_x''.c. abcliste