1. The problem statement, all variables and given/known data
Im trying to work how fast the probe is moving whit an observer placed on earth. Se the image for details http://ift.tt/1pJt1dh
u=u'="speed relativity to earth and spacecraft"=0.9c
v=u''="speed relativity to probe and spacecraft"=0.7c
I started whit the "Einstein Velocity Addition"
[itex]u=\frac{u'+u''}{1+\frac{u'u''}{c^2}}=\frac{0.9c+0.7c}{1+\frac{0.9c*0.7c }{c^2}}=0.9815c[/itex]
This i think is correct?
But i wanned to try to use the lorentz transformations to sove the problem.
2. Relevant equations
[itex]x=\gamma'(x'+u't')[/itex]
[itex]t=\gamma'(t'+\frac{u'x'}{c^2})[/itex]
3. The attempt at a solution
[itex]x=\gamma'(x'+u't'),\quad x'=\gamma''(x'+u''t'')[/itex]
[itex]t=\gamma'(t'+\frac{u'x'}{c^2}),\quad t'=\gamma''(t''+\frac{u''x''}{c^2})[/itex]
Suplementing x' into x, and t' into t
[itex]x=\gamma'\gamma''x''+\gamma'\gamma''u''t''+\gamma'u't'[/itex]
[itex]t=\gamma'\gamma''t''+\gamma'\gamma''\frac{u''x''}{c^2}+\gamma'\frac{u'x '}{c^2}[/itex]
Differentiation:
[itex]dx=\gamma'\gamma''dx''+\gamma'\gamma''u''dt''+\gamma'u'dt[/itex]
[itex]dt=\gamma'\gamma''dt''+\gamma'\gamma''\frac{u''dx''}{c^2}+\gamma'\frac{ u'dx'}{c^2}[/itex]
Express u as dx/dt
[itex]u=\frac{dx}{dt}=\frac{\gamma'\gamma''dx''+\gamma'\gamma''u''dt''+\gamma 'u'dt'}{\gamma'\gamma''dt''+\gamma'\gamma''\frac{u''dx''}{c^2}+\gamma'\ frac{u'dx'}{c^2}}*\frac{\frac{1}{\gamma' dt''}}{\frac{1}{\gamma' dt''}}=\frac{2\gamma''u''+u'\frac{dt'}{dt''}}{\gamma''(1+\frac{u''^2}{c ^2})+\frac{u'^2}{c^2}\cdot\frac{dx'}{dt''}}[/itex]
I take a closer look at dt/dt'' and dx'/dt''
[itex]\frac{dt'}{dt''}=\frac{\gamma''(dt''+\frac{u''}{c^2}dx'')}{dt''}=\gamma ''(1+\frac{u''^2}{c^2}),\quad u''=\frac{dx''}{dt''}[/itex]
[itex]\frac{dx'}{dt''}=\frac{\gamma''(dx''+u''dt'')}{dt''}=2\gamma''u''[/itex]
Preplacing in the "orignal expression"
[itex]u=\frac{2\gamma''u''+\gamma''u'(1+\frac{u''^2}{c^2})}{\gamma''(1+\frac{ u''^2}{c^2})+2\frac{u'u''}{c^2}\gamma''}=\frac{2''u''+u'(1+ \frac{u''^2}{c^2})}{(1+\frac{u''^2}{c^2})+2\frac{u'u''}{c^2}}=\frac{2'' u''+u'(1+\frac{u''^2}{c^2})}{(1+\frac{u''^2}{c^2})+2\frac{u'u''}{c^2}}[/itex]
As far as i can tell this is not equal to the equation given by the "Einstein Velocity Addition", so where does it all go wrong for me. Is there someone that can give me some advise?
Im trying to work how fast the probe is moving whit an observer placed on earth. Se the image for details http://ift.tt/1pJt1dh
u=u'="speed relativity to earth and spacecraft"=0.9c
v=u''="speed relativity to probe and spacecraft"=0.7c
I started whit the "Einstein Velocity Addition"
[itex]u=\frac{u'+u''}{1+\frac{u'u''}{c^2}}=\frac{0.9c+0.7c}{1+\frac{0.9c*0.7c }{c^2}}=0.9815c[/itex]
This i think is correct?
But i wanned to try to use the lorentz transformations to sove the problem.
2. Relevant equations
[itex]x=\gamma'(x'+u't')[/itex]
[itex]t=\gamma'(t'+\frac{u'x'}{c^2})[/itex]
3. The attempt at a solution
[itex]x=\gamma'(x'+u't'),\quad x'=\gamma''(x'+u''t'')[/itex]
[itex]t=\gamma'(t'+\frac{u'x'}{c^2}),\quad t'=\gamma''(t''+\frac{u''x''}{c^2})[/itex]
Suplementing x' into x, and t' into t
[itex]x=\gamma'\gamma''x''+\gamma'\gamma''u''t''+\gamma'u't'[/itex]
[itex]t=\gamma'\gamma''t''+\gamma'\gamma''\frac{u''x''}{c^2}+\gamma'\frac{u'x '}{c^2}[/itex]
Differentiation:
[itex]dx=\gamma'\gamma''dx''+\gamma'\gamma''u''dt''+\gamma'u'dt[/itex]
[itex]dt=\gamma'\gamma''dt''+\gamma'\gamma''\frac{u''dx''}{c^2}+\gamma'\frac{ u'dx'}{c^2}[/itex]
Express u as dx/dt
[itex]u=\frac{dx}{dt}=\frac{\gamma'\gamma''dx''+\gamma'\gamma''u''dt''+\gamma 'u'dt'}{\gamma'\gamma''dt''+\gamma'\gamma''\frac{u''dx''}{c^2}+\gamma'\ frac{u'dx'}{c^2}}*\frac{\frac{1}{\gamma' dt''}}{\frac{1}{\gamma' dt''}}=\frac{2\gamma''u''+u'\frac{dt'}{dt''}}{\gamma''(1+\frac{u''^2}{c ^2})+\frac{u'^2}{c^2}\cdot\frac{dx'}{dt''}}[/itex]
I take a closer look at dt/dt'' and dx'/dt''
[itex]\frac{dt'}{dt''}=\frac{\gamma''(dt''+\frac{u''}{c^2}dx'')}{dt''}=\gamma ''(1+\frac{u''^2}{c^2}),\quad u''=\frac{dx''}{dt''}[/itex]
[itex]\frac{dx'}{dt''}=\frac{\gamma''(dx''+u''dt'')}{dt''}=2\gamma''u''[/itex]
Preplacing in the "orignal expression"
[itex]u=\frac{2\gamma''u''+\gamma''u'(1+\frac{u''^2}{c^2})}{\gamma''(1+\frac{ u''^2}{c^2})+2\frac{u'u''}{c^2}\gamma''}=\frac{2''u''+u'(1+ \frac{u''^2}{c^2})}{(1+\frac{u''^2}{c^2})+2\frac{u'u''}{c^2}}=\frac{2'' u''+u'(1+\frac{u''^2}{c^2})}{(1+\frac{u''^2}{c^2})+2\frac{u'u''}{c^2}}[/itex]
As far as i can tell this is not equal to the equation given by the "Einstein Velocity Addition", so where does it all go wrong for me. Is there someone that can give me some advise?
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