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Phynos
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Homework Statement
(a) A light signal is fired at ##60^o## North of West. Calculate the west-east velocity component of the signal according to an observer traveling due East at 0.5c. State your answer as a multiple of c.
(b) Calculate the North-South velocity component of the light signal according to an observer traveling due East at 0.5c. State your answer as a multiple of c.
(c) Calculate the West-East velocity component of the light source according to an observer traveling due West at 0.5c. State your answer as a multiple of c.
(d) Calculate the North-South velocity components of this light source according to an observer traveling due West at 0.5c. State your answer as a multiple of c.
Homework Equations
## U'_x = \Large \frac{U_x - V}{1 - \frac{U_x V}{c^2}} ##
## U'_y = \Large \frac{U_y}{\gamma(1 - \frac{U_y V}{c^2})} ##
## \gamma = \Large \frac{1}{ \sqrt{1 - \frac{v^2}{c^2}}} ##
The Attempt at a Solution
I'm getting mixed up about what sign each term should have. There must be a easier way to figure this out than trial and error?
For the cases where the observer is moving east at 0.5c...
(a) ## U'_x = \Large \frac{0.5c + cos60*c}{1 + \frac{0.5c*cos60*c}{c^2}} \normalsize = 0.8c ##
Here I have v negative because the observer is moving in the opposite direction as the east-west component of the light.
(b) ## \gamma = \Large \frac{1}{ \sqrt{1 - \frac{(0.5c)^2}{c^2}}} \normalsize = 1.1547 ##
## U'_y = \Large \frac{sin60*c}{1.1547*(1 - \frac{sin60*c*0.5*c}{c^2})} \normalsize = 1.322c ##
Which is nonsense so I assume that sign in the denominator should be positive?
## U'_y = \Large \frac{sin60*c}{1.1547*(1 + \frac{sin60*c*0.5*c}{c^2})} \normalsize = 0.5234c ##
Assuming that's correct, what's the reason for the sign swap? That's not the equation I was given...
For the cases where the observer is moving west at 0.5c...
(c) ## U'_x = \Large \frac{0.5c - cos60*c}{1 + \frac{0.5c*cos60*c}{c^2}} \normalsize = 0 ##
(d) ## \gamma = 1.1547 ##
## U'_y = \Large \frac{sin60*c}{1.1547*(1 + \frac{sin60*c*0.5*c}{c^2})} \normalsize = 0.5234c ##
What kind of result is that!? No velocity along the x axis, all along the y axis, except the y component of the velocity is less than c. This is nonsense.
Also, hopefully the Latex is more readable, it took forever to get it right.
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