- #1
wahaj
- 156
- 2
Homework Statement
A light source moves away from an observer at a speed vs that is small compared to c. show that the fractional shift in the observed wavelength can be approximated by
[tex]\frac{\Delta \lambda}{\lambda} \approx \frac{v_2 }{c}[/tex]
Homework Equations
[tex] f' = \frac{\sqrt{1+ \frac{v}{c} } } {\sqrt{1- \frac{v}{c} } } f [/tex]
[tex] v = f \lambda [/tex]
The Attempt at a Solution
First I know that v = vs. if I put this into the above formula I get
[tex] f' = \frac{\sqrt{1+ \frac{v_s}{c} } } {\sqrt{1- \frac{v_s}{c} } } f [/tex]
Since vs is small compared to c the terms inside the square root signs will be approximately equal to 1 so I can say that
[tex] f' \approx f [/tex]
[tex] f = \frac{v}{\lambda} [/tex]
Since we are working with light, when it leave the light source it travels at v = c. So
[tex] f = \frac{c}{\lambda} [/tex]
so far I have
[tex] f' = \frac{c}{\lambda} [/tex]
this is where I get lost. I have no idea what to do next because I can't seem to quite figure out what f' would be. f' is the frequency of the light observed by the observer. In the observer's frame of reference this would simply be f given above. the equation I derived before implies that I am working in the source's frame of reference so I will need to adjust for f'. But I have no idea how to do that. I know that the velocity of light is fixed so the only way the observed frequency changes is because lambda changes by an amount say Δλ. but this still gets me no where. Could someone give me a hint?