Question

what would the speed of an observer be if a red(4.688x10^14 Hz) traffic light appeared green (5.5555x10^14 Hz) to the observer ?

Answer 1

Let's apply the expression for the relativistic transverse Doppler effect:

$$\dfrac{f_t}{f_o}= \sqrt{\dfrac{1+\beta}{1-\beta}}$$

where $f_t = 4.688\times10^{14} Hz$ is the frequncy of the traffic light, $f_o = 5.5555\times10^{14} Hz$ is the frequncy than the observer sees, $\beta = v/c$, where $v$ is the relative speed, and $c = 3\times 10^8m/s$ is the speed of light in vacuum.

Expressing the $v$ from the formula above, obtain:

$$v = c\dfrac{f_t^2 - f_o^2}{f_t^2 + f_o^2}\\ v = 3\times 10^8\dfrac{(4.688\times10^{14})^2 - (5.5555\times10^{14})^2}{(4.688\times10^{14})^2 + (5.5555\times10^{14})^2}\approx \\ \approx -0.5\times 10^8m/s$$

Answer. -0.5*10^8 m/s.