Dopper Effect in Special Relativity

[yukcream]

Suppose there is a planet, the traffic law there is~
car should stop if the traffic light in infrared region,i.e wavelength ~10^-5m, otherwise the driver will be fined.
There is a driver claimed that he saw the traffic light still in the X-ray region i.e wavelength ~10^-10m when passing the traffic light so he hadn't violated the law
If possible, what will be the speed of the car~

However, I found it is impossible for the driver to have this dopper shift, as the speed of the car will be about 600c ~

I did in following step:
by the formula of dopper effect for SR
[(1-v/c)/(1+v/c)]^1/2 = L0/L = 10^-5/10^-10where L is the wavelength
L0/L = 10^5
the RHS value is so large~
so I expand the LHS as:

(0.5 - 0.5V/c) (0.5+ 0.5v/c) than I get the value as v~600c!

What's going wrong, can anyone help me!

yukyuk

 

[Doc Al]

For one thing, the doppler formula is:
$$f = f' \sqrt{\frac{1 + v/c}{1 - v/c}}$$

As far as your expansion, I don't understand what you are doing. (Realize that v/c is not necessarily small compared to 1.)

 

[yukcream]

f is stand for frequency here I use wavelength instead~

 

[Doc Al]

f is stand for frequency here I use wavelength instead~

You can convert from one to the other: $f = c/\lambda$.