Answer
The fine is 200 million dollars.
Work Step by Step
We can find the speed of the car as:
$\lambda' = \lambda_0~\frac{\sqrt{1-v/c}}{\sqrt{1+v/c}}$
$\frac{\lambda'}{\lambda_0} = \frac{\sqrt{1-v/c}}{\sqrt{1+v/c}}$
$(\frac{\lambda'}{\lambda_0})^2 = \frac{1-v/c}{1+v/c}$
$(\frac{\lambda'}{\lambda_0})^2(c+v) = c-v$
$v~[1+(\frac{\lambda'}{\lambda_0})^2] = [1-(\frac{\lambda'}{\lambda_0})^2]~c$
$v = \frac{1-(\frac{\lambda'}{\lambda_0})^2}{1+(\frac{\lambda'}{\lambda_0})^2}~c$
$v = \frac{1-(\frac{540~nm}{650~nm})^2}{1+(\frac{540~nm}{650~nm})^2}~(3.0\times 10^8~m/s)$
$v = 5.5\times 10^7~m/s$
We then convert the speed to km/h:
$v = (5.5\times 10^7~m/s)(\frac{1~km}{1000~m})(\frac{3600~s}{1~h})$
$v = 2.0\times 10^8~km/h$
The speed of the car is $2.0\times 10^8~km/h$. Since the fine is 1 dollar for every 1 km/h over the speed limit of 50 km/h, the fine is $2.0\times 10^8~dollars$ which is 200 million dollars.
