Most people who studied physics in college have heard of the traffic light problem. A motorist is pulled over for running a red light. He tells the cop that due to the relativistic doppler shift, the red light appeared green. Suppose the officer believes him and doesn't ticket him for running the light. But the officer decides instead to write him up for speeding since the motorist admitted his own guilt. If it was a 35 MPH zone, by how much was the motorist speeding?

First some data:
green: wavelength 550nm, frequency 5.45x10^14 Hz
red: wavelength 650nm, frequency 4.26x10^14 Hz
If red appears green it has shifted (5.45 / 4.26) = 28% higher in frequency.

According to any physics textbook, electromagnetic doppler shift is predicted as follows:
Fobserved = Fsource * sqrt( (1 + B) / (1 - B) )
where B = V / c

In our case we know that: Fobserved = Fsource * 1.28.
so
sqrt( (1+B) / (1-B) ) = 1.28
(1+B) / (1-B) = 1.6384
(1 + B) = 1.6384 * (1 - B)
B + 1 = 1.6384 - 1.6384 * B
B = 0.6384 - 1.6384 * B
2.6384 * B = 0.6384
B = 0.6384 / 2.6384
B = 0.242
V / C = 0.242
V = 0.242 * C

What does 24% of C look like on the speedometer in miles per hour?
C = 3 x 10^8 meters per second, so 0.24C = 7.2 x 10^7 m/s.
With 1,609 meters per mile and 3,600 seconds per hour we have 7.2e7 * 3600 / 1609 = 1.6e8 miles per hour.

That's 160 million miles per hour - the motorist should have kept his mouth shut and taken his red light ticket.