A physics professor claims in court that the reason he went through the red light (λ = 660 nm) was that, due to his motion, the red color was Doppler shifted to green (λ = 510 nm). How fast was he going?

As say in the exercise, the Doppler effect must be applied, in this case because it is an electromagnetic radiation with speed 3 10⁸ m/s and nothing can go faster we must use the relativistic Doppler effect

fo = fe √[(c-v) /(c + v)]

Where fo is the observed frequency, fe the emitted frequency, c the speed of light and v the relative speed of the observer and emit, it is positive move away

The light fulfills the relationship

c = λ f

f = c / λ

In this case v is negative since the source and the observer approach

Substituting in the equation

c /λo = c /λe √[ (c + v) / (c-v)]

λo = 510 10⁻⁹ m

λe = 610 10⁻⁹ m

We calculate the speed

(λe/λo)² = (c + v) / (c-v)

(λe /λo)² (c-v) = c + v

v +(λe /λo)² v = (λe /λo)² c -c

v [1 +(λe /λo)²] = c [(λe /λo)²-1]

v = c [(λe /λo)² -1] / [1 +(λe /λo)²]

v = 3 10⁸ [(610/510)² -1] / [1+ (610/510)²]

v = 3 10⁸ [1.43-1] / [1 + 1.43]

v = 0.53 10⁸ m / s

A little more about the speed of light