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?

the speed is 0.53 10⁸ m / s

Explanation:

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