Problem — A red traffic light seen as green

Problem

A traffic light emits red light at $\lambda=700$ nm. How fast must a driver move towards the light to see it as green ($\lambda'=450$ nm)? Use the longitudinal relativistic Doppler effect.

Solution

Longitudinal Doppler formula, approaching. Moving towards the source, the light is shifted towards the blue (shorter wavelength, $\lambda' < \lambda$). With $\beta = v/c$: $\frac{\lambda'}{\lambda} = \sqrt{\frac{1-\beta}{1+\beta}}$

Numerical setup. The observed ratio is $\frac{\lambda'}{\lambda} = \frac{450}{700} = 0.643$ Squaring: $\frac{1-\beta}{1+\beta} = \left(\frac{450}{700}\right)^2 = 0.413$

Solving for $\beta$. Cross-multiplying: $1-\beta = 0.413\,(1+\beta)$, hence $1 - \beta = 0.413 + 0.413\,\beta$, from which $1 - 0.413 = \beta\,(1 + 0.413) \quad\Rightarrow\quad \beta = \frac{0.587}{1.413} \approx 0.42$

Velocity. $v = \beta\,c \approx 0.42\cdot 3\cdot 10^8\ \text{m/s}$

$\ev{v \approx 0.42\,c \approx 1.26\cdot 10^8\ \text{m/s}}$

Interpretation. It takes about $126\,000$ km/s (over 400 million km/h) to turn a red light green: no speeding ticket could ever be justified by the excuse “I saw it as green because of the Doppler effect”.

Links

Topics: Relatività ristretta Concepts: Effetto Doppler Skills: Impostazione simbolica