Tell It to the Judge. (a) How fast must you be approaching a red traffic light (λ = 675 nm) for it to appear yellow (λ = 575 nm)? Express your answer in terms of the speed of light. (b) If you used this as a reason not to get a ticket for running a red light, how much of a fine would you get for speeding? Assume that the fine is \$1.00 for each kilometer per hour that your speed exceeds the posted limit of 90 km/h.

Verified step by step guidance

1. Step 1: Understand the Doppler effect for light. The Doppler effect describes how the wavelength of light changes due to the relative motion between the source and the observer. For this problem, the observer (you) is moving toward the source (the red traffic light). The formula for the observed wavelength λ' is: ${\lambda }_{\text{'}}={\lambda }_0(\frac{1-\frac{v}{c}}{)}$, where λ₀ is the original wavelength, v is the velocity of the observer, and c is the speed of light.
2. Step 2: Rearrange the formula to solve for the velocity v. Substitute the given values for the wavelengths: λ₀ = 675 nm (red light) and λ' = 575 nm (yellow light). The rearranged formula is: $v=c\left(\frac{{\lambda }_0-{\lambda }_{\text{'}}}{{\lambda }_{\text{'}}}\right)$. This will allow you to calculate the velocity as a fraction of the speed of light.
3. Step 3: Convert the velocity from the fraction of the speed of light to kilometers per hour (km/h). Since the speed of light c is approximately 3 × 10⁸ m/s, you can multiply the fraction by c and then convert the result to km/h using the conversion factor: 1 m/s = 3.6 km/h.
4. Step 4: Determine the excess speed over the posted limit. Subtract the posted speed limit of 90 km/h from the calculated velocity in km/h to find how much faster you are traveling than the limit.
5. Step 5: Calculate the fine. Multiply the excess speed (in km/h) by \$1.00 per km/h to determine the total fine for speeding. This will give you the monetary penalty based on your calculated speed.

Key Concepts

Doppler Effect

The Doppler Effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. In this context, as a vehicle approaches a red traffic light, the light's wavelength shifts due to the vehicle's speed, causing it to appear yellow instead of red. This effect is crucial for understanding how motion affects the perception of light.

Wavelength and Color Perception

Wavelength is a fundamental property of light that determines its color. The visible spectrum ranges from approximately 380 nm (violet) to 750 nm (red). In this scenario, the red light at 675 nm appears yellow at 575 nm due to the Doppler shift, illustrating how changes in wavelength can alter color perception based on the observer's speed.

Speed and Fine Calculation

The calculation of fines for speeding is typically based on how much a driver's speed exceeds the posted limit. In this case, if the speed exceeds 90 km/h, the fine is $1.00 for each kilometer per hour over the limit. Understanding this calculation is essential for determining the financial consequences of exceeding speed limits, even when using the Doppler Effect as a defense.