Question

(a) How fast must you be approaching a red traffic (\(\lambda=\) 675 nm) for it to appear yellow (\(\lambda=\) 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.

Short answer

(a) Speed: 0.075c; (b) Fine: $80,999,910.

Step-by-step solution

Understanding the Problem

To solve this problem, we need to use the Doppler effect principle for light. The observed wavelength changes due to the relative motion between the observer and the source. We will calculate this speed using the formula for the Doppler effect in light.

Applying the Doppler Effect Formula for Light

For light, the relativistic Doppler effect is given by the formula: \( \frac{\lambda'}{\lambda} = \sqrt{\frac{1+\beta}{1-\beta}} \), where \(\lambda'\) is the observed wavelength, \(\lambda\) is the original wavelength, and \(\beta = \frac{v}{c}\) is the velocity as a fraction of the speed of light \(c\).

Plugging in Given Values

We are given \(\lambda = 675 \text{ nm}\) and \(\lambda' = 575 \text{ nm}\). We plug these into the formula: \( \frac{575}{675} = \sqrt{\frac{1+\beta}{1-\beta}} \).

Solving for \(\beta\)

Square both sides to eliminate the square root: \(\left( \frac{575}{675} \right)^2 = \frac{1+\beta}{1-\beta} \). Simplify and solve for \(\beta\) using algebraic manipulation. You'll get: \(\beta \approx 0.075 \).

Converting \(\beta\) to Velocity

The velocity \(v\) in terms of the speed of light \(c\) is \(v = \beta c\). So, \(v = 0.075c\).

Calculating Actual Speed in km/h

Since \(c = 3 \times 10^5 \text{ km/s}\), the speed \(v = 0.075 \times 3 \times 10^5 = 22,500 \text{ km/s}\). To convert this into km/h, multiply by 3600 (seconds per hour): \(v = 22,500 \times 3600 = 81,000,000 \text{ km/h}\).

Determining the Speed Over Limit

The speed limit is 90 km/h. Your speed is \(81,000,000 \text{ km/h}\). So, you are exceeding the speed limit by \(81,000,000 - 90 = 80,999,910 \text{ km/h}\).

Calculating the Fine

The fine is \\(1.00 for each km/h over the speed limit. Thus, the fine will be \\)80,999,910.

Key concepts

Relativistic Doppler Effect

The relativistic Doppler effect is a specific phenomenon that applies to light and other electromagnetic waves. It occurs when the source and observer move relative to each other at speeds close to the speed of light. In these conditions, the familiar Doppler effect we experience with sound waves needs to be adjusted for the effects of special relativity.
Unlike sound waves, where the medium (air, water) influences wave speed, light waves move through the vacuum of space at a constant speed. The formula used to describe the relativistic Doppler effect for light is:

• \(\frac{\lambda'}{\lambda} = \sqrt{\frac{1+\beta}{1-\beta}}\)

where \(\lambda'\) is the observed wavelength, \(\lambda\) is the source wavelength, and \(\beta = \frac{v}{c}\) is the velocity as a fraction of the speed of light \(c\).
This formula accounts for the relativistic effects due to high velocity. When calculating such changes for fast-moving objects, this formula ensures observations accurately reflect the phenomenon's relativistic nature.

Wavelength Change

Wavelength change in the context of the Doppler effect describes how the color of light alters when the source or observer moves at significant speed. For example, as explained in the exercise, a red traffic light with a wavelength of 675 nm might appear yellow (575 nm) to someone moving toward it.
To solve problems involving wavelength change due to relative motion, we use the ratio \(\frac{\lambda'}{\lambda}\). This division gives us the change factor, influenced by the velocity of the motion, which can be plugged into the relativistic Doppler effect formula. The alteration in wavelength is why distant galaxies often appear redshifted, or shifted toward longer wavelengths, if they move away from us at high speed.
This effect is crucial in astrophysics to measure the universe's expansion and observe celestial objects' motion.

Speed of Light

The speed of light, denoted as \(c\), is a fundamental constant of nature. It is approximately \(3 \times 10^8 \text{ m/s}\) in a vacuum. Light's consistent speed forms the backbone of Einstein's theory of relativity, influencing both time and space's behavior.
In problems like the original exercise, the speed of light assists in calculating speed as a fraction of \(c\). For instance, using \(\beta = \frac{v}{c}\) helps determine how close a velocity is to light speed. When we say a speed is a percentage of \(c\), such as 7.5% in the exercise, it contextualizes within the maximum possible speed limit set by the universe.
The unwavering pace of light unifies electromagnetic wave observations, ensuring that any calculations related to the relativistic Doppler effect remain consistent and grounded in reality. This constancy allows light's properties to be predictable and reliable across numerous scientific endeavors.