Question

Consider the tale of the physicist who is ticketed for running a red light and argues that because he was approaching the intersection, the red light was Doppler shifted and appeared green. How fast would he have to have been going? \(\left(\lambda_{\text {red }} \approx 650 \mathrm{nm} \text { and } \lambda_{\text {green }} \approx 530 \mathrm{nm} .\right)\)

Short answer

The physicist must travel at approximately 0.603 times the speed of light towards the traffic light.

Step-by-step solution

Understand the Doppler Effect

The Doppler Effect describes the change in wavelength (or frequency) of a wave in relation to an observer moving relative to the wave source. In this case, the physicist argues the light wave from a red light signal seems green due to this effect while approaching it.

Express the Relativistic Doppler Shift Equation

The relativistic Doppler shift equation for light is given by \( \lambda_{o} = \lambda_{s} \sqrt{\frac{1 + \beta}{1 - \beta}} \), where \( \lambda_{o} \) is the observed wavelength, \( \lambda_{s} \) is the source wavelength, and \( \beta = \frac{v}{c} \) with \( v \) as the speed of the observer and \( c \) the speed of light.

Substitute Known Values into the Equation

Set \( \lambda_{o} = 530 \, \text{nm} \) (green) and \( \lambda_{s} = 650 \, \text{nm} \) (red). Substitute these into the equation: \[ 530 = 650 \sqrt{\frac{1 + \beta}{1 - \beta}} \]

Isolate the Relativistic Factor \( \beta \)

First, divide both sides by 650:\[ \frac{530}{650} = \sqrt{\frac{1 + \beta}{1 - \beta}} \]Simplify this to:\[ 0.8154 = \sqrt{\frac{1 + \beta}{1 - \beta}} \]Square both sides to remove the square root:\[ 0.6651 = \frac{1 + \beta}{1 - \beta} \]

Solve for \( \beta \)

Write the equation: \( 0.6651 (1 - \beta) = 1 + \beta \). Expand and rearrange the terms to solve for \( \beta \):\[ 0.6651 - 0.6651\beta = 1 + \beta \]Combine like terms:\[ 0.6651 = 1.6651\beta + 1 \]\[ 0.6651 - 1 = 1.6651\beta \]\[ -0.3349 = 1.6651\beta \]\[ \beta = \frac{-0.3349}{1.6651} \approx -0.201 \].

Calculate Speed \( v \)

The speed \( v \) is calculated from \( \beta = \frac{v}{c} \), where \( c \approx 3 \times 10^8 \, \text{m/s} \). Thus, \( v = \beta \times c \), \[ v \approx -0.201 \times 3 \times 10^8 \approx -0.603 \times 10^8 \, \text{m/s} \].The negative sign indicates direction towards the source.

Conclusion

The physicist would need to travel at approximately 0.603 times the speed of light towards the source for the Doppler shift to occur as argued.

Key concepts

Relativistic Doppler Shift

The relativistic Doppler shift is a concept from physics that describes how the wavelength of light changes when the source of the light is moving relative to an observer.
When either the observer or the source is moving at a speed close to the speed of light, we use the relativistic Doppler shift formula to calculate the observed changes in wavelength.
This phenomenon can occur with all waves, but it is often associated with light waves and has important implications in fields like astrophysics and cosmology. In this particular scenario, a physicist claims that a red light appeared green due to this effect while moving towards the light source. In other words, due to the motion, the observed wavelength differs from the emitted wavelength. This happens because the light waves bunch up if the observer (or source) is moving toward the other, resulting in a shift to shorter wavelengths.

Wavelength Change

Wavelength change is an integral part of understanding the Doppler Effect.
When an object moves towards a source of light, the wavelengths become shorter, causing a shift towards the blue end of the spectrum (blue-shift).
Conversely, when an object moves away, wavelengths lengthen, resulting in a red-shift.For the physicist to perceive the red light as green, the wavelength of the perceived light must match that of green light, which in this case, is shorter than the original red wavelength.
This requires calculating the observed wavelength using the relativistic Doppler shift equation, \[ \lambda_{o} = \lambda_{s} \sqrt{\frac{1 + \beta}{1 - \beta}}. \]Here, \( \lambda_{o} \) is the green light's wavelength \( 530 \, \text{nm} \), and \( \lambda_{s} \) is the red light's wavelength \( 650 \, \text{nm} \).
Consequently, the change from red to green light is a shift to shorter wavelengths.

Light Waves

Understanding light waves is crucial for comprehending how the Doppler Effect operates on light.
Light waves are electromagnetic waves that travel through a vacuum at a constant speed, known as the speed of light.
These waves possess different wavelengths corresponding to various colors in the visible spectrum, ranging from reds with longer wavelengths to blues and purples with shorter wavelengths. In the scenario involving the physicist, the red light with a wavelength of approximately 650 nanometers appears green at about 530 nanometers.
This transformation occurs as the physicist moves towards the light source, resulting in a compression of the light waves in the direction of movement.
This compression changes the wavelength and alters the color perceived by the observer due to the Doppler Effect.

Speed of Light

The speed of light is a fundamental constant in physics, denoted by \( c \), and is approximately \( 3 \times 10^8 \, ext{m/s} \).
It is crucial in the calculation of the relativistic Doppler shift.In the given problem, the speed of light comes into play when calculating how fast the physicist must travel to cause the requested wavelength change.
The formula used involves the term \( \beta = \frac{v}{c} \) where \( v \) is the velocity of the observer relative to the speed of light.
Understanding this relationship is vital for comprehending how velocity affects wave perception.To make the red light appear green, the physicist would need to travel at a high fraction of the speed of light, illustrating the extreme velocities required for significant relativistic wavelength transformation.
Such speeds are typically beyond the capacity of everyday travel, underscoring the impossibility of achieving this color change with current technology.