A physicist drives through a stop light. When he is pulled...

A physicist drives through a stop light. When he is pulled over, he tells the police officer that the Doppler shift made the red light of wavelength 650 $\mathrm{nm}$ appear green to him, with a wavelength of 520 $\mathrm{nm}$ . The police officer writes out a traffic citation for speeding. How fast was the physicist traveling, according to his own testimony?

Key Concepts

Speed of Light and Relative Velocity

The speed of light is a fundamental constant in nature that acts as a limiting velocity in the universe. When analyzing Doppler shifts in light, especially under relativistic conditions, comparing the relative velocity of the source and observer to the speed of light is essential to determine the magnitude of the wavelength shift and to predict outcomes consistent with special relativity.

Wavelength and Frequency Relationship

Wavelength and frequency are inversely related properties of a wave, connected by the equation ?? = c, where c is the speed of light. Changes in the observed wavelength due to the Doppler effect correspond to proportional changes in frequency, thereby altering the perceived color of light in optical scenarios.

Doppler Effect

The Doppler effect is a phenomenon where the wavelength and frequency of waves (such as sound or light) change for an observer moving relative to the source. This change can make the waves appear to shift in color (for light) or pitch (for sound), depending on the direction and speed of the motion.

Relativistic Doppler Effect

When the velocities involved are a significant fraction of the speed of light, the classical Doppler effect must be modified to include the effects predicted by special relativity. This relativistic Doppler effect incorporates time dilation and other relativistic factors, leading to a more pronounced and accurately described shift in the observed wavelength or frequency.

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