Suppose that you are a traffic officer and you stop a driver...

Suppose that you are a traffic officer and you stop a driver for going through a red light. Further suppose that the driver draws a picture for you (Figure $16-23$ ) and explains that the light looked green because of the Doppler effect when he went through it. Explain to him using the Doppler shift equation just how fast he would have had to be going for the red light $(\lambda=645 \mathrm{nm}),$ to appear green $(\lambda=545 \mathrm{nm}) .$ Hint: For the purpose of this problem, assume that the Doppler shift equation is valid at this speed. (FIGURE CAN'T COPY)

Key Concepts

Doppler Effect

The Doppler effect refers to the phenomenon where the observed frequency or wavelength of a wave changes when there is relative motion between the source and the observer. It applies to all types of waves, including sound and electromagnetic radiation, and is essential in explaining how movement can cause a shift in the color (wavelength) of light as perceived by an observer.

Relativistic Doppler Shift

When dealing with electromagnetic waves such as light, especially when the relative speeds approach a significant fraction of the speed of light, the relativistic Doppler shift formula must be used. This formula accounts for the effects of special relativity, particularly time dilation, and accurately predicts how the wavelength or frequency of light shifts due to high-speed motion.

Wavelength-Frequency Relationship

Wavelength and frequency are inversely related for all waves through the speed of propagation. For light, since the speed is constant (the speed of light), a decrease in wavelength corresponds to an increase in frequency, and vice versa. This relationship is crucial for understanding how a change in observed wavelength can be explained by changes in the relative motion of the source and observer.

Velocity Relative to the Speed of Light

The analysis of wavelength shifts using the Doppler effect involves comparing the speed of the object to the speed of light. Even small fractional changes relative to the speed of light can result in significant shifts in wavelength or frequency when using the relativistic Doppler formula. This concept highlights that for substantial color changes to occur via Doppler shifting, the relative speed must be an appreciable fraction of the speed of light.

Please give Ace some feedback