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The Doppler shift formula is an essential tool when it comes to understanding how the observed frequency or wavelength of a wave changes relative to the motion of the source or observer. In the realm of light, this formula is given by:
\[\begin{equation} v = c \times \bigg(\frac{\lambda_{\text{o}}}{\lambda_{\text{s}}} - 1\bigg)\end{equation}\]
where:

• \(v\) represents the velocity of the observer relative to the source (it will be negative if approaching the source),
• \(c\) stands for the speed of light in a vacuum (3.00 \times 10^8 m/s),
• \(\lambda_{\text{o}}\) is the wavelength of the light as observed, and
• \(\lambda_{\text{s}}\) is the wavelength of the light as emitted by the source.

This formula allows us to calculate the speed at which an observer must move to perceive a change in color of light, which is due to the shift in the frequency of the observed light. In practical terms, if approaching a red traffic light quickly enough, it could appear yellow, much like in the exercise provided.

While the classical Doppler shift formula is suitable for everyday speeds, extreme velocities close to the speed of light require the relativistic Doppler effect to be considered. This comes into play because, according to Einstein's theory of relativity, as one's velocity increases towards the speed of light, time dilation and length contraction occur. These relativistic effects alter the perceived wavelength and frequency of light.
In such cases, the formula is modified to accommodate these effects:
\[\begin{equation} \lambda_{\text{o}} = \lambda_{\text{s}} \sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}\end{equation}\]
Although not necessary for everyday conditions, such as the traffic light scenario, it's crucial for astronomical observations and GPS satellite calculations where relativistic speeds are at play.

The speed of light, denoted by \(c\), is a fundamental constant in physics, clocking in at approximately 3.00 \times 10^8 meters per second (m/s) in a vacuum. It represents the maximum speed at which all conventional matter, energy, and information in the universe can travel. In terms of the Doppler effect, \(c\) is a critical component of the formulas used to calculate changes in wavelengths as a consequence of relative motion. The constancy of the speed of light is the cornerstone of the theory of relativity, which reconciles mechanics with electromagnetism.

Wavelength and frequency are inversely related in the context of a wave's properties, where the wavelength is the distance between two consecutive peaks of a wave, and frequency is the number of waves that pass a point per unit of time. This relationship is expressed by the equation:
\[\begin{equation} c = \lambda \times f\end{equation}\]
where \(\lambda\) is the wavelength, \(f\) is the frequency, and \(c\) remains the speed of light. When applying this to light, if the source or the observer moves, this relationship illustrates why the observed frequency and wavelength change, leading to the Doppler shift we observe in phenomena like a changing traffic light color or the redshift observed in distant galaxies.