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The Doppler effect is a phenomenon where the observed frequency of a wave changes based on the relative motion between the source and the observer. This effect is familiar in sound waves, such as the changing pitch of a passing siren. However, in physics, we also consider the Doppler effect with light waves, notably when dealing with high velocities close to the speed of light. This is called the relativistic Doppler effect. Unlike the classical Doppler effect, the relativistic version considers the effects of special relativity, which introduces corrections for observers moving at significant fractions of the speed of light. This is crucial in understanding phenomena observed in astrophysics, such as the redshift of light from distant galaxies. The fundamental equation for the relativistic Doppler effect is: \[ \frac{u'}{u} = \frac{\text{sqrt}(1 + \beta)}{\text{sqrt}(1 - \beta)} \] Here, \( u' \) is the observed frequency, \( u \) is the source frequency, and \( \beta = \frac{v}{c} \) where \( v \) is the velocity of the observer relative to the source and \( c \) is the speed of light.
Understanding this equation shows how motion affects the wavelength and, consequently, the color of light observed.

The exercise involves a wavelength shift because of the Doppler effect. Here, the professor claims that the red light (650 nm wavelength) was shifted to green (550 nm wavelength) due to his motion.
This shift from one color to another is a direct result of changes in wavelength caused by movement. When an object moves toward the source of light, the wavelengths shorten (blue shift). Conversely, moving away lengthens the wavelengths (redshift).
In this scenario, the professor’s motion towards the red light caused the wavelength to shorten and shift to green. For light, this shift can be calculated using the wavelength form of the Doppler effect:
\[ \frac{\text{λ}'_{\text{o}}}{\text{λ}_{\text{s}}} = \text{sqrt} \bigg(\frac{1 + \beta}{1 - \beta}\bigg) \]
Where:

• \( \text{λ}'_{\text{o}} \) is the observed wavelength (550 nm)
• \( \text{λ}_{\text{s}} \) is the source wavelength (650 nm)
• \( \beta = \frac{v}{c} \) and needs to be solved for the speed \( v \)

Plugging in these values will illustrate the relationship between speed and observed wavelength change.

To find the professor's speed, we need to solve for \( \beta \) in the wavelength shift formula:
\[ \frac{550 \text{nm}}{650 \text{nm}} = \text{sqrt} \bigg(\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}\bigg) \]Firstly, simplify the fraction:
\[ \frac{11}{13} = \text{sqrt} \bigg(\frac{1 + \beta}{1 - \beta}\bigg) \] Square both sides to remove the square root:
\[ \bigg(\frac{11}{13}\bigg)^2 = \bigg(\frac{1 + \beta}{1 - \beta}\bigg) \] Solve for \( \beta \):
\[ \frac{121}{169} = \frac{1 + \beta}{1 - \beta} \] Cross-multiply to clear the fraction:
\[ 121 (1 - \beta) = 169 (1 + \beta) \]
\[ 121 - 121 \beta = 169 + 169 \beta \] Combine like terms:
\[ 121 - 169 = 169 \beta + 121 \beta \]\[ -48 = 290 \beta \]
Solve for \( \beta \):
\[ \beta = \frac{-48}{290} = -0.1655 \]
Since \( \beta = \frac{v}{c} \), find the speed \( v \):
\[ v = \beta c = 0.1655 \times 3 \times 10^8 \text{m/s} \]
\[ v \backsim 49,650,000 \text{m/s} \]
This high speed (approximately 16.55% the speed of light) confirms the professor's claim that the red light shifted to green due to his motion. This calculation showcases how the Doppler effect can quantify the impact of high-speed movement on observed wavelengths.