You are given a traffic ticket for going through a red light (wavelength \(700 \mathrm{~nm}\) ). You tell the police officer that because you were approaching the light, the Doppler effect caused a blueshift that made the light appear green (wavelength \(500 \mathrm{~nm}\) ). How fast would you have had to be going for this to be true? Would the speeding ticket be justified? Explain.

The phenomenon of wavelength shift is central to our understanding of the Doppler Effect in the context of light. This effect occurs when there is relative motion between a light source and an observer. If the source and the observer are moving closer together, the observed wavelengths of light become shorter; if they move apart, the wavelengths become longer. This change in wavelength directly affects the color of the light perceived.

For instance, let's consider a police car's red lights, which normally have a wavelength of around 700 nm. If you're swiftly moving towards the police car, the actual wavelength could shift downwards due to this effect. This hypothetical scenario leads to the question: how fast must you drive for a red light to appear green, which has a shorter wavelength around 500 nm? The exercise at hand explores this concept by calculating the necessary velocity invoking this wavelength shift.

Blueshift is a specific type of wavelength shift that occurs when an object emitting light is moving towards an observer or when an observer is moving toward the light source. In this situation, the waves of light are compressed, and the observed wavelengths become shorter. The term 'blueshift' is derived from the way the color of the light shifts towards the blue end of the spectrum, which is characterized by shorter wavelengths.

In the given exercise, you're asked to calculate the speed at which you would need to approach a traffic light for the red light, which typically has a longer wavelength, to be blueshifted into the green spectrum. Contrary to redshift, where the wavelengths lengthen and shift toward the red end of the spectrum, blueshift can theoretically turn a red light green if one travels fast enough, an intriguing, if not practically improbable scenario.

The Doppler formula for light provides a mathematical relationship to quantify the effect of relative motion on the perceived wavelength of light. The formula is expressed as:
\[ \lambda' = \lambda(1 - \frac{v}{c}) \]
where \( \lambda' \) is the perceived wavelength, \( \lambda \) is the wavelength of the light at rest, \( v \) is the velocity of the observer relative to the light source, and \( c \) is the speed of light in a vacuum.

In practical terms, if you are approaching a traffic light at a significant fraction of the speed of light, this formula allows us to calculate the blueshift that would occur. However, given the enormous magnitude of the speed of light, any speed a car is capable of reaching under normal circumstances is minuscule in comparison, and thus the effect on wavelength shift is negligible. The exercise asks us to suspend disbelief and determine the hypothetical speed that would exhibit a noticeable change.

Perceived frequency, which is the number of wave crests passing a point per unit time, is another way of expressing changes due to the Doppler Effect. As the wavelength of light shortens (blueshift), the frequency increases; conversely, as the wavelength lengthens (redshift), the frequency decreases. So, when an object approaches a light source or vice versa, the frequency of light that the observer detects will be higher than if the object was stationary.

In our traffic ticket scenario, the officer would not be measuring frequency directly but instead relying on the color of the light, which corresponds to a specific frequency range. While you might argue about the perceived color due to the Doppler Effect, the velocity required to actually shift a 700 nm wavelength (red) to a 500 nm wavelength (green) would be significantly beyond the realm of road vehicles – which implies, even with a generous engagement in theoretical physics, the speeding ticket would indeed be justified.