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In physics, especially when dealing with light waves, calculating the observed wavelength is essential. The Doppler Effect is particularly important for understanding how wavelengths change based on the relative motion between a source and an observer. When a source moves away from an observer, the waves stretch, increasing the observed wavelength. This scenario is specifically described by the relativistic Doppler Effect.

The formula to calculate the observed wavelength (\( \lambda_o \)) when the source is moving away is:

• \( \lambda_o = \lambda_s \sqrt{\frac{1 + \beta}{1 - \beta}} \)

where:

• \( \lambda_s \) is the wavelength emitted by the source.
• \( \beta = \frac{v}{c} \), with \( v \) being the velocity of the source relative to the observer and \( c \) the speed of light.

This formula helps calculate the apparent wavelength detected by an observer on Earth when a spaceship moving away emits light.

The given problem uses this formula to determine how the emission from a spaceship changes as it moves at a significant fraction of light speed.

Light emission is a fascinating process, particularly when considering how different observers perceive light emission events. These events occur when an object, such as a spaceship, emits photons at a specific wavelength. In physics, wavelength is the spatial period of a wave—the distance over which the wave's shape repeats.

In a stationary frame, light emitted at 450 nm appears as blue-violet, a part of the visible spectrum signifying a high-energy photon. However, relativistic speeds can significantly alter the appearance of this light due to the Doppler Effect. As the spaceship moves away from the observer, the frequency of this emitted light decreases, reflecting longer wavelengths.

The change in wavelength, or shift, is due to light's constant speed in a vacuum, emphasizing the relativistic nature of light and showcasing the effect of velocity on wave phenomena.

The visible spectrum is a range of wavelengths associated with light that is visible to the human eye. This spectrum is part of the electromagnetic spectrum and includes wavelengths approximately from 380 nm to 750 nm. Each wavelength corresponds to a specific color, ranging from violet to red.

In the case of the spaceship problem, the emitted light at 450 nm falls within this spectrum, characterizing it as blue light initially. After applying the relativistic Doppler shift due to the spaceship's velocity, the observed wavelength becomes 551 nm.
This shift places the light in the green region of the visible spectrum.

The visible spectrum is crucial because it allows us to identify color changes due to shifts, helping us understand how motion impacts perceived light. This concept is not only important in academic exercises but also in real-world applications like astronomy and even radar technology.