Suppose a rocketship is receding from the earth at a speed of 2/10th the velocity of light. A light in the rocketship appears blue to the passengers on the ship. What colour would it appear to an observer on the earth?

1. The Electromagnetic Spectrum and VIBGYOR (basic)

To understand the universe, we must first understand Electromagnetic (EM) Radiation—the primary way information travels across the cosmos. Light is not just what we see; it is a wave of energy that exists in a vast range of sizes called the Electromagnetic Spectrum. These waves are categorized by their wavelength (the distance between two peaks). A fundamental rule to remember is that wavelength and frequency are inversely proportional: waves with the longest wavelengths have the lowest energy, while those with the shortest wavelengths pack the most energy. At the 'long' end of the spectrum, we find Radio waves, which can be larger than our planet. As we move toward shorter wavelengths, we pass through Microwaves, Infrared, Visible Light, Ultraviolet, X-rays, and finally, Gamma rays. Interestingly, our atmosphere acts as a filter; for instance, the ionosphere reflects certain high-frequency radio waves back to Earth, allowing for long-distance communication Physical Geography by PMF IAS, Earths Atmosphere, p.279. However, it absorbs others, like microwaves, which cannot be used for ground-to-ground skywave propagation due to high energy losses Physical Geography by PMF IAS, Earths Atmosphere, p.278. Visible Light is the tiny slice of this spectrum that human eyes can detect. When white light (like sunlight) passes through a prism, it splits into a beautiful band of colors known as a spectrum Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.167. We remember this sequence using the acronym VIBGYOR:

• Violet (Shortest wavelength, Highest energy)
• Indigo
• Blue
• Green
• Yellow
• Orange
• Red (Longest wavelength, Lowest energy)

This difference in wavelength has practical consequences. For example, Red light has a wavelength roughly 1.8 times longer than Blue light. Because blue light has a shorter wavelength, it strikes gas molecules in our atmosphere and scatters more easily, which is why the sky appears blue to us Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.169.

Sources: Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.167, 169; Physical Geography by PMF IAS, Earths Atmosphere, p.278, 279

2. Core Wave Mechanics: Wavelength and Frequency (basic)

To understand the vast universe, we must first master the language of light. Light travels as a wave, and two fundamental properties define its character: wavelength (denoted by the Greek letter lambda, λ) and frequency (denoted by f or the Greek letter nu, ν). Wavelength is the physical distance between two consecutive peaks (crests) of a wave. In contrast, frequency is a measure of 'tempo'—it tells us how many wave cycles pass a fixed point in one second, measured in Hertz (Hz).

The relationship between these two is governed by a simple but profound rule: they are inversely proportional. Because the speed of light (c) in a vacuum is a universal constant (approximately 3 × 10⁸ m s⁻¹), if the wavelength increases, the frequency must decrease to maintain that constant speed. This is expressed by the formula: c = λ × f. As noted in Physical Geography by PMF IAS, Chapter 1, p.279, it is this specific range of wavelength and frequency that determines how waves interact with matter, such as radio waves reflecting off the ionosphere.

In the visible spectrum, we perceive these physical differences as colors. High-frequency waves have short wavelengths and appear toward the blue/violet end of the spectrum. Conversely, low-frequency waves have long wavelengths and appear toward the red end. In fact, red light has a wavelength approximately 1.8 times longer than blue light Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.169. Understanding this 'stretch and squeeze' of waves is the secret to unlocking how astronomers determine if a galaxy is moving toward or away from us.

Property	Long Wavelength (e.g., Red Light)	Short Wavelength (e.g., Blue Light)
Frequency	Lower	Higher
Energy	Lower	Higher
Example	Radio waves, Infrared	X-rays, Ultraviolet

Sources: Physical Geography by PMF IAS, Chapter 1: The Universe, p.279; Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.169; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148

3. The Big Bang Theory and Universal Expansion (intermediate)

The Big Bang Theory, also known as the Expanding Universe Hypothesis, is the leading scientific explanation for how our universe began. It suggests that around 13.8 billion years ago, the entire universe was concentrated in a single point of infinite density and heat called a singularity. This point began to expand rapidly, a process that continues even today. It is crucial to understand that the Big Bang was not an explosion in space, but rather the expansion of space itself. As time passes, galaxies move further and further apart, much like points on the surface of an inflating balloon FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Geography as a Discipline, p.13.

The primary evidence for this expansion comes from the Redshift phenomenon, first described by Edwin Hubble in the 1920s. When a light source moves away from an observer, the light waves are stretched, shifting them toward the longer, redder wavelengths of the electromagnetic spectrum. Conversely, if an object moves closer, the light is compressed (Blueshift). By observing that light from distant galaxies is almost universally redshifted, scientists concluded that these galaxies are receding from us Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.3. For instance, if a spaceship emits a blue light (at ~450 nm) while moving away at 20% the speed of light, an Earth observer would see that light shifted to a yellowish hue (at ~551 nm) due to this longitudinal Doppler effect.

To quantify this expansion, we use Hubble’s Law, which states that the velocity at which a galaxy moves away is directly proportional to its distance from us. This relationship is defined by the Hubble Constant, a critical unit of measurement used to estimate the universe's expansion rate and age Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.5. Beyond light, modern science uses Cosmic Microwave Background Radiation (CMBR)—the faint afterglow of the Big Bang—and Gravitational Waves as "cosmic sirens" to confirm that the universe is not just expanding, but doing so at an accelerating pace Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.3-5.

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Geography as a Discipline, p.13; Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.3; Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.5

4. Stellar Spectroscopy and Chemical Fingerprints (intermediate)

Since we cannot physically visit stars to take samples, we rely on Stellar Spectroscopy—the study of light broken down into its constituent colors—to understand their secrets. When you look at a star through a spectroscope, you don't just see a perfect rainbow. Instead, you see a Continuous Spectrum (a full rainbow) interrupted by thousands of dark, narrow lines. These are called Fraunhofer Lines, or absorption lines. They are the "chemical fingerprints" of the universe because every chemical element absorbs light at very specific, unique wavelengths.

How does this happen? Every star has a hot, dense core that produces a continuous spectrum of light. As this light travels outward, it passes through the star's cooler, less dense outer atmosphere. The atoms in this atmosphere—such as Hydrogen and Helium—absorb specific photons of light that match the energy required to move their electrons to higher energy levels Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.9. Because these energy jumps are precisely fixed for each element, the resulting dark gaps in the spectrum tell us exactly what the star is made of. For example, if we see a specific pattern of lines in the red and blue-green regions, we know for certain that Hydrogen is present, as stars primarily fuse Hydrogen into Helium in their cores Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.10.

Spectroscopy also acts as a celestial speedometer through the Doppler Effect. If a star is moving away from us, its spectral lines shift toward the red end of the spectrum (Redshift), meaning the wavelengths are stretched. Conversely, if a star moves toward us, the lines shift toward the blue end (Blueshift). This shift is critical for understanding the expansion of the universe and the movement of galaxies. Furthermore, a broad absorption range allows us to see how stars utilize their internal energy efficiently, much like how renewable energy technologies aim for a broad absorption range to utilize the solar spectrum Environment, Shankar IAS Academy (ed 10th), Renewable Energy, p.289.

Spectrum Type	Appearance	Physical Cause
Continuous	Unbroken rainbow	Hot, dense solids or gases (the star's core).
Absorption	Rainbow with dark lines	Cooler gas absorbing specific light (the star's atmosphere).
Emission	Bright lines on black	Hot, thin gas emitting light at specific frequencies.

Sources: Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.9; Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.10; Environment, Shankar IAS Academy (ed 10th), Renewable Energy, p.289

5. The Doppler Effect: Light vs. Sound (exam-level)

The Doppler Effect is the shift in the frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. While we experience this daily with sound—like the rising and falling pitch of a passing siren—its application in astronomy regarding light is what allows us to map the expansion of the universe. To understand this, we must first distinguish between the two types of waves. Sound is a mechanical wave that travels through the compression and rarefaction of a medium; consequently, its velocity actually increases as the density of the medium increases Physical Geography by PMF IAS, Earths Magnetic Field, p.64. Light, however, is an electromagnetic wave that can travel through a vacuum. Interestingly, as the density of a medium increases, light's velocity decreases due to a higher refractive index Physical Geography by PMF IAS, Earths Magnetic Field, p.64.

In the vacuum of space, the Doppler effect for light manifests as Redshift and Blueshift. When an object (like a galaxy or a star) moves away from an observer, the light waves are 'stretched,' leading to a longer wavelength (Redshift). Conversely, if the object moves closer, the waves are compressed into shorter wavelengths (Blueshift). Because light often behaves as both a wave and a particle—a concept reconciled by modern quantum theory—we can precisely measure these shifts to determine celestial velocities Science Class X (NCERT 2025), Light – Reflection and Refraction, p.134.

When dealing with extreme speeds in astrophysics, we must account for relativistic effects. For example, if a spacecraft emitting blue light (wavelength ~450 nm) moves away from Earth at 20% the speed of light (β = 0.2), the light doesn't just look slightly different; the wavelength increases by approximately 22.5%. This shifts the observed light to roughly 551 nm, meaning an observer on Earth would see the ship glowing yellow-green instead of blue. This phenomenon is a cornerstone of the Big Bang Theory, providing the primary evidence that galaxies are receding from us.

Feature	Sound Waves	Light Waves
Wave Type	Mechanical / Longitudinal Fundamentals of Physical Geography Class XI (NCERT 2025), p.20	Electromagnetic / Transverse Physical Geography by PMF IAS, p.64
Medium Effect	Velocity increases with density	Velocity decreases with density
Recession Result	Lower Pitch (Frequency)	Redshift (Longer Wavelength)

Sources: Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64; Science Class X (NCERT 2025), Light – Reflection and Refraction, p.134; Fundamentals of Physical Geography Class XI (NCERT 2025), The Origin and Evolution of the Earth, p.20

6. Calculating Redshift and Color Shifts (exam-level)

When we observe light from a celestial object, its color isn't just determined by its temperature, but also by its relative velocity to us. This phenomenon is the Doppler Effect applied to light. If a source is moving away from an observer, the light waves are "stretched," leading to an increase in wavelength and a shift toward the red end of the spectrum, known as Redshift Physical Geography by PMF IAS, Chapter 1, p.3. Conversely, if the source moves closer, the waves are compressed into a Blueshift.

To calculate these shifts at high velocities, we must account for Special Relativity. We define the recession speed as β = v/c (where v is the object's velocity and c is the speed of light). The ratio between the observed wavelength (λ_obs) and the emitted wavelength (λ_emit) is determined by the relativistic longitudinal Doppler factor:

λ_obs / λ_emit = √((1 + β) / (1 − β))

Consider a practical scenario: a spacecraft moving away from Earth at 20% of the speed of light (β = 0.2). Using our formula, the shift factor is √((1 + 0.2) / (1 − 0.2)) = √(1.2 / 0.8) = √1.5 ≈ 1.225. This means every wavelength is stretched by about 22.5%. If the spacecraft shines a blue light (~450 nm), the Earth observer calculates: 450 nm × 1.225 ≈ 551 nm. Consulting the visible spectrum, 551 nm falls within the yellow-green range. Thus, what was blue at the source appears yellow to us!

Direction of Motion	Wavelength Effect	Frequency Effect	Observed Color Change
Moving Away	Increases (Stretches)	Decreases	Shifted toward Red
Moving Toward	Decreases (Compresses)	Increases	Shifted toward Blue

It is important to remember that light speed remains constant in a vacuum regardless of the source's motion Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148; however, the energy of the light changes. Since energy is inversely proportional to wavelength, a redshifted photon (longer wavelength) has less energy than when it was originally emitted.

Sources: Physical Geography by PMF IAS, Chapter 1: The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.3; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148

7. Solving the Original PYQ (exam-level)

This question perfectly integrates your knowledge of the Relativistic Doppler Effect and the Visible Light Spectrum. In your learning path, you discovered that when a light source and an observer move away from each other, the observed wavelength "stretches," a phenomenon known as Redshift. As noted in Physical Geography by PMF IAS, this principle is the same one used to prove the expansion of the universe. Here, the rocket acts as a moving laboratory, requiring you to apply conceptual physics to a practical, astronomical scenario.

To arrive at the correct answer, you must track the shift along the VIBGYOR scale. Since the ship is receding (moving away), the light must shift from Blue toward the red end of the spectrum (longer wavelengths). A recession speed of 2/10th the velocity of light ($0.2c$) results in a wavelength increase of approximately 22.5%. If we take a standard blue wavelength of 450 nm and increase it by this factor, we land at roughly 551 nm. This wavelength moves past the green band and settles into the Yellow part of the spectrum. Therefore, (C) Yellow is the correct observation for the stationary Earth inhabitant.

UPSC often uses options like (B) Orange and (D) Yellow-orange as traps to see if you understand the magnitude of the shift. While these colors are in the correct "redward" direction, they would require the rocket to be moving at a significantly higher fraction of the speed of light than $0.2c$. Conversely, (A) Blue is the "stationary trap," which would only be correct if there were no relative motion between the ship and Earth. Success here depends on knowing that recession equals redshift and having a feel for the sequential order of colors in the visible spectrum.