A jet plane flies through air with a velocity of 2 Mach. While the velocity of sound is 332 m/s the air speed of the plane is :

1. Physics of Sound: Longitudinal Waves and Propagation (basic)

To understand the physics of sound, we must first look at how energy moves through matter. Sound is a mechanical wave, meaning it requires a medium (solid, liquid, or gas) to travel. It begins with a vibration—like a bell ringing or a string plucking. When a material like metal vibrates, it produces a clear, ringing sound due to a property called sonority Science-Class VII . NCERT(Revised ed 2025), The World of Metals and Non-metals, p.46. This vibration pushes against the surrounding air molecules, starting a chain reaction of energy transfer.

Sound travels as a longitudinal wave (also known as a compression wave). In this type of wave, the particles of the medium vibrate back and forth in a direction parallel to the direction in which the wave travels. This is identical to the behavior of Primary waves (P-waves) in seismology, which are the fastest seismic waves because they transmit energy by pushing and pulling the medium in the same direction the wave moves Physical Geography by PMF IAS, Earths Interior, p.61. As the wave passes, it creates regions of high density called compressions and regions of low density called rarefactions.

Feature	Longitudinal Waves (Sound/P-waves)	Transverse Waves (Light/S-waves)
Particle Motion	Parallel to wave direction	Perpendicular to wave direction
Medium Type	Solids, Liquids, and Gases	Primarily Solids (for mechanical types)
Key Characteristic	Compressions and Rarefactions	Crests and Troughs

Because sound relies on particle-to-particle interaction, its speed depends heavily on the density and elasticity of the medium. It travels fastest in solids, where atoms are packed tightly, and slowest in gases like air. This distinguishes it from transverse waves, such as S-waves, which distort the medium by moving particles up and down or side-to-side, making them generally slower and unable to pass through liquids Physical Geography by PMF IAS, Earths Interior, p.62.

Sources: Science-Class VII . NCERT(Revised ed 2025), The World of Metals and Non-metals, p.46; Physical Geography by PMF IAS, Earths Interior, p.61; Physical Geography by PMF IAS, Earths Interior, p.62

2. Speed of Sound: Role of Medium, Temperature, and Humidity (intermediate)

To understand how sound moves, we must first view it as a mechanical wave. Unlike light, sound cannot travel through a vacuum; it requires a medium because it propagates through the compression and rarefaction of particles Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. The efficiency of this movement depends entirely on how those particles are packed and how they interact. In solids, particles are held by strong interparticle forces and are very close together, allowing the vibration to pass almost instantly Science, Class VIII . NCERT, Particulate Nature of Matter, p.113. In gases, particles are far apart with negligible attraction, making the energy transfer much slower. Thus, the general rule of thumb is: Speed in Solids > Liquids > Gases.

Environmental factors like temperature play a massive role in this speed. When we increase the temperature of a medium, we are essentially giving its particles more kinetic energy. At higher temperatures, particles move and vibrate more vigorously Science, Class VIII . NCERT, Particulate Nature of Matter, p.115. This heightened activity allows the sound wave to be handed off from one particle to the next much faster. For instance, in air, the speed of sound increases by roughly 0.6 meters per second for every degree Celsius rise in temperature.

Finally, let's look at humidity, which is a frequent point of confusion in competitive exams. You might assume that humid air is 'heavier' and would slow sound down, but the chemistry tells a different story. Water vapor (H₂O) is actually lighter (less dense) than the Nitrogen (N₂) and Oxygen (O₂) molecules it replaces in the atmosphere. Because sound travels faster through less dense gases, sound travels faster in humid air than in dry air.

Summary of Factors:

Factor	Change	Effect on Speed of Sound
Medium Density	Solid vs. Gas	Faster in high-elasticity solids
Temperature	Increase (↑)	Increases (↑)
Humidity	Increase (↑)	Increases (↑)

Sources: Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64; Science, Class VIII . NCERT, Particulate Nature of Matter, p.113; Science, Class VIII . NCERT, Particulate Nature of Matter, p.115

3. Wave Phenomena: The Doppler Effect (intermediate)

The Doppler Effect is a fundamental wave phenomenon characterized by a change in the perceived frequency of a wave when there is relative motion between the source of the wave and the observer. Imagine standing on a sidewalk as an ambulance approaches with its siren blaring. As it moves toward you, the pitch sounds higher; as it passes and moves away, the pitch drops significantly. It is important to understand that the source is emitting the same frequency the entire time; the shift occurs because the motion of the source causes the wave fronts to "bunch up" in front of it and "spread out" behind it.

Since sound is a mechanical wave that travels through the compression and rarefaction of a medium, its speed is influenced by the density and elasticity of that medium Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. When the source of the sound moves at speeds approaching the velocity of sound, we enter the realm of transonic and supersonic physics. A critical tool for measuring this is the Mach Number, which is the ratio of the speed of an object to the local speed of sound in that medium:

When an object reaches Mach 1, it is traveling at the speed of sound. If it exceeds this (supersonic), it outpaces its own sound waves, creating a conical shock wave known as a sonic boom. In practical applications, the India Meteorological Department utilizes Doppler-Radars in the Himalayas to monitor cloudbursts Geography of India by Majid Husain, Contemporary Issues, p.35. These radars function by bouncing radio waves off moving water droplets; by measuring the frequency shift of the returning signal, meteorologists can determine the velocity and direction of storms with high precision.

Scenario	Observed Frequency	Reasoning
Source moving toward observer	Higher (High Pitch)	Wave fronts are compressed (shorter wavelength).
Source moving away from observer	Lower (Low Pitch)	Wave fronts are elongated (longer wavelength).
Source at Mach 1+	Shock Wave	Object travels faster than the waves it produces.

Sources: Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64; Geography of India by Majid Husain, Contemporary Issues, p.35

4. Defense Technology: Classification of Missiles by Speed (exam-level)

To understand how we classify missiles, we first need to understand the yardstick we use: the Speed of Sound. In the world of acoustics and defense technology, we measure an object's speed relative to sound using the Mach Number. This is a dimensionless ratio where Mach 1 represents the local speed of sound (approximately 343 m/s in dry air at 20°C). As noted in the study of atmospheric phenomena, when an object exceeds this speed, it moves at a supersonic rate, creating intense pressure changes known as shock waves Physical Geography by PMF IAS, Thunderstorm, p.349.

Missiles are generally categorized into four speed regimes based on their Mach number. This classification is vital for defense strategy because higher speeds require more advanced propulsion systems and materials capable of withstanding extreme heat caused by air friction. The push for modernization in India's defense sector, which began gaining significant momentum in the 1990s with the induction of missiles like the Prithvi, has consistently aimed at moving from subsonic to high-supersonic and hypersonic capabilities A Brief History of Modern India (2019 ed.), After Nehru..., p.745.

The standard classification is as follows:

Category	Speed Range	Characteristics
Subsonic	< Mach 0.8	Slower than the speed of sound. These missiles (like the Nirbhay) are often fuel-efficient and can fly at low altitudes to avoid radar.
Transonic	Mach 0.8 to 1.2	The critical range where the missile transitions through the sound barrier, experiencing significant aerodynamic stress.
Supersonic	Mach 1.2 to 5.0	Faster than sound. India’s BrahMos is a prime example, traveling at roughly Mach 2.8 to 3.0.
Hypersonic	> Mach 5.0	Extremely fast (over 1.7 km per second). These are incredibly difficult to intercept due to their speed and maneuverability.

Sources: Physical Geography by PMF IAS, Thunderstorm, p.349; A Brief History of Modern India (2019 ed.), After Nehru..., p.745

5. Understanding Mach Number and Sonic Booms (intermediate)

The Mach Number is a dimensionless quantity representing the ratio of the speed of an object to the local speed of sound in the surrounding medium. Mathematically, it is expressed as:

Mach Number (M) = Speed of Object / Speed of Sound

Because the speed of sound varies depending on the medium and temperature (e.g., sound travels faster in warmer air or denser water), the Mach number is a 'relative' measure. For instance, if a jet is flying at Mach 2, it is traveling at exactly twice the speed of sound. If the local speed of sound is 332 m/s, the jet's airspeed would be 664 m/s. We categorize these speeds into regimes: Subsonic (M < 1), Transonic (M ≈ 1), Supersonic (M > 1), and Hypersonic (M > 5).

When an object travels at supersonic speeds, a fascinating phenomenon called a Sonic Boom occurs. Normally, sound waves from a moving object spread out in all directions. However, as the object reaches the speed of sound, these waves cannot 'get out of the way' of the object; they pile up at the nose, forming a high-pressure shock wave. This is similar to how a thunderstorm generates thunder: the intense heat of lightning causes a plasma channel to expand at a supersonic rate, creating a shock wave that eventually decays into the sound we hear Physical Geography by PMF IAS, Thunderstorm, p.349.

To a listener on the ground, the passing of this compressed shock wave cone is heard as a thunder-like 'boom.' While we use instruments like anemometers to measure standard wind speeds in km/h for weather monitoring Exploring Society: India and Beyond, NCERT Class VII, Understanding the Weather, p.37, Mach numbers are the essential metric for high-speed aviation and ballistics where the compressibility of air becomes a critical factor.

Regime	Mach Range	Physical Characteristic
Subsonic	M < 0.8	Smooth airflow; sound waves move faster than the object.
Supersonic	M > 1.2	Shock waves form; object outpaces its own sound.
Hypersonic	M > 5.0	Extreme heat and chemical changes in the air (plasma).

Sources: Physical Geography by PMF IAS, Thunderstorm, p.349; Exploring Society: India and Beyond, NCERT Class VII, Understanding the Weather, p.37

6. Calculating Airspeed using the Mach Formula (intermediate)

In physics and aviation, we often measure the speed of high-velocity objects, such as fighter jets or rockets, relative to the local speed of sound. This relative measure is known as the Mach number. While we typically define speed as the total distance covered divided by the total time taken (Science-Class VII, Measurement of Time and Motion, p.113), the Mach number is a dimensionless ratio. It compares the true airspeed of an object to the speed of sound in the medium through which it is traveling. Because the speed of sound varies with temperature and atmospheric conditions, the Mach number provides a crucial reference for how the air will behave around the moving object.

The calculation of airspeed using the Mach formula is straightforward. It is expressed as:

Airspeed = Mach Number × Local Speed of Sound

When an object's Mach number is greater than 1, it is traveling at supersonic speeds. At these velocities, the object moves faster than the pressure waves it creates, leading to the formation of shock waves (Physical Geography by PMF IAS, Thunderstorm, p.349). For example, if a jet is flying at Mach 2, it is traveling at exactly twice the speed of sound. If the speed of sound in the surrounding air is 332 m/s, the plane's airspeed is 664 m/s (2 × 332 m/s). This tells us the aircraft has comfortably breached the sound barrier, which occurs at Mach 1.

Sources: Science-Class VII, Measurement of Time and Motion, p.113; Physical Geography by PMF IAS, Thunderstorm, p.349

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental properties of sound waves and their propagation, this question allows you to apply the concept of the Mach number in a practical scenario. As we discussed in our study of NCERT Class 9 Science, the Mach number is a dimensionless ratio that compares the speed of an object to the local speed of sound in the same medium. This specific problem requires you to synthesize your understanding of ratios with the classification of speeds—moving from subsonic to supersonic thresholds.

To arrive at the correct answer, simply rearrange the core formula: Speed of Object = Mach Number × Speed of Sound. By substituting the given values, you multiply the Mach number of 2 by the speed of sound, 332 m/s, resulting in 664 m/s. This indicates the plane is traveling at exactly twice the speed of sound. In the context of UPSC Prelims, always ensure you are performing a simple multiplication rather than getting distracted by the complexity of the term 'Mach'.

Understanding why the other options are incorrect is crucial for avoiding UPSC 'traps.' Option (A) 166 m/s is a division trap, where a candidate might mistakenly divide the speed of sound by the Mach number. Option (C) 332 m/s is simply the speed of sound itself (Mach 1), failing to account for the plane's actual velocity. Most importantly, option (B) 66.4 km/s is a unit conversion trap; while the digits look familiar, the unit 'km/s' makes the value 100 times larger than the correct airspeed. Therefore, (D) 664 m/s is the only logically and mathematically sound choice.