Sound waves travel at 300 m/s. Sound produced at a point is heard by a person after 5 seconds while the same sound is heard by another person after 6 seconds. What could be the maximum and minimum distances between the two persons?

1. Nature of Sound: Mechanical and Longitudinal Waves (basic)

To understand sound, we must first look at how it travels. Sound is a mechanical wave, which means it requires a material medium—such as air, water, or a solid—to propagate. Unlike light, which can travel through the vacuum of space, sound relies on the physical interaction of particles. When an object vibrates, it pushes against the surrounding molecules, creating a series of compressions (high-pressure regions where particles are crowded) and rarefactions (low-pressure regions where particles are spread apart) Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. This constant 'push and pull' is why sound is also termed a pressure wave.

Specifically, sound is a longitudinal wave. In this type of wave, the displacement of the medium's particles is parallel to the direction in which the wave travels. Imagine a Slinky being pushed and pulled lengthwise; the energy moves forward, and the coils vibrate back and forth in that same line. This is exactly how Primary waves (P-waves) work during an earthquake—they are longitudinal waves that compress and stretch the ground as they move Physical Geography by PMF IAS, Earths Interior, p.60. Because these waves apply force directly in the direction of propagation, they transmit energy very efficiently and are typically the fastest waves in a given medium.

The speed of sound is not constant; it depends heavily on the properties of the medium it is passing through. Generally, sound travels faster in media that are more elastic and dense. In a dense medium, particles are packed closer together, making it easier for the 'compression' signal to pass from one molecule to the next Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. This is why sound travels faster in steel than in water, and faster in water than in air.

Feature	Longitudinal Waves (Sound, P-waves)	Transverse Waves (Light, S-waves)
Particle Motion	Parallel to wave direction	Perpendicular to wave direction
Medium Requirement	Requires a medium (Mechanical)	Can travel in vacuum (Electromagnetic*)
Mechanism	Compression and Rarefaction	Crests and Troughs

*Note: While S-waves are transverse, they are mechanical and still require a medium, unlike light.

Sources: Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64; Physical Geography by PMF IAS, Earths Interior, p.60; Physical Geography by PMF IAS, Earths Interior, p.61

2. Characteristics of Sound: Frequency, Wavelength, and Speed (basic)

To understand sound, we must visualize it as a sequence of disturbances traveling through a medium. When an object vibrates, it creates regions of high pressure (compressions) and low pressure (rarefactions). The three pillars of any wave, including sound, are its wavelength, frequency, and speed.

Wavelength (λ) is the physical distance between two consecutive identical points in a wave, such as the distance from one compression to the next. In broader physical terms, it is the horizontal distance between two successive crests Physical Geography by PMF IAS, Tsunami, p.192. Frequency (f), on the other hand, represents the 'tempo' of the wave—it is the number of complete waves that pass a fixed point in one second, measured in Hertz (Hz) Physical Geography by PMF IAS, Tsunami, p.192. Crucially, wavelength and frequency share an inverse relationship: for a constant speed, as frequency increases, the wavelength must decrease Physical Geography by PMF IAS, Earths Atmosphere, p.279.

Speed (v) is the rate at which the sound energy travels through the medium. It is calculated by the product of frequency and wavelength (v = f × λ). While light speed is famously constant in a vacuum, the speed of sound is highly dependent on the medium it travels through (typically faster in solids and slowest in gases) and environmental factors like temperature. For practical calculations in many physics problems, we use the basic relationship of Distance = Speed × Time to determine how far a sound has traveled Science-Class VII, Measurement of Time and Motion, p.112.

Characteristic	Definition	Standard Unit
Wavelength (λ)	Distance between two successive peaks/compressions.	Metre (m)
Frequency (f)	Number of oscillations per second.	Hertz (Hz)
Speed (v)	The distance the wave travels per unit time.	Metres per second (m/s)

Sources: Physical Geography by PMF IAS, Tsunami, p.192; Physical Geography by PMF IAS, Earths Atmosphere, p.279; Science-Class VII NCERT, Measurement of Time and Motion, p.112

3. Factors Affecting the Speed of Sound (intermediate)

To understand how fast sound travels, we must look at the medium it moves through. Sound is a mechanical wave that relies on the interaction of particles. Two fundamental properties of a medium dictate this speed: Elasticity (how quickly particles return to their original position) and Density (how heavy or 'sluggish' the particles are). While we often think of solids as being too dense for movement, their particles are closely packed and held by strong interparticle interactions Science, Class VIII . NCERT(Revised ed 2025), Particulate Nature of Matter, p.113. This high elasticity or 'stiffness' allows sound to travel much faster in solids (like iron) than in liquids or gases.

Temperature plays a crucial role, especially in gases. As the temperature rises, particles gain kinetic energy and move more vigorously Science, Class VIII . NCERT(Revised ed 2025), Particulate Nature of Matter, p.115. This increased motion allows the sound vibrations to be passed from one molecule to the next much more rapidly. In air, for every 1°C increase in temperature, the speed of sound increases by approximately 0.6 m/s. This is why sound travels faster on a hot summer afternoon than on a cold winter night.

Finally, we must consider Humidity. It is a common misconception that humid air is 'heavier' or denser. In reality, water vapor (H₂O) molecules are lighter than the Nitrogen (N₂) and Oxygen (O₂) molecules they displace. As relative humidity increases, the overall density of the air decreases Physical Geography by PMF IAS, Hydrological Cycle, p.327. Since sound travels faster in less dense gases, it actually travels faster in humid air than in dry air. Interestingly, changes in Atmospheric Pressure do not affect the speed of sound, provided the temperature remains constant, because the change in pressure is perfectly balanced by a corresponding change in density.

Factor	Change	Effect on Speed of Sound
Temperature	Increase	Increases
Humidity	Increase	Increases (due to lower air density)
Medium Density	Increase (same phase)	Decreases
Pressure	Increase	No Change (at constant temperature)

Sources: Science, Class VIII. NCERT (Revised ed 2025), Particulate Nature of Matter, p.113, 115; Physical Geography by PMF IAS, Hydrological Cycle (Water Cycle), p.327-328

4. Acoustic Applications: Echo and SONAR (intermediate)

At its heart, an echo is simply the reflection of sound. Just as a mirror reflects light to create an image, a hard surface reflects sound waves back to our ears. However, our brains have a unique property called the persistence of hearing: we can only distinguish a reflected sound as a separate echo if it reaches us at least 0.1 seconds after the original sound. If it arrives sooner, the sounds blur together. To calculate the distance an echo has traveled, we use the fundamental relationship: Distance = Speed × Time Science-Class VII, NCERT(Revised ed 2025), Measurement of Time and Motion, p.115. Because the sound must travel to the obstacle and bounce back, the total distance covered is twice the actual distance (2d) from the source to the wall.
SONAR (Sound Navigation and Ranging) is the high-tech application of this simple principle. It uses ultrasonic waves (sounds with frequencies too high for humans to hear) to map the unseen. A transmitter on a ship sends out a pulse, which travels through the water, hits an object like the seabed or a submarine, and reflects back to a detector. By measuring the time interval between transmission and reception, and knowing the speed of sound in seawater, we can precisely calculate depth or distance. This is vital for navigating ocean currents and understanding the vertical motions of water bodies Physical Geography by PMF IAS, Tsunami, p.192.
When solving problems involving acoustics, remember that the relative position of the observers matters. If two people hear a sound from the same source at different times, their maximum separation occurs if the source is between them (distances add), while their minimum separation occurs if they are on the same side of the source (distances subtract).

Sources: Science-Class VII, NCERT(Revised ed 2025), Measurement of Time and Motion, p.115; Physical Geography by PMF IAS, Tsunami, p.192

5. Electromagnetic Waves vs Sound: Lightning and Thunder (basic)

When a thunderstorm strikes, we experience a fascinating natural race between two different types of waves: Electromagnetic waves (light) and Mechanical waves (sound). While both are triggered by the same physical event—a massive electrical discharge between clouds or the ground—they reach our senses at very different times due to their fundamental properties.

Lightning is the visual signal. It consists of light, which travels at a staggering speed of approximately 3 × 10⁸ m s⁻¹ in a vacuum and only marginally slower in air Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148. Because this speed is so high, the flash of lightning reaches your eyes almost instantaneously, regardless of whether the storm is 1 km or 10 km away. On the other hand, Thunder is the auditory signal. The intense heat from the lightning (reaching up to 30,000°C) causes the surrounding air to expand explosively, creating a shock wave that we hear as sound Environment and Ecology, Majid Hussain (3rd ed.), Natural Hazards and Disaster Management, p.52. Sound is a mechanical wave and travels through air at a much more "leisurely" pace—roughly 330 to 340 m/s (depending on temperature and humidity).

Feature	Lightning (Light Wave)	Thunder (Sound Wave)
Type	Electromagnetic (Transverse)	Mechanical (Longitudinal)
Medium	Does not require a medium	Requires a medium (Air)
Speed in Air	~300,000,000 m/s	~330-340 m/s

Because light travels about a million times faster than sound, there is a noticeable time lag between seeing the flash and hearing the boom Environment and Ecology, Majid Hussain (3rd ed.), Natural Hazards and Disaster Management, p.52. This lag allows us to calculate how far away the storm is using the simple formula: Distance = Speed × Time. Since the light arrives almost at time zero, we only need to multiply the speed of sound by the number of seconds we waited to hear the thunder.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148, 150; Environment and Ecology, Majid Hussain (3rd ed.), Natural Hazards and Disaster Management, p.52; Physical Geography by PMF IAS, Thunderstorm, p.349

6. Geometry of Displacement: Maximum and Minimum Separation (intermediate)

To understand the physical distance between two observers who hear the same sound at different times, we must first establish the relationship between speed, time, and distance. As defined in basic physics, speed is the distance covered by an object in a unit of time Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113. By rearranging this, we can determine the distance an object (or a sound wave) has traveled if we know its constant speed and the time elapsed: Distance = Speed × Time Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115. If two people hear a sound at different times, they are essentially at different radial distances from the source.

The geometry of displacement comes into play when we determine how far these two observers are from each other. Their relative positions are like coordinates on a map Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), Latitudes and Longitudes, p.250. There are two extreme geometric configurations to consider:

Scenario	Geometric Arrangement	Calculation Method
Maximum Separation	The two observers are on opposite sides of the source in a straight line.	Add the distances: D₁ + D₂
Minimum Separation	The two observers are on the same side of the source in a straight line.	Subtract the distances: |D₂ - D₁|

For any other arrangement (like an L-shape or an angle), the distance would fall somewhere between these two extremes. In a linear model, the maximum distance occurs when the source is the 'middle' point, pushing the observers as far apart as possible. Conversely, the minimum distance occurs when they are 'stacked' on the same path, leaving only the difference in their travel distances as the gap between them.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115; Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), Latitudes and Longitudes, p.250

7. Solving the Original PYQ (exam-level)

This question perfectly synthesizes the fundamental relationship between speed, distance, and time with the principles of spatial geometry. Having mastered the basic formula (Distance = Speed × Time), you can now see how UPSC elevates a simple calculation into a conceptual challenge. By calculating the individual distances—1500 meters for the first person and 1800 meters for the second—you establish the radii of two concentric circles centered on the sound source. The problem then becomes a test of your ability to visualize relative displacement in a physical space.

To arrive at the correct answer, we must consider the two extreme geometric scenarios. The maximum distance occurs when the two individuals are positioned diametrically opposite each other, with the sound source sitting directly in the middle; in this case, we simply add the distances (1.5 km + 1.8 km = 3.3 km). Conversely, the minimum distance is achieved when both individuals are standing on the same line originating from the source, meaning we find the difference between their positions (1.8 km - 1.5 km = 0.3 km). This logical deduction leads us firmly to Option (D), as explained in NCERT Class 9 Science: Sound.

UPSC often includes "distractor" options to penalize calculation haste or unit conversion errors. For instance, options like (A) or (B) might attract students who misplace a decimal point during the meter-to-kilometer conversion or those who try to apply the Pythagorean theorem unnecessarily, assuming a right-angled positioning. Remember, in "maximum/minimum" problems, the examiners are testing your linear reasoning—don't let complex-looking numbers distract you from the simple logic of alignment and opposition.