Three identical vessels A, B and C are filled with water, mercury and kerosene respectively up to an equal height. The three vessels are provided with identical taps at the bottom of the vessels. If the three taps are opened simultaneously, then which vessel is emptied first?

1. Fundamentals of Fluid Statics: Density and Pressure (basic)

Welcome to your first step in mastering fluid mechanics! To understand how fluids (liquids and gases) behave, we must start with two foundational pillars: Density and Pressure. These concepts explain why a heavy iron ship floats while a small pebble sinks, and why your ears might pop when you fly in an airplane.

Density is a measure of how much "stuff" (mass) is packed into a specific amount of space (volume). Mathematically, it is expressed as Density = Mass / Volume (Science, Class VIII . NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.140). While a substance's density is independent of its size or shape, it can change with temperature and pressure. For instance, if you heat a gas, its particles move apart, increasing the volume and thus decreasing the density. However, liquids are nearly incompressible, meaning their density changes very little even under high pressure (Science, Class VIII . NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.148). To simplify comparisons between substances, we often use Relative Density, which is the ratio of a substance's density to the density of water at the same temperature (Science, Class VIII . NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.141).

Pressure, on the other hand, is defined as the force acting per unit area (P = F/A). In the International System of Units (SI), it is measured in Pascals (Pa) or Newtons per square metre (N/m²) (Science, Class VIII . NCERT, Pressure, Winds, Storms, and Cyclones, p.94). A key characteristic of fluids is that they exert pressure in all directions—against the walls of their container and on any object submerged within them. In fluid statics, the pressure at the bottom of a container depends directly on the density of the fluid and the height of the liquid column above it. This is why a column of mercury (which is very dense) exerts much more pressure at the bottom than a column of water of the same height.

Here is a quick look at how pressure affects the density of different states of matter:

State of Matter	Effect of Increasing Pressure on Density	Reasoning
Solids	Negligible	Particles are already very close together.
Liquids	Small/Minor	Liquids are considered nearly incompressible.
Gases	Significant Increase	Particles move closer together, decreasing volume (Science, Class VIII . NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.148).

Sources: Science, Class VIII . NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.140, 141, 148; Science, Class VIII . NCERT, Pressure, Winds, Storms, and Cyclones, p.94

2. Hydrostatic Paradox and Pressure at Depth (basic)

To understand why fluids behave the way they do, we must start with the fundamental principle of hydrostatic pressure. Imagine a column of liquid; the pressure at the bottom is simply the result of the weight of the liquid pressing down from above. This leads us to a crucial formula: P = hρg, where h is the depth, ρ (rho) is the density of the liquid, and g is the acceleration due to gravity. As the height of the water column increases, the pressure at the bottom also increases Science, Class VIII NCERT, Pressure, Winds, Storms, and Cyclones, p.84. This is exactly why overhead water tanks are placed on roofs—to use height to create enough pressure for a strong stream in your taps.

The Hydrostatic Paradox is the surprising observation that the pressure at a certain depth in a liquid is completely independent of the shape or total volume of the container. Whether you have a wide vat or a thin tube, if the liquid reaches the same vertical height, the pressure at the base is identical. However, we must distinguish between pressure (force per unit area) and the total force exerted on the bottom. If two vessels have the same liquid height but different base areas, the pressure (P) at the bottom is the same, but the total force (F) will be greater in the vessel with the larger base area because Force = Pressure × Area Science, Class VIII NCERT, Pressure, Winds, Storms, and Cyclones, p.95.

When we compare different liquids, like water and mercury, density (ρ) becomes the deciding factor. Even if the height (h) is the same, a denser liquid like mercury exerts significantly more pressure because it has more mass packed into the same volume. In dynamic situations, such as liquid flowing out of a hole (efflux), this higher density translates to greater hydrostatic pressure at the opening and higher momentum. While Torricelli’s Law (v = √2gh) suggests the theoretical exit velocity depends only on height, the higher density and inertia of a fluid like mercury often allow it to overcome resistive forces more effectively, influencing how quickly a vessel empties.

Sources: Science, Class VIII NCERT, Pressure, Winds, Storms, and Cyclones, p.84; Science, Class VIII NCERT, Pressure, Winds, Storms, and Cyclones, p.95

3. Viscosity: Internal Friction in Fluids (intermediate)

When we think of friction, we usually imagine two solid surfaces rubbing against each other. However, fluids (liquids and gases) also experience a form of friction internally. This property is known as viscosity. While we know that particles in a liquid are free to move past one another (Science, Class VIII NCERT, Particulate Nature of Matter, p.113), they don't move with absolute freedom. As one layer of fluid slides over another, the interparticle forces of attraction create a resistance to that motion. You can think of viscosity as the "thickness" or "stickiness" of a fluid.

To visualize this, imagine a fluid as a series of horizontal layers. When the top layer is pushed, it tries to drag the layer below it due to cohesive forces. The measure of this resistance to flow is what we define as viscosity. For example, honey has a much higher viscosity than water because its particles exert stronger internal resistance against moving past each other. This is a fundamental aspect of how liquids behave when they take the shape of their container (Science, Class VIII NCERT, Particulate Nature of Matter, p.104); their ability to flow into that shape is governed by how "viscous" they are.

The behavior of viscosity changes significantly with temperature. In liquids, as the temperature rises, the particles gain kinetic energy and move more vigorously. This movement weakens the interparticle forces of attraction (Science, Class VIII NCERT, Particulate Nature of Matter, p.105), making it easier for the layers to slide past each other. Consequently, the viscosity of a liquid decreases as temperature increases. This is why cold engine oil is thick and sluggish, but becomes thin and flows easily once the car engine warms up.

Feature	Low Viscosity (e.g., Water)	High Viscosity (e.g., Molasses)
Flow Rate	Fast and easy	Slow and resistant
Internal Friction	Low	High
Interparticle Drag	Weak	Strong

Sources: Science, Class VIII NCERT, Particulate Nature of Matter, p.113; Science, Class VIII NCERT, Particulate Nature of Matter, p.104; Science, Class VIII NCERT, Particulate Nature of Matter, p.105

4. Surface Tension and Capillary Action (intermediate)

At the heart of fluid mechanics lies the concept of Surface Tension, a property where the surface of a liquid acts like a stretched elastic membrane. This occurs because of the particulate nature of matter: molecules inside a liquid are pulled equally in all directions by neighboring molecules, but those at the surface have no neighbors above them. This creates a net inward force, making the liquid contract to the smallest possible surface area—which is why raindrops are spherical. Science Class VIII, Particulate Nature of Matter, p.111. This tension is so strong that it can support the weight of small insects or allow a needle to float if placed carefully.

Capillary Action is the logical extension of these molecular forces when a liquid meets a solid surface. It is governed by the tug-of-war between two forces: Cohesion (attraction between like molecules, like water-to-water) and Adhesion (attraction between unlike molecules, like water-to-glass). When adhesive forces are stronger than cohesive forces, the liquid 'climbs' the walls of a narrow tube. This is how plants transport water from roots to leaves and how paper towels soak up spills. Conversely, if cohesion is stronger—as is often the case with mercury—the liquid will actually depress or move downward in a narrow tube, forming a convex meniscus.

The behavior of fluids is also influenced by their density and viscosity. For instance, while water is a common solvent used in traditional Indian medicine, other substances like oils, ghee, and milk are used because their different surface tensions and molecular structures help dissolve specific medicinal extracts. Science Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.138. Similarly, the unique properties of mercury, such as its high density and high surface tension, make it behave very differently from water in both industrial and natural settings. Environment Shankar IAS, Environment Issues and Health Effects, p.413.

Sources: Science Class VIII (Revised ed 2025), Particulate Nature of Matter, p.111; Science Class VIII (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.138; Environment Shankar IAS (10th ed), Environment Issues and Health Effects, p.413

5. Fluid Dynamics: Bernoulli’s Principle (intermediate)

At its heart, Bernoulli’s Principle is an expression of the Law of Conservation of Energy for flowing fluids. It tells us that in a steady flow, the sum of all forms of energy in a fluid along a streamline is the same at all points. These energy forms typically include static pressure, kinetic energy (due to motion), and potential energy (due to height). When a fluid moves faster, its kinetic energy increases; to keep the total energy constant, its static pressure or potential energy must decrease. As noted in meteorological contexts, within a horizontal flow of fluid, points of higher fluid speed will have less pressure than points of slower fluid speed Physical Geography by PMF IAS, Tropical Cyclones, p.358. To visualize this, imagine water flowing through a horizontal pipe that narrows in the middle. Because water is practically incompressible Science, Class VIII, Particulate Nature of Matter, p.107, the same volume of water must pass through the narrow section as the wide section in the same amount of time. This forces the water to speed up in the narrow part. According to Bernoulli, because the speed (kinetic energy) went up, the internal pressure in that narrow section must drop. This is why a fast-moving wind over a roof can create enough of a pressure drop to lift the roof off a house during a storm. The mathematical representation of this principle is:
P + ½ρv² + ρgh = constant
Where P is the static pressure, ρ (rho) is the fluid density, v is the velocity, and h is the elevation. This equation explains everything from how heavy airplanes stay aloft (the air moves faster over the curved top of the wing, creating lower pressure) to Torricelli’s Law, which describes how fast a liquid leaks out of a hole in a tank. In the case of a leaking tank, the potential energy of the water at the top is converted into the kinetic energy of the water shooting out the bottom.

Sources: Physical Geography by PMF IAS, Tropical Cyclones, p.358; Science, Class VIII, Particulate Nature of Matter, p.107

6. Reynolds Number and Flow Regimes (exam-level)

When we observe fluids in motion—whether it is water in a pipe or air circulating around a low-pressure center—the flow behavior can change drastically based on a single dimensionless value: the Reynolds Number (Re). This number acts as a scales-tipper, helping us predict whether a fluid will flow in a smooth, orderly fashion or become chaotic and swirling. Scientifically, the Reynolds Number represents the ratio of inertial forces (the fluid's tendency to keep moving due to its momentum) to viscous forces (the internal 'stickiness' or friction of the fluid).

The formula for Reynolds Number is Re = (ρ × v × L) / μ, where ρ (rho) is the density, v is the flow velocity, L is a characteristic length (like the diameter of a tube), and μ (mu) is the dynamic viscosity. In simple terms, if a fluid is very dense or moving very fast, its inertia dominates, resulting in a high Reynolds number. Conversely, if a fluid is very 'thick' or viscous (like honey), the viscous forces dominate, leading to a low Reynolds number. This balance is critical because it determines the flow regime: Laminar flow (low Re) is characterized by smooth, parallel layers, while Turbulent flow (high Re) is marked by eddies and fluctuations. This relates to Bernoulli's principle, which notes that within a horizontal flow, higher speed correlates with lower pressure Physical Geography by PMF IAS, Tropical Cyclones, p.358.

In practical scenarios, such as liquid discharging from a vessel, the Reynolds number explains why denser fluids often behave differently than lighter ones. A fluid with high density, like mercury, possesses significant inertia. Even if its theoretical velocity (based on depth) is the same as water, its high Reynolds number means that the relative impact of viscous drag is reduced compared to its massive momentum. This allows the fluid to overcome resistance more effectively. In meteorology, similar mechanics of flow and pressure are seen when centripetal acceleration creates circular patterns (vortices) around pressure centers, defining the movement of cyclones and anticyclones Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309.

Sources: Physical Geography by PMF IAS, Tropical Cyclones, p.358; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309

7. Torricelli’s Law and Speed of Efflux (exam-level)

When you puncture a hole in a container filled with liquid, the liquid gushes out with a certain velocity. This phenomenon is known as efflux, and the speed at which it exits is governed by Torricelli’s Law. To understand this from first principles, imagine a small droplet of liquid at the surface. As it falls through a height h to reach the opening (orifice), its potential energy is converted into kinetic energy. According to the principle of conservation of energy, the speed of efflux (v) is expressed by the formula:

v = √(2gh)

where g is the acceleration due to gravity and h is the depth of the orifice below the liquid surface. Remarkably, this formula is identical to the final velocity of a solid object falling freely from the same height. Just as the speed of a falling stone doesn't depend on its mass, Torricelli's Law suggests that the theoretical speed of efflux is independent of the liquid's density. Whether it is water or a much denser liquid, the initial speed depends solely on the depth of the liquid column. This is a fundamental concept in fluid mechanics, similar to how we calculate average speed as the ratio of total distance to total time Science, Class VII, Measurement of Time and Motion, p.119.

However, in the real world, the rate of discharge (how fast the vessel actually empties) is influenced by the physical properties of the fluid, such as density and viscosity. While the speed formula v = √(2gh) seems density-neutral, a denser fluid exerts greater hydrostatic pressure at the same depth. Furthermore, the Reynolds number—a dimensionless value that represents the ratio of inertial forces to viscous forces—tends to be higher for denser fluids. A higher Reynolds number means that the fluid's inertia dominates over the viscous drag (friction). Just as seismic wave velocities change based on the density and composition of the medium through which they travel Physical Geography by PMF IAS, Earths Interior, p.63, the behavior of a discharging fluid is a result of the interplay between its weight, its internal friction, and its momentum.

Factor	Impact on Efflux
Liquid Depth (h)	Primary driver; higher depth equals higher velocity (v ∝ √h).
Gravity (g)	Constant on Earth; higher gravity increases efflux speed.
Density (ρ)	Theoretically no effect on speed, but increases hydrostatic pressure and reduces the relative impact of drag.
Viscosity	Acts as internal friction; higher viscosity slows down the actual flow rate.

Sources: Science, Class VII, Measurement of Time and Motion, p.119; Physical Geography by PMF IAS, Earths Interior, p.63

8. Solving the Original PYQ (exam-level)

This question masterfully connects Torricelli’s Law with the practical nuances of fluid dynamics. While you have learned that the theoretical velocity of efflux ($v = \sqrt{2gh}$) depends only on the height of the liquid, this problem requires you to apply the concepts of density, inertia, and the Reynolds number. In a real-world scenario, the time taken to empty a vessel isn't just about initial velocity; it is about how the fluid's mass and momentum overcome the resistance at the tap.

To arrive at the correct answer, consider the unique properties of mercury in Vessel B. Because mercury is significantly denser than water or kerosene, it exerts much higher hydrostatic pressure at the bottom. Furthermore, mercury's high density leads to a higher Reynolds number, which implies that inertial forces dominate over viscous forces. This allows mercury to maintain its flow more effectively against the frictional drag of the tap's edges. Therefore, Vessel B will be the first to empty because its high density translates into greater momentum and a more efficient discharge rate.

UPSC often includes Option B as a classic formula trap; it targets students who memorize $v = \sqrt{2gh}$ but fail to consider that different fluids behave differently under the same gravity due to their viscosity and density. Options C and D are incorrect because water and kerosene have lower densities, resulting in lower momentum and a greater relative impact from viscous drag, which slows down their exit compared to the heavy, high-pressure flow of mercury.