A parachutist jumps from a height of 5000 metres. The relationship between his falling speed ‘v’ and the distance fallen through ‘d’ is best represented as

1. Newton’s Laws of Motion and Force (basic)

To master mechanics, we must first understand the concept of **Force**. At its simplest, a force is a push or a pull acting upon an object. In our daily lives, we see force in action everywhere: when you kick a football, you are applying a force to change its state of motion. According to Science, Class VIII, Exploring Forces, p.64, a force can cause a moving object to stop, change its speed, alter its direction of motion, or even change the very shape of an object. To quantify this, we use the SI unit called the **newton (N)** Science, Class VIII, Exploring Forces, p.65. It is important to distinguish between mass and **weight**; while mass is the amount of matter, weight is actually the gravitational force with which the Earth pulls an object toward its center, and thus weight is also measured in newtons Science, Class VIII, Exploring Forces, p.72. Newton summarized the behavior of these forces through three fundamental laws. **Newton’s First Law (Law of Inertia)** states that an object will remain at rest or continue to move at a constant velocity unless acted upon by an external net force. **Newton’s Second Law** provides the mathematical backbone: **Force = mass × acceleration (F = ma)**. This tells us that the more mass an object has, the more force is required to accelerate it. Finally, **Newton’s Third Law** explains that forces always exist in pairs; for every action force, there is an equal and opposite reaction force. Whether it is the wind creating circular patterns around pressure systems Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309 or a ball rolling on the floor, these laws govern every movement in the universe.

Sources: Science, Class VIII, Exploring Forces, p.64, 65, 72; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309

2. Free Fall and Gravitational Acceleration (basic)

Gravity is the fundamental force of attraction that exists between any two masses. In the context of Earth, it is the primary force that keeps us grounded and "switches on" the movement of all surface materials, from falling rain to massive landslides FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Geomorphic Processes, p.38. When an object falls toward the Earth solely under the influence of this force—without air resistance or other forces acting on it—it is said to be in Free Fall. During free fall, all objects, regardless of their mass, accelerate at the same constant rate, known as the acceleration due to gravity (g).

On Earth, the average value of g is approximately 9.8 m/s². This means that for every second an object falls, its velocity increases by 9.8 meters per second. However, this value is not uniform across the universe or even across the Earth's surface. For instance, the Moon’s surface gravity is only 1.62 m/s², while the Sun’s immense mass results in a surface gravity of 274 m/s², which is 28 times that of Earth Physical Geography by PMF IAS, The Solar System, p.23.

Even on Earth, the value of g fluctuates based on your location. Because the Earth is an oblate spheroid (bulging at the equator and flattened at the poles), a person at the poles is actually closer to the Earth's center than a person at the equator. Consequently, gravity is greater at the poles and less at the equator FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19. Furthermore, the uneven distribution of mass within the Earth's crust creates small variations in gravity known as gravity anomalies, which geologists use to map the Earth's interior Physical Geography by PMF IAS, Earths Interior, p.58.

Location/Body	Approx. Gravity (g)	Reason for Difference
Earth (Poles)	~9.83 m/s²	Closer to Earth's center of mass.
Earth (Equator)	~9.78 m/s²	Farther from center; centrifugal force.
Moon	1.62 m/s²	Lower mass than Earth.

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Geomorphic Processes, p.38; Physical Geography by PMF IAS, The Solar System, p.23; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19; Physical Geography by PMF IAS, Earths Interior, p.58

3. Kinetic Energy and Distance Relationships (intermediate)

To understand how Kinetic Energy (KE) relates to the distance an object travels, we must first look at the Work-Energy Theorem. In its simplest form, work done on an object (Force × Distance) results in a change in its kinetic energy. In a vacuum, if a constant force like gravity acts on a falling object, its velocity increases steadily, and its kinetic energy grows linearly with the distance fallen. This is a form of non-uniform linear motion because the speed is constantly changing as the object covers more ground Science-Class VII, NCERT (Revised ed 2025), Measurement of Time and Motion, p.117.

However, real-world mechanics often involve resistive forces, such as air resistance or fluid drag. As an object accelerates through a fluid (like air), the drag force pushing back against it increases with its speed. We see this principle in how wind turbines capture energy; the kinetic energy available in the wind depends heavily on the wind speed Environment, Shankar IAS Academy (ed 10th), Renewable Energy, p.290. In a free-fall scenario, there comes a point where the upward drag force exactly matches the downward gravitational force. At this precise moment, the net force becomes zero.

Once the forces are balanced, the object reaches what is known as terminal velocity. From this point forward, the object continues to move at a constant speed regardless of how much further it falls. Consequently, its Kinetic Energy (KE = ½mv²) also becomes constant. If you were to graph this relationship, you would see the KE rise sharply at first as the object accelerates, and then flatten out into a horizontal line as the object reaches terminal velocity, showing that distance is increasing but energy is no longer changing.

Sources: Science-Class VII, NCERT (Revised ed 2025), Measurement of Time and Motion, p.117; Environment, Shankar IAS Academy (ed 10th), Renewable Energy, p.290

4. Fluid Dynamics: Viscosity and Drag Force (intermediate)

When we think of friction, we usually imagine two solid surfaces rubbing together. However, fluids—a term that includes both liquids and gases—also exert a resistive force on any object moving through them. This internal resistance to flow is known as viscosity, and the resulting frictional force exerted by the fluid is called drag force Science, Class VIII NCERT, Exploring Forces, p.68.

The magnitude of this drag force depends on three primary factors: the nature of the fluid (its thickness or viscosity), the shape of the object (which is why airplanes and fish are streamlined), and, most critically, the speed of the object. Unlike the friction between two solids, which is relatively constant, the drag force in a fluid increases as the object moves faster. This creates a fascinating dynamic when an object falls through the atmosphere under the influence of gravity.

Imagine an object dropped from a height. Initially, the only significant force is gravity, causing the object to accelerate downward (F = ma). As its velocity increases, the upward drag force (air resistance) also begins to climb. Eventually, the object reaches a speed where the upward drag force exactly equals the downward gravitational pull Fundamentals of Physical Geography, Class XI NCERT, Atmospheric Circulation and Weather Systems, p.78. At this precise moment, the net force becomes zero, acceleration ceases, and the object continues to fall at a steady, maximum speed called terminal velocity.

Phase of Fall	Force Balance	Velocity Trend
Initial Drop	Gravity > Drag	Rapidly Increasing
Mid-Fall	Gravity > Drag (but Drag is rising)	Increasing slowly
Terminal State	Gravity = Drag	Constant (Steady State)

Sources: Science, Class VIII NCERT, Exploring Forces, p.68; Fundamentals of Physical Geography, Class XI NCERT, Atmospheric Circulation and Weather Systems, p.78

5. Atmospheric Layers and Air Density (intermediate)

To understand the atmosphere, we must first view it as a fluid governed by gravity. Imagine a tall column of air stretching from the ground to the edge of space. Because air is compressible, the weight of the upper layers presses down on the lower layers, packing the molecules closer together. This is why air density is highest at sea level and decreases rapidly as we ascend. In fact, in the lower atmosphere, the pressure drops at an average rate of about 34 millibars for every 300 metres of height Physical Geography by PMF IAS, Pressure Systems and Wind System, p.305. By the time you reach the summit of Mt. Everest, the air pressure is nearly two-thirds less than at sea level, meaning there are far fewer oxygen molecules in every breath you take. Despite this thinning of the air, the chemical composition remains remarkably consistent in the lower 80 km of the atmosphere, a region known as the Homosphere. While the total number of molecules decreases, the proportion of nitrogen to oxygen remains stable Environment and Ecology by Majid Hussain, Basic Concepts of Environment and Ecology, p.7. This thinning air has significant mechanical consequences: less density means less aerodynamic drag. This is why meteors usually only begin to burn up due to friction when they hit the slightly denser gases of the Mesosphere Physical Geography by PMF IAS, Earths Atmosphere, p.280. We often forget that we live at the bottom of a heavy 'ocean' of air. The atmospheric pressure exerted on a small 15 cm × 15 cm area is equivalent to the force of gravity on a 225 kg mass Science Class VIII NCERT, Pressure, Winds, Storms, and Cyclones, p.87. We aren't crushed because our internal body fluids and gases exert an equal outward pressure, maintaining a delicate equilibrium with the environment.

Altitude Range	Density & Pressure Status	Key Characteristic
Sea Level	Maximum	Highest concentration of mass; high drag.
0 - 80 km (Homosphere)	Rapidly Decreasing	Uniform mix of gases despite thinning.
High Altitude (e.g., Everest)	Very Low	Reduced pressure (1/3rd of sea level).

Sources: Physical Geography by PMF IAS, Pressure Systems and Wind System, p.305; Environment and Ecology by Majid Hussain, Basic Concepts of Environment and Ecology, p.7; Physical Geography by PMF IAS, Earths Atmosphere, p.280; Science Class VIII NCERT, Pressure, Winds, Storms, and Cyclones, p.87

6. The Concept of Terminal Velocity (exam-level)

Imagine you drop a small stone from a tall building. Initially, the gravitational force (the Earth's pull) is the dominant force acting on it, causing the stone to accelerate downwards Science, Class VIII. NCERT(Revised ed 2025), Exploring Forces, p.72. However, as the stone gains speed, it begins to collide with air molecules. This creates a resistive force called drag or air resistance, which acts in the opposite direction to the motion. Unlike gravity, which remains constant, this drag force increases rapidly as the object's velocity increases.

The journey to Terminal Velocity is essentially a tug-of-war between these two forces. In the beginning, gravity is much stronger than air resistance, so the object accelerates quickly. But as the object moves faster, the air resistance grows until it becomes exactly equal in magnitude to the downward gravitational force. At this precise moment, the net force acting on the object becomes zero. According to the laws of motion, when the net force is zero, the acceleration also becomes zero. The object doesn't stop moving; instead, it stops speeding up and continues to fall at a steady, maximum speed.

This concept is why raindrops don't hit us with the speed of a bullet. If we were to graph this motion (Velocity vs. Time or Velocity vs. Distance), we would see a sharp upward curve at the start representing the initial acceleration. As the object approaches its limit, the curve begins to flatten out, eventually becoming a horizontal straight line. This horizontal line represents the terminal velocity—the point where velocity remains constant regardless of how much further the object falls.

Phase of Fall	Force Balance	Velocity Status
Initial Release	Gravity > Drag	Accelerating rapidly
Mid-fall	Gravity > Drag (but Drag is rising)	Accelerating slowly
Terminal Phase	Gravity = Drag	Constant Velocity (Zero Acceleration)

Sources: Science, Class VIII. NCERT(Revised ed 2025), Exploring Forces, p.72

7. Interpreting Motion Graphs (v-d vs. v-t) (exam-level)

To master mechanics, we must look beyond simple Velocity-Time (v-t) graphs and understand how variables relate spatially. In a standard graph, the independent variable (the cause) is plotted on the horizontal axis, while the dependent variable (the effect) is on the vertical axis Microeconomics NCERT class XII 2025 ed., Theory of Consumer Behaviour, p.22. When analyzing a falling object like a parachutist, we often track Velocity (v) against Distance (d). Initially, gravity is the dominant force, causing the object to accelerate; on a v-d graph, this appears as a curve moving upward as the distance from the starting point increases. However, real-world motion involves fluid dynamics. As an object falls through the atmosphere, it encounters air resistance or drag. This is similar to how friction in the upper troposphere affects the velocity of jet streams Physical Geography by PMF IAS, Jet streams, p.386. As the velocity increases, the upward drag force also increases until it perfectly balances the downward pull of gravity. At this precise moment, the net force becomes zero, and the object reaches terminal velocity. Graphically, terminal velocity is represented by a horizontal plateau. Whether the horizontal axis represents time or distance, a horizontal line signifies that the velocity is no longer changing (Δv = 0). Unlike a linear equation where a constant slope indicates a steady rate of change Macroeconomics NCERT class XII 2025 ed., Determination of Income and Employment, p.58, a flat line in a motion graph indicates that the acceleration has dropped to zero.

Graph Feature	Velocity-Time (v-t)	Velocity-Distance (v-d)
Slope represents	Acceleration (a = Δv/Δt)	Rate of change of velocity over space
Horizontal Line	Constant Velocity (Zero Acceleration)	Constant Velocity (Terminal Phase)
Upward Curve	Increasing Velocity (Acceleration)	Increasing Velocity as object falls further

Sources: Microeconomics NCERT class XII 2025 ed., Theory of Consumer Behaviour, p.22; Physical Geography by PMF IAS, Jet streams, p.386; Macroeconomics NCERT class XII 2025 ed., Determination of Income and Employment, p.58

8. Solving the Original PYQ (exam-level)

This question perfectly synthesizes the concepts of gravitational acceleration and fluid dynamics you have just mastered. To solve this, you must look beyond simple equations of motion and consider the real-world impact of air resistance. As the parachutist jumps, they initially experience free fall where gravity causes a rapid increase in velocity (v) relative to the distance fallen (d). However, as taught in NCERT Class 11 Physics, air exerts a drag force that increases with speed. The "building block" here is the realization that this motion is not infinite; it is governed by a balancing act between downward weight and upward air friction.

Your reasoning should follow the timeline of the jump: In the initial phase, the velocity rises sharply, which eliminates any graph that is flat from the start. As the speed increases, the upward drag eventually equals the downward gravitational force. At this equilibrium, the acceleration becomes zero, and the parachutist reaches Terminal Velocity. In a graph of velocity (v) vs. distance (d), this physical limit is represented by a horizontal line—indicating that no matter how much further the parachutist falls, the speed stays constant. Therefore, Graph IV is the only representation that correctly shows an initial acceleration followed by a definitive velocity plateau.

UPSC often includes "ideal world" traps to test your conceptual depth. Graph I is a classic trap; it represents constant acceleration in a vacuum, which ignores air resistance entirely. Graphs II and III are incorrect because they suggest that velocity continues to increase indefinitely or linearly, failing to account for the saturation point defined by terminal velocity. When tackling Science & Tech PYQs, always ask yourself: "Is there a resistive force or a limit that changes the trend?" In this case, that limit makes Option (D) the correct choice.