The variation of displacement [d] with time [t] in the case of a particle falling freely under gravity from rest is correctly

1. Foundations: Distance, Displacement, and Velocity (basic)

In basic mechanics, we start by distinguishing between how far an object has traveled and where it ended up relative to its starting point. Distance is a scalar quantity representing the total path length covered during motion. For example, when the Suez Canal opened in 1869, it significantly altered the travel path, reducing the distance between Europe and India by 7,000 km CONTEMPORARY INDIA-I, Geography Class IX, India Size and Location, p.2. Because it is a scalar, distance only cares about 'how much ground' was covered, regardless of direction.

Displacement, however, is a vector quantity. It measures the shortest straight-line distance from the initial position to the final position and must include a direction. Even if the longitudinal extent of a country like India is roughly 30°, the actual distance measured in kilometers varies because the physical 'gap' between longitudes decreases as you move toward the poles INDIA PHYSICAL ENVIRONMENT, Geography Class XI, India — Location, p.2. In a complete circle, your distance might be large, but your displacement would be zero because you returned to your starting point.

Building on these, we define Velocity as the rate of change of displacement (displacement divided by time). Unlike speed, velocity is a vector; it tells us both how fast and in what direction an object is moving. When an object undergoes free fall from rest, its displacement (s) is not a simple linear progression. Instead, it follows the kinematic equation s = ½ g t² (where g is acceleration due to gravity). This means the displacement increases in proportion to the square of time, creating a parabolic relationship where the object covers more distance in each successive second.

Feature	Distance	Displacement
Type	Scalar (Magnitude only)	Vector (Magnitude + Direction)
Path Dependence	Depends on the actual path taken	Depends only on start and end points
Can it be zero?	Only if no motion occurred	Yes, if the object returns to the start

Sources: CONTEMPORARY INDIA-I, Geography Class IX, India Size and Location, p.2; INDIA PHYSICAL ENVIRONMENT, Geography Class XI, India — Location, p.2

2. Understanding Acceleration (basic)

To understand acceleration, we must first look at how an object moves. In uniform linear motion, an object moves at a constant speed in a straight line, covering equal distances in equal intervals of time Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117. However, if the speed of that object keeps changing — like a car covering 60 km in the first hour and 70 km in the next — the motion is non-uniform Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.119. Acceleration is simply the physical quantity that describes this rate of change of velocity over time.

It is vital to remember that acceleration is a vector quantity, meaning it involves both magnitude (speed) and direction. You accelerate not just when you 'speed up' or 'slow down' (deceleration), but also when you change direction. For instance, air flowing around a low-pressure center in a cyclone experiences centripetal acceleration; even if the wind speed were constant, the inward-directed force continuously changes the wind's direction to create a circular vortex Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309.

The standard formula for average acceleration is a = (v - u) / t, where 'v' is final velocity, 'u' is initial velocity, and 't' is time. A special case we often see in geography and physics is constant acceleration. A prime example is an object in free-fall near the Earth's surface. Here, the object's velocity increases by approximately 9.8 m/s every single second due to gravity (g). Because the speed is increasing at a steady rate, the distance it covers grows faster and faster over time, following a quadratic relationship (proportional to t²).

Type of Motion	Velocity Status	Acceleration Status
Uniform Motion	Constant (Speed & Direction)	Zero Acceleration
Non-Uniform Motion	Changing Speed	Non-Zero Acceleration
Circular Motion	Changing Direction	Non-Zero (Centripetal)

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117, 119; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309

3. Newton’s Equations of Motion (intermediate)

When an object moves along a straight line, we categorize its motion based on whether its speed changes. In uniform linear motion, an object covers equal distances in equal intervals of time Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117. However, when an object accelerates—meaning its velocity changes over time—we enter the realm of non-uniform motion. To describe this mathematically, we use Newton’s Equations of Motion (also known as kinematic equations), which specifically apply to objects moving with constant acceleration.

The most vital equation for understanding how far an object travels over time is the displacement formula: s = ut + ½at². Here, s is the displacement, u is the initial velocity, a is the constant acceleration, and t is the time elapsed. Unlike uniform motion where distance is directly proportional to time (d ∝ t), in accelerated motion, the distance is proportional to the square of time (t²). This quadratic relationship means that as time passes, the object covers significantly more distance in each subsequent second because it is constantly speeding up.

A classic real-world application is free fall. When you drop a ball from rest, its initial velocity (u) is zero. Gravity provides a constant acceleration (g ≈ 9.8 m/s²). Substituting these into our equation, we get s = ½gt². Because the displacement varies with the square of time, the graph of this motion is not a straight line but a parabola that starts at the origin. This explains why an object falling for two seconds travels four times as far as an object falling for only one second Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118

4. Force of Gravity and 'g' (intermediate)

Gravity is the fundamental force that governs the motion of celestial bodies and shapes the very landscape of our planet. As a force, it acts as the primary driver for all geomorphic processes; without gravity and the resulting gradients, there would be no mobility of material, no erosion, and no deposition FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geomorphic Processes, p.38. While Isaac Newton’s law of gravitation provided the mathematical framework for this force Themes in world history, Changing Cultural Traditions, p.119, the practical manifestation we observe daily is the constant acceleration it provides to falling objects, denoted as 'g'. On Earth, g is approximately 9.8 m/s². This value varies significantly across the solar system; for instance, the Sun’s surface gravity is a staggering 274 m/s² (28 times that of Earth), while the Moon’s is a mere 1.62 m/s² Physical Geography by PMF IAS, The Solar System, p.23. When an object undergoes free fall—meaning it is moving solely under the influence of gravity—its velocity increases linearly every second. However, the distance it covers does not follow a simple linear path. To understand how far an object falls over time, we look at the kinematic equation: s = ut + ½at². For an object dropped from rest, the initial velocity (u) is 0, and the acceleration (a) is g. This simplifies the formula to:

s = ½gt²

This equation reveals a quadratic relationship: the distance (s) is directly proportional to the square of time (t²). If you were to plot this on a graph with time on the horizontal axis and distance on the vertical axis, the result would be a parabola that starts at the origin (0,0) with a slope of zero. This curved line represents how the object 'speeds up' its accumulation of distance as it falls faster and faster.

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geomorphic Processes, p.38; Themes in world history, Changing Cultural Traditions, p.119; Physical Geography by PMF IAS, The Solar System, p.23

5. Terminal Velocity and Air Resistance (exam-level)

When an object is dropped from a height, it initially undergoes free fall. In this early stage, the only significant force acting on it is gravitational force Physical Geography by PMF IAS, Pressure Systems and Wind System, p.306. According to the laws of motion, the displacement (s) of such an object is given by the equation s = ut + ½at². Since the object starts from rest (initial velocity u = 0) and moves under gravity (a = g), the equation simplifies to s = ½gt². This tells us that the distance fallen is proportional to the square of time (t²), which mathematically represents a parabola starting from the origin with a zero slope. However, as the object picks up speed, it begins to interact with the surrounding air. This interaction creates air resistance (or drag), a type of fluid friction that acts in the opposite direction to gravity. In the study of atmospheric dynamics, we see that friction is one of the primary factors affecting the movement of air and particles Physical Geography by PMF IAS, Pressure Systems and Wind System, p.306. The faster the object falls, the stronger this upward air resistance becomes. Eventually, a point is reached where the upward force of air resistance exactly balances the downward pull of gravity. At this precise moment, the net force on the object becomes zero. According to Newton's First Law, if the net force is zero, the object stops accelerating and continues to move at a constant speed. This maximum, steady speed is known as Terminal Velocity.

Phase of Motion	Force Balance	Velocity Trend	Graph Shape (s vs t)
Initial Fall	Gravity > Air Resistance	Increasing rapidly	Parabolic (Curve upwards)
At Terminal Velocity	Gravity = Air Resistance	Constant speed	Linear (Straight line)

Sources: Physical Geography by PMF IAS, Pressure Systems and Wind System, p.306; Science, Class VIII NCERT, Pressure, Winds, Storms, and Cyclones, p.85

6. Kinematics of Free Fall from Rest (intermediate)

When we talk about Free Fall, we are describing the motion of an object falling solely under the influence of gravity, without any air resistance or other forces acting upon it. In a standard physics context, such as when an object is dropped from a height, it follows a straight vertical path downwards (Science Class VIII, Exploring Forces, p.72). The defining characteristic of this motion is constant acceleration, denoted as g (approximately 9.8 m/s²), which causes the object's speed to increase steadily as it falls.

To understand the kinematics of an object released from rest, we look at the second equation of motion: s = ut + ½at². Here, "from rest" is a critical technical clue—it tells us that the initial velocity (u) is zero. When we substitute u = 0 and set the acceleration (a) to the acceleration due to gravity (g), the equation simplifies beautifully to:

s = ½gt²

This derivation reveals a fundamental principle: the displacement (s) of a falling object is directly proportional to the square of the time (t²) it has been falling. This means that if you double the time of the fall, the object doesn't just travel twice as far; it travels four times as far (2² = 4). Graphically, this relationship is represented by a parabola that starts at the origin (0,0) with a flat slope, representing the moment of release where the velocity is still zero.

Sources: Science Class VIII, Exploring Forces, p.72; Science Class VIII, Exploring Forces, p.78

7. Graphical Analysis: s-t and v-t Graphs (exam-level)

To master mechanics, we must look beyond numbers and visualize motion through Graphical Analysis. Graphs allow us to see the relationship between variables like displacement (s), velocity (v), and time (t) at a glance. In an s-t graph, the vertical axis represents the position and the horizontal axis represents time. If an object moves with uniform linear motion, it covers equal distances in equal intervals of time Science-Class VII, Measurement of Time and Motion, p.117, resulting in a straight, upward-sloping line. However, when an object is accelerating—such as in free fall—its speed is constantly changing, leading to non-uniform motion Science-Class VII, Measurement of Time and Motion, p.118. In such cases, the s-t graph is not a straight line but a curve. When a particle is released from rest (initial velocity u = 0) and falls under gravity, its displacement is governed by the kinematic equation s = ut + ½at². Substituting u = 0 and acceleration a = g, we get s = ½gt². This mathematical relationship tells us that displacement is proportional to the square of time (s ∝ t²). Graphically, this produces a parabola that starts at the origin. Because the velocity is increasing, the slope of this curve becomes steeper over time, reflecting an upward-sloping function where the rate of change is positive Microeconomics (NCERT class XII), Theory of Consumer Behaviour, p.22. Understanding the distinction between s-t and v-t graphs is vital for interpreting physical phenomena, whether it is the movement of a simple pendulum Science-Class VII, Measurement of Time and Motion, p.118 or the rapid earthflow of saturated materials down a slope Physical Geography by PMF IAS, Geomorphic Movements, p.87.

Feature	s-t Graph (Displacement-Time)	v-t Graph (Velocity-Time)
Slope represents	Velocity	Acceleration
Area under curve represents	N/A (No physical meaning)	Displacement / Distance
Constant Acceleration	A Parabolic Curve	A Straight Sloping Line

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118; Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.22; Physical Geography by PMF IAS, Geomorphic Movements, p.87

8. Solving the Original PYQ (exam-level)

Review the concepts above and try solving the question.