The displacement-time graph of a particle acted upon by a constant force is

1. Newton’s Laws of Motion: The Origin of Force (basic)

In physics, we define Force as a push or a pull resulting from an object's interaction with another object Science, Class VIII, Exploring Forces, p.77. It is the fundamental 'agent of change' in the universe. Without force, an object at rest stays at rest, and an object in motion continues moving at the same speed in the same direction. When a force is applied, it can change an object's speed, its direction of motion, or even its physical shape Science, Class VIII, Exploring Forces, p.65. These interactions can occur through direct physical contact, like muscular force or friction, or from a distance through non-contact forces like gravity and magnetism Science, Class VIII, Exploring Forces, p.77. The standard unit we use to measure this 'push or pull' is the newton (N).

The true 'origin' of force in classical mechanics is best understood through its relationship with acceleration. According to Newton’s Second Law, when a constant net force (F) acts on an object of constant mass (m), it produces a constant acceleration (a), expressed by the famous equation F = ma. In terms of motion, this means the object's velocity is changing at a steady rate. If an object is moving along a straight line with a speed that keeps changing due to such a force, it is in non-uniform linear motion Science, Class VII, Measurement of Time and Motion, p.118.

To visualize this motion, we look at the relationship between displacement (s) and time (t). For an object starting with an initial velocity (u) under constant acceleration (a), the displacement is given by the kinematic equation: s = ut + ½ at². Because the time variable (t) is squared, the relationship is quadratic rather than linear. While a displacement-time graph for constant velocity is a straight line, the graph for an object under a constant force is a parabola. This curve is the mathematical 'signature' of a force acting steadily on an object, showing that the distance covered increases more rapidly in each successive second.

Sources: Science, Class VIII, Exploring Forces, p.65, 77; Science, Class VII, Measurement of Time and Motion, p.118

2. Kinematics: Displacement, Velocity, and Acceleration (basic)

To understand how objects move, we must master the trio of kinematics: displacement, velocity, and acceleration. Displacement is the straight-line distance between a starting and ending point. When an object covers equal distances in equal intervals of time, it is in uniform linear motion, meaning its velocity is constant (Science-Class VII . NCERT, Measurement of Time and Motion, p.117). However, most motion in daily life is non-uniform, where the speed or direction changes over time (Science-Class VII . NCERT, Measurement of Time and Motion, p.119).
When a constant net force is applied to an object, it experiences constant acceleration. This means its velocity changes by the same amount every second. In such cases, the relationship between displacement (s) and time (t) is no longer linear; it becomes quadratic. This is captured by the second equation of motion:

s = ut + ½at²
Where 'u' is initial velocity and 'a' is acceleration. Because displacement is proportional to the square of time (t²), the motion is represented by a parabola on a displacement-time graph, rather than a straight line.
Comparing these states of motion helps us visualize physical changes quickly:

Type of Motion	Acceleration	Displacement-Time Graph
Uniform Motion	Zero	Straight Line (Slope = Velocity)
Non-Uniform (Constant Accel.)	Constant	Curved Line (Parabola)

Sources: Science-Class VII . NCERT, Measurement of Time and Motion, p.117; Science-Class VII . NCERT, Measurement of Time and Motion, p.119

3. The Equations of Motion for Uniform Acceleration (intermediate)

To understand how objects move under a steady push or pull, we look at Uniform Acceleration. This occurs when the velocity of an object changes at a constant rate over time. While basic uniform motion involves covering equal distances in equal intervals of time Science-Class VII NCERT (Revised ed 2025), Measurement of Time and Motion, p.119, accelerated motion is more complex because the velocity itself is evolving. According to Newton’s Second Law, this steady change happens whenever a constant net force is applied to a mass.
In kinematics, we use three primary equations to describe this motion (where u is initial velocity, v is final velocity, a is acceleration, s is displacement, and t is time):

• v = u + at (The Velocity-Time relation)
• s = ut + ½at² (The Displacement-Time relation)
• v² = u² + 2as (The Velocity-Displacement relation)

The second equation, s = ut + ½at², is vital for predicting where an object will be at any given moment. Mathematically, it reveals a quadratic relationship: because time is squared (t²), the displacement grows faster and faster as time passes. This is why a falling object covers much more distance in its third second of fall than in its first.
Visualizing this on a graph is a favorite topic for competitive exams. If you plot displacement (s) against time (t):

Type of Motion	Acceleration	s-t Graph Shape
Uniform Velocity	Zero (a = 0)	Straight Line
Uniform Acceleration	Constant (a > 0)	Parabola (Curve)

This parabolic path is the signature of constant forces at work, such as the force of gravity or the pole-fleeing centrifugal force caused by Earth's rotation Physical Geography by PMF IAS, Tectonics, p.95. Understanding this curve helps us differentiate between a car cruising at a steady speed and one floor-it at a green light.

Sources: Science-Class VII NCERT (Revised ed 2025), Measurement of Time and Motion, p.119; Physical Geography by PMF IAS, Tectonics, p.95

4. Connected Concept: Motion Under Gravity (intermediate)

When we talk about Motion Under Gravity, we are essentially looking at the real-world application of Newton’s Second Law. At its core, gravity is an attractive, non-contact force that the Earth exerts on all objects Science Class VIII NCERT, Exploring Forces, p.72. Because the Earth's mass is so vast, this force is relatively constant near the surface, leading to a constant acceleration (commonly denoted as 'g'). This means that every second an object falls, its velocity increases by a steady amount.

To understand the mathematics of this motion, we look at the second equation of kinematics: s = ut + ½at². In this equation, 's' represents displacement, 'u' is initial velocity, 'a' is acceleration (which is 'g' in this context), and 't' is time. Because the time variable 't' is squared, the relationship between displacement and time is quadratic. This is why, if you were to plot the movement of a falling stone on a displacement-time (s-t) graph, you wouldn't see a straight line; instead, you would see a parabola. A straight line would imply constant velocity (zero acceleration), but gravity ensures the object is constantly speeding up.

This gravitational pull is the fundamental "engine" behind many Earth processes. For instance, in geography, we see that gravity is the force that "switches on" the movement of all surface material, enabling erosion, transportation, and deposition Fundamentals of Physical Geography Class XI NCERT, Geomorphic Processes, p.38. Whether it is a ball slowing down as it is thrown upward or a debris fall moving rapidly down a steep slope, the motion is dictated by the constant interplay between the object's mass and the Earth's gravitational field Fundamentals of Physical Geography Class XI NCERT, Geomorphic Processes, p.42.

Sources: Science Class VIII NCERT, Exploring Forces, p.72, 78; Fundamentals of Physical Geography Class XI NCERT, Geomorphic Processes, p.38, 42; Physical Geography by PMF IAS, Tectonics, p.108

5. Connected Concept: Work, Energy, and Power (intermediate)

In the realm of mechanics, Work, Energy, and Power form a trinity that describes how physical changes occur in the universe. While we often use these terms interchangeably in daily life, physics defines them with precision. Work is done when a force acting on an object causes it to move. It is the transfer of energy through motion. For instance, in an electrical circuit, work is done when a charge (Q) is moved across a potential difference (V), expressed as W = VQ Science, class X (NCERT 2025 ed.), Electricity, p.173. If there is no displacement, no matter how much force you apply, the mechanical work done is zero.

Energy is the capacity to do that work. It exists in various forms—kinetic, potential, chemical, or thermal—and is governed by the laws of thermodynamics. The first law reminds us that energy is conserved, but the second law reveals a crucial catch: whenever work is done, some energy is inevitably dissipated as heat, making it unavailable for further work Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14. This is why the flow of energy through our biosphere is unidirectional; as autotrophs (plants) convert sunlight into chemical energy to support life, energy is lost at each step of the food chain Science, class X (NCERT 2025 ed.), Our Environment, p.210.

Finally, Power measures the rate at which this work is performed or energy is transferred. While energy tells us "how much" work can be done, power tells us "how fast." In an electrical context, power (P) is the product of potential difference and current (P = VI), representing the energy supplied to a circuit per unit of time Science, class X (NCERT 2025 ed.), Electricity, p.188. Understanding these concepts allows us to analyze everything from the motion of the Earth driven by gravitational and geothermal forces Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267 to the metabolic efficiency of living organisms.

Concept	Core Definition	Key Formula (Example)
Work	Transfer of energy via force and displacement.	W = F × s (or W = VQ)
Energy	The capacity or "fuel" to perform work.	E = Pt (Power × Time)
Power	The rate at which work is done.	P = W / t (or P = VI)

Sources: Science, class X (NCERT 2025 ed.), Electricity, p.173, 188; Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14; Science, class X (NCERT 2025 ed.), Our Environment, p.210; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267

6. Graphical Analysis of Motion (exam-level)

To master mechanics, we must be able to translate physical laws into visual stories. A **displacement-time (s-t) graph** is a powerful tool that shows how an object's position changes over time. When an object moves with a constant speed, it covers equal distances in equal intervals of time, a state known as **uniform linear motion** Science-Class VII NCERT, Measurement of Time and Motion, p.117. Visually, this results in a straight, upward-sloping line because the rate of change (velocity) is constant. In mathematical terms, this is an **increasing function** where the slope—the steepness of the line—never changes Microeconomics NCERT class XII, Theory of Consumer Behaviour, p.22. However, the real world is rarely that simple. When a **constant net force** is applied to an object, Newton’s Second Law tells us it will experience **constant acceleration**. This changes the mathematical relationship between displacement (s) and time (t). According to the second equation of motion, s = ut + ½at², displacement is proportional to the **square of time** (t²). Because the relationship is quadratic rather than linear, the graph is no longer a straight line; it becomes a **parabola**. The ever-increasing steepness of this curve represents the increasing velocity of the object. Understanding the shape of these graphs is essential for interpreting complex data, much like how scientists analyze arrival times on a seismograph to map the Earth's interior Physical Geography by PMF IAS, Earths Interior, p.63. Just as a sudden change in seismic wave velocity indicates a change in the Earth's density, a change in the slope of a displacement graph indicates a change in the object's velocity (acceleration).

Motion Type	Acceleration	s-t Graph Shape	Slope (Velocity)
Uniform Motion	Zero	Straight Line	Constant
Non-Uniform Motion	Constant (Non-zero)	Parabola	Changing

Sources: Science-Class VII NCERT, Measurement of Time and Motion, p.117; Microeconomics NCERT class XII, Theory of Consumer Behaviour, p.22; Physical Geography by PMF IAS, Earths Interior, p.63

7. Quadratic Relationships and Parabolic Curves (exam-level)

In our study of mechanics, we often encounter two types of relationships: linear and quadratic. While a linear relationship suggests that one variable changes at a constant rate relative to another (like a straight-line consumption function in economics Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58), a quadratic relationship involves a variable being proportional to the square of another. In physics, this is most famously seen when a constant force is applied to an object. According to Newton’s Second Law, a constant force results in constant acceleration. This leads to the kinematic equation: s = ut + ½ at². Because the displacement (s) depends on the square of time (t²), the resulting graph is not a straight line, but a parabola.

To visualize this, imagine an object starting from rest (u = 0). In the first second, it travels a small distance; in the next second, because it is accelerating, it travels a much larger distance. This 'accelerating' rate of change creates a curve. We can compare the different graphical representations of motion to understand how 'curvature' helps us identify physical properties:

Graph Type	Shape for Constant Velocity	Shape for Constant Acceleration
Velocity-Time (v-t)	Horizontal Straight Line	Sloped Straight Line
Displacement-Time (s-t)	Sloped Straight Line	Parabolic Curve

This concept of curvature is a fundamental geometric principle. Just as spherical mirrors are defined by their inward (concave) or outward (convex) curves Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.135, the 'steepness' of a parabolic displacement graph tells us how quickly the velocity is changing. If the curve gets steeper over time, the object is speeding up; if it flattens out, the object is slowing down. Even the Earth's shape—an oblate spheroid bulged at the equator—is a result of forces acting differently across a curved surface Physical Geography by PMF IAS, Latitudes and Longitudes, p.241. In mechanics, the parabola is the 'signature' of motion under a constant net force.

Sources: Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58; Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.135; Physical Geography by PMF IAS, Latitudes and Longitudes, p.241

8. Solving the Original PYQ (exam-level)

This question effectively synthesizes your knowledge of Newton’s Second Law and Kinematics. When you see the term "constant force," your mind should immediately link it to constant acceleration (a = F/m). This is the bridge between dynamics and the geometric representation of motion. By applying the second equation of motion, s = ut + ½at², you are looking at a mathematical relationship where displacement (s) is a quadratic function of time (t). As you learned in the building blocks of motion, the power to which the independent variable is raised determines the geometry of the graph.

To arrive at the correct answer, (C) a parabola, think about the shape that any equation of the form y = ax² + bx + c creates on a coordinate plane. Because the time variable is squared, the rate of change of displacement is not constant; rather, it increases (or decreases) linearly. This curvature is the hallmark of a parabolic path on a displacement-time graph. As noted in Wikipedia: Equations of Motion, this quadratic dependency is a fundamental property of uniformly accelerated motion.

Be careful not to fall for common distractors. Option (A) a straight line represents constant velocity (zero acceleration), which contradicts the presence of a force. Option (B) a circle is a trap because time is a unidirectional dimension; a particle cannot return to a previous point in time. Finally, option (D) is a classic UPSC trap designed to make you doubt the fundamental laws of physics by suggesting unnecessary complexity. While initial conditions like initial velocity (u) might shift the vertex of the curve, they never change the fundamental parabolic nature of the graph under constant acceleration.