Which of the following distance-time graph (x - t) represents one-dimensinal uniform motion?

1. Motion, Rest, and Reference Frames (basic)

In our daily lives, we often classify objects as either being "still" or "moving." However, in physics, Rest and Motion are relative terms. An object is said to be at rest if its position does not change with respect to its surroundings over time. Conversely, it is in motion if its position changes. This "point of view" or surroundings we use for comparison is called a Reference Frame. For instance, while you might feel at rest while reading this, you are actually moving at thousands of miles per hour relative to the Sun because the Earth is rotating and orbiting.

When we look at how objects move, the simplest form is Linear Motion, which occurs when an object travels along a straight path, such as a train moving on a straight track between two stations Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116. We categorize this based on how consistent that movement is:

Type of Motion	Description	Distance/Time Relationship
Uniform Motion	Movement at a constant speed along a straight line.	Covers equal distances in equal intervals of time.
Non-Uniform Motion	Movement where the speed changes over time.	Covers unequal distances in equal intervals of time.

Most real-world movements are non-uniform. A train starting from a station begins slowly, speeds up to a cruising velocity, and then slows down to a halt; only the middle portion where it maintains a steady speed is considered uniform Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117. Beyond straight lines, motion can be Periodic, like a pendulum bob that repeats its path after a fixed interval of time Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109. Understanding these basics is crucial because even complex phenomena, like the circular motion of water particles beneath ocean waves, are built upon these fundamental principles of position and time Physical Geography by PMF IAS, Tsunami, p.192.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109, 116, 117; Physical Geography by PMF IAS, Tsunami, p.192

2. Distance vs. Displacement: The Scalar-Vector Distinction (basic)

Concept: Distance vs. Displacement: The Scalar-Vector Distinction

3. Speed and Velocity: Rates of Change (basic)

In our study of mechanics, Speed and Velocity are the fundamental ways we measure how an object moves. At its simplest, speed is the rate at which an object covers distance. It is a scalar quantity, meaning it only tells us 'how fast' something is moving without regard to its direction. For instance, the high-altitude winds known as jet streams can reach speeds of up to 400 kmph Physical Geography by PMF IAS, Jet streams, p.386. However, when we specify the direction—such as a jet stream flowing from west to east—we are talking about velocity, which is a vector quantity.

To understand motion more deeply, we categorize it based on how consistent that rate of change is. This brings us to the distinction between Uniform and Non-uniform motion:

Type of Motion	Description	Distance/Time Relationship
Uniform Motion	An object moving along a straight line at a constant speed.	Covers equal distances in equal intervals of time.
Non-Uniform Motion	An object whose speed keeps changing as it moves.	Covers unequal distances in equal intervals of time.

In a practical sense, most movement we observe in geography or transport is non-uniform. For example, a train traveling the 1419 km between Delhi and Mumbai Geography of India ,Majid Husain, Transport, Communications and Trade, p.3 rarely maintains a perfectly constant speed because it must slow down for stations or speed up on open tracks. When it moves at a constant pace between two points, it is in uniform linear motion Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117. We can visualize this on a Distance-Time graph: uniform motion appears as a straight diagonal line, where the slope of the line represents the speed. A steeper slope indicates a higher speed, while a horizontal line indicates the object is at rest.

Sources: Physical Geography by PMF IAS, Jet streams, p.386; Geography of India ,Majid Husain, Transport, Communications and Trade, p.3; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117

4. Scalar and Vector Quantities in UPSC Physics (intermediate)

In the realm of physics, we categorize physical quantities based on the information required to describe them completely. At the most fundamental level, we have scalars and vectors. A scalar quantity is one that is fully described by its magnitude (a numerical value and a unit) alone. For instance, when you measure your mass or the time taken for a journey, you don't specify a direction. Common examples include mass, temperature, time, and speed.

A vector quantity, however, requires both magnitude and direction to be fully understood. Think about applying a force to an object; it matters not just how hard you push (magnitude), but also which way you push. As seen in practical tools like the spring balance, which measures weight (a force), the quantity is inherently tied to the direction of gravitational pull toward the Earth's center Science, Class VIII, Exploring Forces, p.74. Other key vectors include displacement, velocity, and acceleration. This distinction is crucial in geography and mechanics, such as when analyzing the rotation and revolution of the Earth, where the axis and direction of movement determine climatic patterns Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267.

To help you distinguish between the two in your UPSC preparation, refer to this comparison table:

Feature	Scalar Quantity	Vector Quantity
Definition	Has only magnitude.	Has both magnitude and direction.
Change	Changes with a change in magnitude.	Changes with a change in magnitude, direction, or both.
Addition	Follows simple algebraic addition.	Follows specific laws of vector addition (e.g., triangle law).
Examples	Distance, Speed, Mass, Energy, Density.	Displacement, Velocity, Force, Momentum, Weight.

Understanding this difference is vital because many complex phenomena, like the refraction of light through different media, involve changes in velocity—a vector quantity—which explains why light bends in specific directions when moving from water to glass Science, Class X, Light – Reflection and Refraction, p.145.

Sources: Science, Class VIII (NCERT), Exploring Forces, p.74; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267; Science, Class X (NCERT), Light – Reflection and Refraction, p.145

5. Newton's First Law: The Physics of Constant Velocity (intermediate)

Newton's First Law, often called the Law of Inertia, states that an object will maintain its state of rest or uniform motion in a straight line unless compelled to change that state by an applied external force. This law introduces the idea that motion does not require a force to maintain it; rather, force is required to change it. This shift in thinking was a cornerstone of the scientific revolution, which reached its climax with Isaac Newton’s theories Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119. To understand this law, we must look closely at the concept of constant velocity.

Uniform motion implies that an object moves with a constant speed, covering equal distances in equal intervals of time. In physics, we use the term velocity because direction matters. If an object is moving at a steady 60 km/h but is turning a corner, its velocity is changing (accelerating) even if its speed is not. Newton’s First Law tells us that if the net force (the sum of all forces acting on an object) is zero, the object’s velocity remains unchanged. The SI unit used to measure these forces is the newton (N) Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.65.

We can visualize these states of motion through distance-time (x-t) graphs. The slope of the line on such a graph represents the speed of the object. For uniform motion, the graph is a straight line inclined to the time axis, indicating a constant, non-zero slope. If the line is horizontal, the slope is zero, meaning the object is at rest. If the line is curved, the slope is changing, which indicates acceleration (non-uniform motion).

Motion Type	Velocity Status	Distance-Time Graph
At Rest	Zero (Constant)	Horizontal straight line
Uniform Motion	Non-zero (Constant)	Inclined straight line
Accelerated Motion	Changing	Curved line (Parabola)

Sources: Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119; Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.65

6. Interpreting Distance-Time (x-t) Graphs: Slopes and Shapes (exam-level)

In mechanics, a Distance-Time (x-t) graph is a visual story of an object’s journey. By plotting time on the x-axis (independent variable) and distance on the y-axis (dependent variable), we can instantly determine the nature of an object’s motion without complex calculations. The most vital concept to grasp here is the slope: in an x-t graph, the steepness or slope of the line directly represents the speed of the object (Science-Class VII . NCERT, Measurement of Time and Motion, p.113). A steeper line indicates that a greater distance is covered in a shorter amount of time, signifying a higher speed.

When an object moves in uniform motion, it covers equal distances in equal intervals of time (Science-Class VII . NCERT, Measurement of Time and Motion, p.117). This is represented by a straight line inclined to the time axis. Because the slope of a straight line is constant, the speed is constant. If the line is horizontal (parallel to the time axis), the slope is zero, meaning the distance is not changing as time passes—the object is at rest. Conversely, non-uniform motion occurs when the speed changes, resulting in a curved line. For example, if the curve gets steeper over time, the object is accelerating; if it flattens out, the object is slowing down (Science-Class VII . NCERT, Measurement of Time and Motion, p.117).

Graph Shape	Type of Motion	Speed Interpretation
Straight line (inclined)	Uniform Motion	Constant Speed
Horizontal line	Object at Rest	Zero Speed
Curved line	Non-uniform Motion	Changing Speed (Acceleration/Deceleration)

It is also helpful to understand the direction of the slope. In a distance-time context, an upward-sloping line indicates an increasing distance from the starting point (Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.22). While distance itself cannot decrease (it is a scalar sum of path traveled), if we were looking at a displacement-time graph, a downward slope would indicate the object is returning toward its starting position.

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.117; Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.22

7. Solving the Original PYQ (exam-level)

Now that you have mastered the building blocks of kinematics, this question brings everything together. You have learned that one-dimensional uniform motion occurs when an object travels in a straight line at a constant speed. In the language of coordinate geometry, if the speed—which is the rate of change of distance over time—is constant, the slope of the distance-time (x-t) graph must also be constant. This is the fundamental link: a constant physical rate always translates to a linear (straight-line) geometric representation.

To solve this like a seasoned aspirant, visualize the slope of each graph provided. Since speed equals the change in distance divided by the change in time, we are looking for a graph where this ratio remains identical at every single point. Only Graph IV depicts a straight line inclined at a constant angle to the time axis. This indicates that as time progresses, the distance increases at a steady, unchanging rate. Consequently, (D) IV is the only representation that satisfies the condition for uniform motion, as the constant slope reflects a constant velocity.

UPSC frequently uses the other options as conceptual traps to test your precision. Graphs showing curved lines represent non-uniform or accelerated motion, where the slope (and thus the speed) is changing at every moment. Another common pitfall is confusing an x-t graph with a v-t (velocity-time) graph; while a horizontal line represents uniform motion on a velocity-time graph, on a distance-time graph, a horizontal line would mean the object is at rest. As noted in NCERT Class 9 Science (Chapter: Motion), the nature of the motion is always dictated by the linearity of the slope in an x-t relationship.