If a body travels half the distance with velocity v and next half with velocity v2, then which one of the following be the average velocity of the

1. Foundation: Distance vs. Displacement (basic)

To understand how objects move in our physical world, we must first distinguish between the actual path taken and the net change in position. Imagine an archaeologist tracing the "roads, paths, and bazaars" of an ancient city like Vijayanagara Themes in Indian History Part II, An Imperial Capital: Vijayanagara, p.189. Every twist and turn they take adds to the Distance. Distance is a scalar quantity, which means it describes "how much ground an object has covered" during its motion, regardless of its starting or ending direction.

In contrast, Displacement is a vector quantity. It doesn't care about the winding path; it only cares about the shortest straight-line gap between the starting point and the ending point. It is defined as the change in position of an object. For example, a mudflow might travel a long, winding distance down a mountain slope Environment and Ecology, Natural Hazards and Disaster Management, p.43, but its displacement is simply the straight arrow pointing from the peak to the point where it eventually stops.

The distinction is vital because displacement can be zero even if distance is very large. If you walk 10 kilometers and end up exactly where you started, your distance is 10 km, but your displacement is zero. This fundamental difference is the reason why "speed" (based on distance) and "velocity" (based on displacement) are often different values in physics.

Sources: Themes in Indian History Part II, An Imperial Capital: Vijayanagara, p.189; Environment and Ecology, Natural Hazards and Disaster Management, p.43

2. Understanding Speed and Velocity (basic)

Welcome back! Now that we have a basic sense of motion, let’s dive into how we actually measure it. In physics, speed is the rate at which an object covers distance. It is calculated by dividing the total distance traveled by the time taken. As per the standard SI system, we express speed in metres per second (m/s), though for vehicles and larger distances, kilometres per hour (km/h) is more common Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113.

In the real world, objects rarely move at a perfectly constant rate. If a train covers equal distances in equal intervals of time, we call it uniform linear motion. However, if the speed keeps changing—like a car in city traffic—it is non-uniform motion Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117. To describe such motion effectively, we use the concept of Average Velocity (or Average Speed). While we often think of an "average" as simply adding two numbers and dividing by two, physics requires a more nuanced approach depending on whether the time or the distance is constant.

The most common trap in competitive exams involves a journey split into two equal distance segments. Suppose you travel the first half of your journey at velocity v₁ and the second half at v₂. You might be tempted to use the arithmetic mean (v₁ + v₂)/2, but that is incorrect because you spend more time traveling during the slower leg of the journey. Instead, the average velocity is the harmonic mean of the two speeds:

v_avg = 2v₁v₂ / (v₁ + v₂)

This formula ensures that the total distance divided by the total time (t₁ + t₂) is accurately represented. If you were to travel for two equal time intervals, then and only then would the simple arithmetic average apply.

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

3. Types of Motion: Uniform and Non-Uniform (basic)

In our journey through mechanics, we must distinguish how objects move over time. When an object moves along a straight path, we call it linear motion. However, not all linear motion is the same. Imagine a train leaving a station: it starts slowly, picks up speed, maintains a steady pace in the middle of the journey, and eventually slows down to stop. This variation defines whether motion is uniform or non-uniform Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116.

Uniform Motion occurs when an object covers equal distances in equal intervals of time. In this state, the object's speed remains constant. For instance, a car on a cruise-controlled highway with no traffic is in uniform motion Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117. Conversely, Non-Uniform Motion is far more common in daily life. Here, the speed keeps changing, meaning the object covers unequal distances in equal time intervals—like a car navigating city traffic or a planet whose orbital speed increases as it nears the sun Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.257.

When dealing with non-uniform motion, we often calculate Average Speed to describe the overall journey. A critical nuance arises when a journey is split into two equal distance segments with different speeds, v₁ and v₂. You might be tempted to simply average them, but the correct formula for average speed in this specific case is 2v₁v₂ / (v₁ + v₂). This is known as the harmonic mean, and it accounts for the fact that the object spends more time traveling the slower segment than the faster one.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116-118; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.257

4. Graphical Analysis: Distance-Time and Velocity-Time (intermediate)

Graphs are powerful visual tools that allow us to interpret the motion of an object at a glance. In mechanics, the most fundamental graph is the Distance-Time (D-T) graph. When we plot distance on the vertical (y) axis and time on the horizontal (x) axis, the slope of the line represents the speed of the object. As observed in Science-Class VII, Measurement of Time and Motion, p.117, a straight line indicates uniform motion—where an object covers equal distances in equal intervals of time. If the graph is a curve, it represents non-uniform motion, meaning the speed is changing.

To deepen our analysis, we look at Velocity-Time (V-T) graphs. Here, the slope represents acceleration. If the graph is upward sloping, the velocity is increasing; if it is downward sloping, the velocity is decreasing Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.22. A crucial property of V-T graphs is that the area under the curve represents the total displacement (or distance in a straight line) covered by the object. This is a common shortcut used in competitive exams to avoid complex integration.

When analyzing a journey composed of different segments, we often need the Average Velocity. While the basic formula is total distance divided by total time Science-Class VII, Measurement of Time and Motion, p.113, a specific "intermediate" scenario frequently arises: equal distance segments. If a body travels the first half of a total distance with velocity v₁ and the second half with velocity v₂, your instinct might be to take the arithmetic mean (v₁ + v₂)/2. However, because the time taken for each segment differs (t = distance/speed), we must use the Harmonic Mean:

v_avg = 2v₁v₂ / (v₁ + v₂)

This formula accounts for the fact that the object spends more time traveling the segment where its velocity is lower. Only when the time intervals are equal do we use the simple arithmetic average.

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

5. Adjacent Concept: Circular Motion and Angular Velocity (intermediate)

In our previous discussions, we explored linear motion, which occurs when an object moves along a straight path, like a train traveling between two stations Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116. However, the world doesn't always move in straight lines. When an object moves along the circumference of a circle, it is in circular motion. A crucial distinction here is that even if the object maintains a constant speed, its velocity is technically changing because its direction is constantly being shifted by a force Science ,Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.64.

To describe how fast something rotates, we use Angular Velocity (ω). While linear velocity measures the change in distance over time, angular velocity measures the change in the angle (measured in radians) over time. Imagine a spinning ceiling fan: every point on the blade completes one full circle in the same amount of time, meaning they all share the same angular velocity. However, a point at the very tip of the blade has to cover a much larger circle than a point near the motor, so its linear (tangential) velocity is much higher.

The relationship between these two is governed by the radius (r) of the path: v = rω. This concept is vital in fields like physical geography; for instance, centripetal acceleration acts on air flowing around centers of circulation, creating the distinctive rotating patterns we see in cyclones and anticyclones Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309. Understanding that rotation involves a constant change in direction helps us grasp why a "force" (centripetal force) is always required to keep an object from flying off in a straight line.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116; Science ,Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.64; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309

6. Calculating Average Velocity: Mathematical Cases (exam-level)

In our previous discussions, we established that speed or velocity is essentially the distance covered in a unit of time Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113. However, in the real world, motion is rarely uniform; objects often speed up or slow down throughout a journey Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117. To describe such motion, we use Average Velocity, which is the total distance covered divided by the total time taken.

A common mistake in competitive exams is to assume that average velocity is always the simple average (arithmetic mean) of the speeds. This is only true if the object travels at each speed for the same amount of time. However, a much more frequent case involves equal distances. For instance, if a vehicle travels the first half of a journey at velocity v₁ and the second half at velocity v₂, it will spend more time on the slower segment. Because the time intervals are not equal, we cannot simply add the velocities and divide by two.

To calculate the average velocity for two equal distance segments, we use the Harmonic Mean formula. If the total distance is d, each half is d/2. By calculating the time for each segment (t = distance/velocity) and then dividing the total distance by the sum of these times, we arrive at the elegant formula: v_avg = 2v₁v₂ / (v₁ + v₂). This approach ensures we account for the extra time spent during the slower portion of the trip.

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

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental definitions of displacement, time, and velocity, this question serves as the perfect litmus test for your conceptual clarity. In the UPSC GS syllabus, questions often move beyond rote memorization to test if you can apply the core definition of Average Velocity: the total distance covered divided by the total time taken. The "half-distance" scenario you see here is a classic application of rate problems where the independent variable is distance, requiring you to synthesize your understanding of the d = v × t relationship by expressing time as a function of distance and velocity.

To arrive at the correct answer, think like a physicist: let the total distance be D. The time taken for the first half is t1 = (D/2) / v1 and for the second half is t2 = (D/2) / v2. When you sum these to find the total time and divide the total distance D by that sum, the distance D cancels out, leaving you with a formula that depends only on the velocities. This algebraic process yields the Harmonic Mean, which is 2v1v2 / (v1 + v2). It is essential to recognize that when distances are equal, the average speed is biased toward the slower velocity because the object spends more time traveling that segment.

The most common trap UPSC sets is the Arithmetic Mean, (v1 + v2) / 2. This is only correct if the body travels for equal time intervals at each velocity, not equal distances. Students often rush to this option because it feels intuitive, but it fails to account for the longer duration spent at the lower speed. Other distractors might involve simple products or squares of velocities, but you can dismiss them quickly by performing a dimensional analysis check—average velocity must maintain the units of distance/time. According to the NCERT Class 11 Physics - Motion in a Straight Line, maintaining this distinction between time-averaging and distance-averaging is key to scoring in mechanics.