A person travelled from one place to another at an average speed of 40 kilometres/hour and back to the original place at an average speed of 50 kilometres/hour. What is his average speed in kilometres/hour during the entire round-trip ?

1. Distance vs Displacement in Kinematics (basic)

To master kinematics, we must first distinguish between how far an object has traveled and how far it has actually moved from its starting point. Distance is the total length of the path covered by an object, regardless of the direction. It is a scalar quantity, meaning it only has magnitude. For instance, if you walk 5 km North and then 5 km South, your total distance is 10 km. Even in geographic measurements, we look at linear distances between latitudes to understand physical separation on the Earth's surface Physical Geography by PMF IAS, Latitudes and Longitudes, p.240.

Displacement, on the other hand, is the shortest straight-line path between the initial and the final position of an object. It is a vector quantity because it requires both a magnitude and a specific direction. In the previous example of walking 5 km North and 5 km South, while your distance was 10 km, your displacement is zero because you returned to your starting point. It is important to distinguish this physical change in position from the apparent displacement we see in optics, such as when a pencil in water looks shifted due to light refraction Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.145.

Feature	Distance	Displacement
Definition	Total path length covered.	Shortest path between start and end.
Type	Scalar (Magnitude only).	Vector (Magnitude + Direction).
Value	Always positive or zero.	Can be positive, negative, or zero.

Sources: Physical Geography by PMF IAS, Latitudes and Longitudes, p.240; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.145

2. Defining Speed and Velocity (basic)

In the world of physics and quantitative aptitude, understanding motion begins with Speed. At its simplest level, speed is a measure of how fast an object is moving. We define it as the distance covered by an object in a unit of time. The fundamental formula you must master is: Speed = Total Distance Covered / Total Time Taken Science-Class VII, Measurement of Time and Motion, p.114.

In real-world scenarios, objects rarely move at a perfectly constant rate; a car might slow down at a turn or speed up on a highway. Because of this variation, we usually calculate the Average Speed. Even if the book simply says 'speed', it is often referring to this average value for the entire journey Science-Class VII, Measurement of Time and Motion, p.115. When we introduce a specific direction to this movement—such as "50 km/h toward the North"—we transition from speed to Velocity. While speed is a scalar quantity (having only magnitude), velocity is a vector quantity (having both magnitude and direction).

Feature	Speed	Velocity
Definition	Distance covered per unit time.	Displacement (distance in a specific direction) per unit time.
Nature	Scalar (only "how much").	Vector ("how much" and "which way").

The Standard International (SI) unit for speed is metre per second (m/s), though kilometre per hour (km/h) is frequently used for larger distances Science-Class VII, Measurement of Time and Motion, p.113. When recording these in your notes or exam, remember to use lowercase symbols (s, min, h) and always leave a space between the number and the unit (e.g., 60 km/h) Science-Class VII, Measurement of Time and Motion, p.111.

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.111; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.114; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.115

3. Uniform vs Non-Uniform Motion (intermediate)

In the world of kinematics, we categorize how things move based on their consistency. An object is said to be in uniform linear motion when it moves along a straight line at a constant speed. This means it covers equal distances in equal intervals of time, no matter how small those intervals are Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.117. For example, a train cruising at a steady 90 km/h on a straight track is in uniform motion. However, in our daily lives, this is an idealization; most objects we observe, like a car weaving through city traffic, exhibit non-uniform motion because their speed keeps changing due to signals, pedestrians, or road conditions Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.118. To better grasp the difference, consider the table below comparing two scenarios:

Feature	Uniform Motion	Non-Uniform Motion
Speed	Remains constant throughout.	Changes over time.
Distance-Time Relation	Equal distances in equal time intervals.	Unequal distances in equal time intervals.
Example	Light traveling in a vacuum.	A marathon runner's pace.

When dealing with non-uniform motion, we often use the concept of average speed to describe the journey as a whole. Since the object might move faster at some points and slower at others, we simply calculate the total distance covered divided by the total time taken Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.115. If a car covers 60 km in the first hour and 40 km in the second, its motion is non-uniform, and its average speed for the trip would be 50 km/h Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.119.

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.117; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.115; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.119; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.118

4. Acceleration and Equations of Motion (intermediate)

In our previous discussions, we explored speed as a simple ratio of distance over time. However, in the real world, objects rarely move at a perfectly constant speed. When you press the gas pedal in a car or when a seismic wave moves from a low-density crust into a high-density mantle, the velocity changes. This change in velocity over time is what we call acceleration. As noted in Science-Class VII, Chapter 8, p.119, most daily motions are non-uniform, meaning the speed is constantly fluctuating.

To master quantitative aptitude, we must go beyond simple averages and understand the Equations of Motion. These three formulas allow us to predict where an object will be or how fast it will be going at any point in time, provided the acceleration is constant. These equations link five key variables: initial velocity (u), final velocity (v), acceleration (a), time (t), and displacement (s).

Equation Name	Formula	What it relates
First Equation	v = u + at	Velocity and Time
Second Equation	s = ut + ½at²	Displacement and Time
Third Equation	v² = u² + 2as	Velocity and Displacement

It is important to remember that acceleration is a vector quantity, meaning direction matters. In the context of Earth's atmosphere, horizontal winds respond to the pressure gradient force and the Coriolis force, which essentially accelerate the air in specific directions FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Chapter 7, p.78. Similarly, seismic waves change velocity as they hit different layers of the Earth, a phenomenon used by geologists to map the interior Physical Geography by PMF IAS, Earths Interior, p.58.

Sources: Science-Class VII, Chapter 8: Measurement of Time and Motion, p.119; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Chapter 7: Atmospheric Circulation and Weather Systems, p.78; Physical Geography by PMF IAS, Earths Interior, p.58

5. Relative Speed and Object Interaction (intermediate)

When we talk about Speed, we are essentially measuring the distance covered by an object in a unit of time, such as a second, a minute, or an hour Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113. However, in competitive exams like the UPSC, we often deal with Object Interaction—where two bodies move relative to each other. This brings us to the concept of Relative Speed. If two objects move toward each other, their relative speed is the sum of their individual speeds; if they move in the same direction, the relative speed is the difference between them.

One of the most frequent traps in quantitative aptitude is calculating the Average Speed for a round trip or a journey with different segments. It is a common mistake to simply take the arithmetic mean (average) of the speeds. Instead, we must always return to the fundamental principle: Average Speed = Total Distance Covered / Total Time Taken Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.114. When an object travels equal distances at different speeds, the average speed is actually the Harmonic Mean of those speeds, because more time is spent traveling at the slower speed.

Scenario	Calculation Logic
Opposite Directions	Add speeds (Speed A + Speed B)
Same Direction	Subtract speeds (Higher Speed - Lower Speed)
Average Speed (Equal Distances)	2xy / (x + y) where x and y are the speeds

For example, if a train travels a certain distance at speed v₁ and returns the same distance at speed v₂, the total time taken is the sum of the time for each leg. Since Time = Distance / Speed, the slower leg contributes more to the total time, which pulls the average speed down below the simple midpoint of the two speeds Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.119.

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.114; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.119

6. Average Speed and the Harmonic Mean (exam-level)

In quantitative aptitude, the concept of Average Speed is often misunderstood as a simple average of two numbers. However, from first principles, speed is defined as the distance covered in a unit of time. As noted in Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p. 113, we calculate the speed of an object by dividing the total distance covered by the total time taken. This becomes critical when an object moves with non-uniform motion—meaning it changes its speed during different parts of the journey.

The most common trap in competitive exams occurs when an object covers equal distances at different speeds (for instance, a round trip). If you travel from point A to B at speed u and return from B to A at speed v, you spend more time travelling at the slower speed. Because average speed is weighted by time, the result is the Harmonic Mean of the speeds, not the Arithmetic Mean. Mathematically, if the distance is d, the total distance is 2d and the total time is (d/u + d/v). Simplifying this gives us the classic formula: 2uv / (u + v).

Scenario	Condition	Formula for Average Speed
Equal Time	Travels at u for 1 hour, then v for 1 hour	(u + v) / 2 (Arithmetic Mean)
Equal Distance	Travels 10km at u, then 10km at v	2uv / (u + v) (Harmonic Mean)

For example, if a car travels at 40 km/h and returns at 50 km/h, the average speed isn't 45 km/h. Using the harmonic mean: (2 × 40 × 50) / (40 + 50) = 4000 / 90 = 44.44 km/h. This illustrates why the average speed will always be slightly closer to the lower speed, as more time is spent in that phase of the journey. As highlighted in Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p. 115, most daily motions are non-uniform, making this calculation essential for real-world accuracy.

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.115

7. Solving the Original PYQ (exam-level)

You have just mastered the fundamental definitions of motion, and this question is a perfect application of those building blocks. According to Science-Class VII . NCERT(Revised ed 2025), average speed is defined as the total distance covered divided by the total time taken. In this specific scenario, even though the numerical distance isn't provided, you should recognize a "constant distance" framework: the outward journey and the return journey cover the exact same path. This allows the unknown distance to cancel out mathematically, leading us to the harmonic mean formula—a specialized tool you learned for situations where distances are equal but speeds vary.

To arrive at the correct answer, visualize the trip in two segments. If we let the one-way distance be d, the time spent going is d/40 and the time spent returning is d/50. By adding these, the total time becomes 9d/200. When you divide the total round-trip distance (2d) by this total time, the d disappears, leaving you with (C) 400/9 km/h. Alternatively, using the shortcut 2xy/(x+y), you calculate (2 × 40 × 50) / (40 + 50), which simplifies to 4000/90 or 400/9. Reasoning through the units ensures that your final value remains a speed (km/h) and not just a raw number.

UPSC frequently uses specific traps in these problems. Option (A) 45 is the most common pitfall; it represents the arithmetic mean [(40+50)/2], which ignores the fact that the traveler spends more time at the slower speed than the faster one. Option (D) is a classic "data sufficiency" lure designed to make you think the problem is unsolvable without a specific distance. However, as your conceptual training showed, average speed in a round-trip depends only on the ratio of speeds, making the actual mileage irrelevant. Recognizing these traps is a vital part of your exam temperament.