Two men are standing on opposite ends of a bridge of 1200 metres long. If they walk towards each other at the rate of 5 m/minute and 10 m/ minute respectively, in how much time will they meet each other?

1. Fundamentals of Motion: Distance, Time, and Speed (basic)

At the heart of every quantitative aptitude problem involving movement lies the fundamental relationship between Distance, Time, and Speed. Speed is essentially the "rate" at which an object covers distance. As defined in Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.113, we call the distance covered by an object in a unit time (such as one second, one minute, or one hour) as the speed of that object. The mathematical relationship is expressed as:

Speed = Total Distance Covered / Total Time Taken

In the real world, motion is rarely perfectly steady. If an object moves along a straight line and its speed keeps changing—like a train pulling out of a station and gradually accelerating—it is said to be in non-uniform linear motion. Conversely, if it maintains a constant speed over a straight path, it is in uniform linear motion (Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.117). In most competitive exams, we use the average speed to simplify these real-world variations into a single workable number.

A vital concept to master early on is Relative Speed. This describes how fast two objects are moving in relation to each other. When two objects move in opposite directions (towards each other), they are closing the distance between them much faster than a single object would. In this scenario, their Relative Speed is the sum of their individual speeds. For example, if two friends are walking toward each other from opposite ends of a 1200-metre path, one at 5 m/min and the other at 10 m/min, they are effectively approaching each other at 15 m/min. To find the time they meet, you simply divide the total distance by this combined speed.

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.117-118

2. The Concept of Average Speed (basic)

In the real world, objects rarely move at a perfectly constant pace. Whether it is a bus navigating city traffic or an athlete running a marathon, their speed fluctuates—speeding up, slowing down, or even stopping briefly. To describe such motion simply, we use the concept of Average Speed. It represents a single, uniform speed that would allow an object to cover the same total distance in the same total amount of time. As defined in Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113, speed is the distance covered by an object in a unit of time, but for journeys with varying speeds, we rely on the total figures. To calculate average speed, we do not simply look at the speedometer at one moment. Instead, we use the fundamental formula: Average Speed = Total Distance Covered ÷ Total Time Taken. This is a crucial distinction for competitive exams; students often make the mistake of simply averaging the different speeds (arithmetic mean), which is mathematically incorrect unless the time spent at each speed is exactly the same. Even if a vehicle remains stationary for a period, that 'stop time' must be included in the total time because the journey is still ongoing Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.115. Understanding the relationship between these variables allows us to derive any one if the other two are known. For instance, if you know the average speed and the time taken, you can find the distance by multiplying them (Distance = Speed × Time). This principle is universal, applying to everything from a walking person to the speed of light passing through different media like glass or water Science , class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.159.

Type of Motion	Description	Speed Characteristic
Uniform Motion	Moving along a straight line at a constant speed.	Actual speed = Average speed at all times.
Non-Uniform Motion	Speed changes during the journey.	Average speed is a representative value for the whole trip.

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113, 115, 119; Science , class X (NCERT 2025 ed.), Chapter 9: Light – Reflection and Refraction, p.159

3. Scalar vs Vector: Understanding Velocity (intermediate)

To master quantitative aptitude, we must first distinguish between Scalar and Vector quantities. A scalar quantity, like speed, is defined solely by its magnitude (how much). We calculate it by dividing the total distance covered by the total time taken Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113. For instance, India's planned High-Speed Rail corridors are designed for speeds of 200 km/h to 350 km/h Indian Economy, Vivek Singh (7th ed. 2023-24), Infrastructure and Investment Models, p.412. This number tells us how fast the train moves, but not where it is going. When an object moves at a constant speed in a straight line, it is in uniform linear motion Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.117.

Velocity, however, is a vector quantity. It includes both speed and direction. In competitive exams, this distinction becomes vital when dealing with Relative Speed. When two objects move towards each other from opposite directions, their velocities are working together to close the gap. To find the 'combined' or relative speed, we add their individual speeds. If Object A moves at 5 m/min and Object B moves at 10 m/min towards it, they effectively close the distance at 15 m/min. Conversely, if they move in the same direction, we subtract the speeds to find how quickly one is gaining on the other.

Feature	Speed (Scalar)	Velocity (Vector)
Definition	Distance covered per unit time.	Displacement per unit time in a specific direction.
Components	Magnitude only.	Magnitude + Direction.
Formula	Distance ÷ Time	Displacement ÷ Time

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.117; Indian Economy, Vivek Singh (7th ed. 2023-24), Infrastructure and Investment Models, p.412

4. Relative Motion in the Same Direction (intermediate)

When we talk about motion in the real world, speed is rarely absolute; it is almost always relative to something else. In the context of quantitative aptitude, Relative Motion in the Same Direction refers to the scenario where two objects are moving along the same path toward the same destination. Imagine you are on a train moving at 60 km/h and another train on a parallel track overtakes you at 80 km/h. To an observer standing on the ground, the second train is moving very fast. However, to you, sitting in your seat, the second train appears to be slowly drifting past at a speed of only 20 km/h. This "perceived speed" is the relative speed.

The mathematical principle is straightforward: when two bodies move in the same direction, their relative speed is the difference between their individual speeds. If Object A moves at speed v₁ and Object B moves at speed v₂ (where v₁ > v₂), the relative speed is (v₁ - v₂). This concept is the foundation for solving "chase" or "overtaking" problems. As noted in fundamental physics, speed is defined as the distance covered per unit of time Science-Class VII, Chapter 8, p.113. When one object chases another, the distance between them is reduced only by this difference in speed each hour or second.

To master these problems, you must follow two critical steps:

• Unit Consistency: Always ensure that the speeds and distances are in the same units (e.g., convert km/h to m/s if the distance is in metres) Science-Class VII, Chapter 8, p.118.
• The "Meeting Time" Formula: The time taken for the faster object to catch the slower one is calculated as:
  Time = (Initial Distance between them) / (Relative Speed).

Sources: Science-Class VII, Measurement of Time and Motion, p.113; Science-Class VII, Measurement of Time and Motion, p.118

5. Train Problems: Crossing Platforms and Poles (intermediate)

When solving train-related problems in aptitude tests, the most critical step is correctly identifying the total distance the train must travel. In basic kinematics, we use the formula Distance = Speed × Time Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115. However, unlike a small car or a person, a train has a significant length that cannot be ignored. The "distance" changes depending on whether the train is crossing a stationary point or an extended object.

To master these problems, you must distinguish between two primary scenarios:

Scenario	Object Being Crossed	Distance Calculation
Point Object	Electric Pole, Signal Post, Stationary Man	Distance = Length of the Train (Lₜ)
Extended Object	Platform, Bridge, Tunnel, Another Train	Distance = Lₜ + Length of the Object (Lₒ)

As noted in studies of uniform linear motion, an object is considered to have crossed another only when its entire length has passed the reference point Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117. For a bridge, this means the front of the engine enters the bridge, but the "crossing" is only complete when the last wagon exits the other side.

When two objects are in motion simultaneously—such as two trains or a train and a moving person—we apply the principle of Relative Speed. If they move towards each other (opposite directions), we add their speeds (S₁ + S₂) because the gap between them closes faster. If they move in the same direction, we subtract the slower speed from the faster one (S₁ - S₂) to find the effective speed at which the distance is being covered Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113.

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

6. Relative Motion in Opposite Directions (exam-level)

When we talk about Relative Motion, we are essentially looking at how one moving object perceives another. In the context of competitive exams like the UPSC CSAT, the most common scenario involves two objects moving in opposite directions (either towards each other or away from each other). From first principles, if you are walking towards a friend at 5 km/h and they are walking towards you at 5 km/h, the distance between you doesn't just shrink by your speed; it shrinks by the combined effort of both your movements. This is why the Relative Speed in opposite directions is always the sum of the individual speeds (V₁ + V₂).

To calculate the time it takes for two objects to meet, we rely on the fundamental relationship: Time = Distance ÷ Speed. Since both objects are contributing to covering the initial gap between them, we divide the total initial distance by the relative speed. As noted in basic kinematics, speed is the distance covered per unit of time Science-Class VII, Measurement of Time and Motion, p.115. If two people are at opposite ends of a 1200-metre bridge, moving at 5 m/min and 10 m/min respectively, they are effectively "closing the gap" at a rate of 15 metres every single minute. Therefore, the time to meet is simply 1200 / 15 = 80 minutes.

Scenario	Relative Speed Formula	Effect on Distance
Moving Towards Each Other	Speed A + Speed B	Distance decreases rapidly until they meet.
Moving Away From Each Other	Speed A + Speed B	Distance increases rapidly as they separate.

Sources: Science-Class VII, Measurement of Time and Motion, p.115

7. Solving the Bridge Meeting Point Problem (exam-level)

Now that you have mastered the fundamental relationship between Distance, Speed, and Time, this question serves as the perfect application of Relative Speed. In the concepts we covered, you learned that when two objects move toward one another, they are effectively "working together" to close the gap between them. Instead of viewing this as two separate journeys, we combine their efforts into a single Relative Speed. As established in Science-Class VII . NCERT(Revised ed 2025), speed is simply the distance covered per unit of time; here, that distance is the 1200-metre bridge being closed from both ends simultaneously.

To solve this like a seasoned aspirant, first identify the combined rate at which the gap is shrinking. By adding the two individual speeds (5 m/min + 10 m/min), we find a Relative Speed of 15 m/min. Next, apply the core formula: Time = Distance / Speed. By dividing the total distance of 1200 metres by the combined rate of 15 m/min, the calculation yields 80 minutes. This logical progression—moving from individual components to a collective rate—is the most efficient way to navigate UPSC CSAT problems involving convergence.

UPSC often includes distractor options to catch students who rush their logic or make simple arithmetic slips. For example, options like 60 or 90 minutes are common traps for those who might perform hasty division or incorrectly apply the speed of only one person. If a student mistakenly subtracted the speeds (thinking the men were moving in the same direction), they would calculate a much slower closure rate. Always remember: when objects move towards each other, their speeds must be added to find the time of meeting. This ensures you confidently arrive at the correct answer: (B) 80 minutes.