Solving a Train Crossing a Man Problem: A Step-by-Step Guide with Various Methods

In this article, we explore the problem of determining the speed of a train based on the time it takes to pass a man running in the opposite direction. We will present multiple methods to solve this problem, ensuring a comprehensive understanding of the underlying concepts.

Problem Statement

A train of length 200 meters takes 12 seconds to cross a man running at a speed of 10 km/hr in the opposite direction. What is the speed of the train?

Method 1: Direct Calculation

Let's assume the speed of the train is a' m/s.

The speed of the man is 10 km/hr 10 * (5/18) m/s 2.7778 m/s. The relative speed of the train with respect to the man is a 2.7778 m/s. In 12 seconds, the train crosses a distance of 200 meters. Hence, the relative speed can be calculated as: Relative speed 200/12 16.67 m/s. Solving for a, we find that a 16.67 - 2.7778 13.8922 m/s. Converting this to km/hr, 13.8922 * (18/5) 50.01 km/hr. Therefore, the speed of the train is approximately 50.01 km/hr.

Method 2: Using Distance and Time

Let the speed of the train be s km/hr.

The relative speed in m/s is s * (5/18) 10/3.6 (5s 10) * (5/18) m/s. The distance covered is the length of the train, which is 200 meters. The time taken is 12 seconds. Equation: Distance Relative speed * Time 200 5s 10 * 10 * 12 * (5/18). Solving this equation, we get s 26 km/hr.

Method 3: Using Speed and Distance Directly

Let the speed of the train be s km/h.

The relative speed of the train with respect to the man, when moving in the same direction, is s - 10 km/h. The speed of the man is converted to km/hr. The distance covered is the length of the train, which is 180 meters. The time taken is 12 seconds. Equation: Relative speed Distance / Time s - 10 180 * 3600 / 12. Solving this equation, we get s 60 km/h.

Method 4: Simplified Relative Speed Calculation

Let the speed of the train be X km/h.

The relative speed of the train with respect to the man is X - 6 km/h (since the man is running in the opposite direction). The distance covered is the length of the train, which is 180 meters (0.18 km). The time taken is 12 seconds (0.0033 hours). Equation: Relative speed Distance / Time X - 6 0.18 * 3600 / 12. Solving this equation, we get X 60 km/h.

Conclusion

Through various methods, we have calculated the speed of the train to be approximately 50 km/hr, 26 km/hr, 60 km/hr, and 60 km/hr. The most consistent and accurate solution is the 60 km/hr variant, as it aligns with both the problem statement and real-world speed limits for a train.

Key Takeaways

Understanding relative speed is crucial in solving such problems. Converting units between km/hr and m/s is essential for accuracy. Adopting different problem-solving methods can provide multiple perspectives and cross-verification of answers.