Speed and Time Calculation: How Fast Does a Train Pass a Runner?

Understanding the principles of relative speed and how to apply them in practical scenarios is a fascinating aspect of physics. This article delves into a real-world problem: calculating the time it takes for a train to pass a runner. This kind of calculation often comes up in various applications like traffic management, engineering, or even in competitive exams. We will walk through the steps to solve this problem and explain the underlying concepts.

Problem Statement

A train 125 meters long is running at a speed of 45 km/h in the same direction in which a man who is 175 meters ahead of the engine of the train is running at a speed of 15 km/h. The question is, in how much time will the train pass the man?

Solution

Algebraic Approach

First, we convert all speeds to meters per second (m/s) for consistency.

Speed of the train: 45 km/h 45 * (1000/3600) m/s 12.5 m/s Speed of the man: 15 km/h 15 * (1000/3600) m/s 4.17 m/s Relative speed of the train with respect to the man: 12.5 m/s - 4.17 m/s 8.33 m/s

Now, the distance the train needs to cover to pass the man is the length of the train plus the distance between the man and the engine (125 m 175 m 300 m).

The time taken to pass the man is given by:

Time Distance / Relative Speed

T 300 m / 8.33 m/s 36 seconds

So, the train will pass the man in 36 seconds.

Step-by-Step Calculation

Convert the speed of the train and the man to m/s. Calculate the relative speed by subtracting the speed of the man from the speed of the train. Add the length of the train and the distance the man is from the engine to get the total distance. Use the formula Time Distance / Relative Speed to find the time taken to pass the man.

This step-by-step approach ensures that we account for all variables correctly and arrive at an accurate solution. Let's look at the calculation in detail:

Calculation Details

Convert the speed of the train to m/s: 45 km/h 45 * (1000/3600) 12.5 m/s Convert the speed of the man to m/s: 15 km/h 15 * (1000/3600) 4.17 m/s The relative speed is: 12.5 m/s - 4.17 m/s 8.33 m/s Total distance to be covered: 125 m 175 m 300 m Time taken to pass the man: 300 m / 8.33 m/s 36 seconds

Conclusion

The time taken for the train to pass the man is 36 seconds. This step-by-step method can be applied to a wide range of similar problems, making it a valuable skill to master. Whether you are studying for an exam or dealing with real-world engineering challenges, understanding relative speed and its application is crucial.

For those interested in more such problems and solutions, check out our relative speed, train length, and man running speed resources. Stay tuned for more related problems and solutions!