Determine the Speed of a Train When Passing a Running Man

Have you ever wondered how we can calculate the speed of a train that passes a man running in the opposite direction? This article will guide you through a step-by-step process to determine the speed of a train, making it an ideal resource for both students and professionals looking to refine their problem-solving skills in physics or mathematics.

Understanding the Problem

Given a scenario where a train 110 meters long passes a man running at a speed of 6 km/h in the opposite direction in 6 seconds, the goal is to find the speed of the train. This problem requires an understanding of basic physics concepts such as speed, distance, time, and the use of relative speed.

Step-by-Step Solution

Step 1: Convert the Speed of the Man from km/h to m/s

The first step is to convert the speed of the man from kilometers per hour (km/h) to meters per second (m/s).

``` 6 text{ km/h} frac{6 text{ km/h} times 1000 text{ m/km}}{3600 text{ s/h}} frac{6000}{3600} frac{5}{3} text{ m/s} approx 1.67 text{ m/s} ```

Step 2: Calculate the Relative Speed of the Train with Respect to the Man

The relative speed of the train with respect to the man adds the speed of the train to the speed of the man, as they are moving in opposite directions.

``` text{Relative speed} V_t frac{5}{3} text{ m/s} ```

Step 3: Use Distance and Time to Find the Relative Speed

The relative speed can also be determined by using the distance and time formula.

``` text{Relative speed} frac{text{Distance}}{text{Time}} frac{110 text{ m}}{6 text{ s}} approx 18.33 text{ m/s} ```

Step 4: Set Up the Equation

Using the formula: text{Relative speed} V_t frac{5}{3} text{ m/s} 18.33 text{ m/s}

Step 5: Solve for (V_t)

First, convert (frac{5}{3}) to a decimal: (frac{5}{3} approx 1.67).

``` V_t 1.67 18.33 V_t 18.33 - 1.67 approx 16.66 text{ m/s} ```

Step 6: Convert the Speed of the Train Back to km/h

To convert the speed of the train from meters per second to kilometers per hour, multiply by (frac{3600 text{ s/h}}{1000 text{ m/km}}).

``` text{Speed in km/h} 16.66 text{ m/s} times frac{3600 text{ s/h}}{1000 text{ m/km}} approx 59.99 text{ km/h} ```

Therefore, the speed of the train is approximately 60 km/h.

Conclusion and Verification

The problem can also be solved using alternative methods, such as the one provided, which results in a speed of 35 km/h. However, this is likely due to a miscalculation or a different interpretation of the problem. The method detailed above is correct and provides the accurate speed of the train.

Examining Alternative Solutions

Another solution proposes that the width of the person and the length of the train are 110 meters, and the relative velocity of the train with respect to the person is 66 km/h. Further calculations show that the time to cross the man is indeed 6 seconds. This reaffirms the accuracy of the first solution.

Key Takeaways

1. **Conversion of Units**: Understanding the conversion between km/h and m/s is crucial for solving such problems. 2. **Relative Speed**: When two objects are moving in opposite directions, their relative speed is the sum of their individual speeds. 3. **Distance and Time**: The formula (text{Speed} frac{text{Distance}}{text{Time}}) is fundamental in solving such problems.

Conclusion

Mastering the concepts of speed, distance, and time, along with the application of relative speed, can significantly enhance one's problem-solving capabilities in related fields. By practicing such problems, you can develop a deeper understanding of the underlying physics and improve your ability to solve real-world scenarios.