Solving Train Length: Train X Crosses Man in Train Y
This problem involves calculating the length of Train X based on its speed, the speed of Train Y, and the time it takes for Train X to cross a man standing on Train Y. Both trains are moving in the same direction.
Understanding Relative Speed
When two objects move in the same direction, their relative speed is the difference between their individual speeds. This relative speed is crucial for calculating the time taken to overtake or cross each other.
- Speed of Train X (\(S_X\)): 100 km/hr
- Speed of Train Y (\(S_Y\)): 60 km/hr
- Direction: Same
The relative speed (\(S_{rel}\)) of Train X with respect to the man in Train Y is:
\(S_{rel} = S_X - S_Y\)
Converting Units for Calculation
The speeds are given in kilometers per hour (km/hr), but the time is in seconds (s), and the required length is in meters (m). We need to convert the speeds to meters per second (m/s).
The conversion factor is: \(1 \text{ km/hr} = \frac{5}{18} \text{ m/s}\).
- Converting Speed of Train X:
\(S_X = 100 \text{ km/hr} = 100 \times \frac{5}{18} \text{ m/s} = \frac{500}{18} \text{ m/s} = \frac{250}{9} \text{ m/s}\)
- Converting Speed of Train Y:
\(S_Y = 60 \text{ km/hr} = 60 \times \frac{5}{18} \text{ m/s} = \frac{300}{18} \text{ m/s} = \frac{150}{9} \text{ m/s} = \frac{50}{3} \text{ m/s}\)
Calculating Relative Speed
Now, we calculate the relative speed using the converted values:
\(S_{rel} = S_X - S_Y = \frac{250}{9} \text{ m/s} - \frac{150}{9} \text{ m/s} = \frac{100}{9} \text{ m/s}\)
Calculating the Length of Train X
When Train X crosses a man standing on Train Y, the distance covered by Train X relative to the man is equal to the length of Train X (\(L_X\)). We use the formula: Distance = Speed × Time.
- Time taken (\(t\)): 9 seconds
- Relative Speed (\(S_{rel}\)): \(\frac{100}{9}\) m/s
Therefore, the length of Train X is:
\(L_X = S_{rel} \times t\)
\(L_X = \left( \frac{100}{9} \text{ m/s} \right) \times (9 \text{ s})\)
\(L_X = 100 \text{ m}\)
Conclusion
The length of Train X is calculated to be 100 meters.
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