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.
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.
The relative speed (\(S_{rel}\)) of Train X with respect to the man in Train Y is:
\(S_{rel} = S_X - S_Y\)
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: \(1 \text{ km/hr} = \frac{5}{18} \text{ m/s}\).
\(S_X = 100 \text{ km/hr} = 100 \times \frac{5}{18} \text{ m/s} = \frac{500}{18} \text{ m/s} = \frac{250}{9} \text{ m/s}\)
\(S_Y = 60 \text{ km/hr} = 60 \times \frac{5}{18} \text{ m/s} = \frac{300}{18} \text{ m/s} = \frac{150}{9} \text{ m/s} = \frac{50}{3} \text{ m/s}\)
Now, we calculate the relative speed using the converted values:
\(S_{rel} = S_X - S_Y = \frac{250}{9} \text{ m/s} - \frac{150}{9} \text{ m/s} = \frac{100}{9} \text{ m/s}\)
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.
Therefore, the length of Train X is:
\(L_X = S_{rel} \times t\)
\(L_X = \left( \frac{100}{9} \text{ m/s} \right) \times (9 \text{ s})\)
\(L_X = 100 \text{ m}\)
The length of Train X is calculated to be 100 meters.
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