Train and Platform Length Calculation

This problem involves calculating the length of a train and a platform using concepts of speed, time, and distance, including relative speed.

Convert Speeds to Meters per Second (m/s)

First, convert the given speeds from kilometers per hour (km/h) to meters per second (m/s) because the time is given in seconds. The conversion factor is $1 \, \text{km/h} = \frac{5}{18} \, \text{m/s}$.

• Train speed: $54 \, \text{km/h} = 54 \times \frac{5}{18} \, \text{m/s} = 3 \times 5 \, \text{m/s} = 15 \, \text{m/s}$
• Man's speed: $9 \, \text{km/h} = 9 \times \frac{5}{18} \, \text{m/s} = \frac{1}{2} \times 5 \, \text{m/s} = 2.5 \, \text{m/s}$

Calculate Train Length

When a train passes a man running in the same direction, the relative speed is the difference between their speeds. The distance the train covers relative to the man is its own length ($L_{\text{train}}$).

• Relative speed = Train speed - Man's speed $ \text{Relative Speed} = 15 \, \text{m/s} - 2.5 \, \text{m/s} = 12.5 \, \text{m/s} $
• The time taken to pass the man is 14 seconds.
• Distance = Speed $\times$ Time $ L_{\text{train}} = 12.5 \, \text{m/s} \times 14 \, \text{s} $ $ L_{\text{train}} = 175 \, \text{m} $

The length of the train is 175 meters.

Calculate Platform Length

When a train passes a platform, the total distance covered is the sum of the train's length ($L_{\text{train}}$) and the platform's length ($L_{\text{platform}}$). The train travels at its own constant speed.

• Train speed = $15 \, \text{m/s}$
• Time taken to pass the platform is 25 seconds.
• Total Distance = Train speed $\times$ Time $ L_{\text{train}} + L_{\text{platform}} = 15 \, \text{m/s} \times 25 \, \text{s} $ $ L_{\text{train}} + L_{\text{platform}} = 375 \, \text{m} $
• Substitute the calculated train length ($L_{\text{train}} = 175 \, \text{m}$): $ 175 \, \text{m} + L_{\text{platform}} = 375 \, \text{m} $ $ L_{\text{platform}} = 375 \, \text{m} - 175 \, \text{m} $ $ L_{\text{platform}} = 200 \, \text{m} $

The length of the platform is 200 meters.

Final Answer

The length of the train is 175 meters and the length of the platform is 200 meters.