Two trains are travelling in the same direction at 22.5 km/h and 7.5 km/h respectively. The faster train crosses a man in the slower train in 18 seconds. What is the length of the faster train?

Options

A. 87.5 m
B. 75 m
C. 122.5 m
D. none of these

Answer

To solve the problem step by step, we need to find the length of the faster train based on the given speeds and the time taken to cross a man in the slower train.

### Step 1: Identify the speeds of the trains
- Speed of the faster train = 22.5 km/h
- Speed of the slower train = 7.5 km/h

### Step 2: Calculate the relative speed
Since both trains are moving in the same direction, the relative speed is calculated by subtracting the speed of the slower train from the speed of the faster train.

\[ \text{Relative Speed} = \text{Speed of Faster Train} - \text{Speed of Slower Train} \]

\[ \text{Relative Speed} = 22.5 \text{ km/h} - 7.5 \text{ km/h} = 15 \text{ km/h} \]

### Step 3: Convert the relative speed from km/h to m/s
To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \).

\[ \text{Relative Speed in m/s} = 15 \text{ km/h} \times \frac{5}{18} = \frac{75}{18} \text{ m/s} \approx 4.17 \text{ m/s} \]

### Step 4: Use the time taken to cross the man
The time taken to cross the man in the slower train is given as 18 seconds.

### Step 5: Calculate the distance (length of the faster train)
The distance (length of the faster train) can be calculated using the formula:

\[ \text{Distance} = \text{Speed} \times \text{Time} \]

Substituting the values we have:

\[ \text{Distance} = \text{Relative Speed in m/s} \times \text{Time} \]

\[ \text{Distance} = 4.17 \text{ m/s} \times 18 \text{ s} = 75 \text{ meters} \]

### Conclusion
The length of the faster train is **75 meters**.

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