Two trains are running on parallel lines in the same direction at a speed of 50 km. and 30 km per hour respectively. The faster train crosses a man in slower train in 18 seconds. The length of the faster train is:

Options

A. 98 metres
B. `7(1)/(5)` metres
C. 100 metres
D. 85 metres

Answer

To solve the problem of finding the length of the faster train, we can follow these steps:

### Step 1: Determine the speeds of the trains
The speeds of the two trains are given as:
- Speed of the faster train = 50 km/h
- Speed of the slower train = 30 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:
\[ \text{Relative Speed} = \text{Speed of Faster Train} - \text{Speed of Slower Train} = 50 \text{ km/h} - 30 \text{ km/h} = 20 \text{ km/h} \]

### Step 3: Convert the relative speed from km/h to m/s
To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \):
\[ \text{Relative Speed in m/s} = 20 \text{ km/h} \times \frac{5}{18} = \frac{100}{18} \text{ m/s} \approx 5.56 \text{ m/s} \]

### Step 4: Use the formula for distance
The distance covered by the faster train while crossing the man in the slower train is equal to the length of the faster train. The formula for distance is:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Here, the time taken to cross the man is given as 18 seconds.

### Step 5: Calculate the length of the faster train
Substituting the values into the distance formula:
\[ \text{Length of Faster Train} = \text{Relative Speed} \times \text{Time} = 5.56 \text{ m/s} \times 18 \text{ s} = 100 \text{ m} \]

### Final Answer
The length of the faster train is **100 meters**.

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