A train travelling at 48 km/h completely crosses an another train having half length of first train and travelling in opposite directions at 42km/h in 12s.It also passes are ailway platfor min 45 s. The length of the platform is

Options

A. 400 m
B. 450 m
C. 560 m
D. 600 m

Answer

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Understand the problem
We have two trains:
- Train A is traveling at 48 km/h and has a length of 2L.
- Train B is traveling at 42 km/h and has a length of L (half of Train A).
- They cross each other in 12 seconds.

### Step 2: Convert speeds from km/h to m/s
To work with the time in seconds, we need to convert the speeds from km/h to m/s using the conversion factor \( \frac{5}{18} \).

- Speed of Train A in m/s:
\[ 48 \text{ km/h} = 48 \times \frac{5}{18} = \frac{240}{18} = 13.33 \text{ m/s} \]

- Speed of Train B in m/s:
\[ 42 \text{ km/h} = 42 \times \frac{5}{18} = \frac{210}{18} = 11.67 \text{ m/s} \]

### Step 3: Calculate the relative speed
Since the trains are moving in opposite directions, their speeds add up.

\[ \text{Relative speed} = 13.33 + 11.67 = 25 \text{ m/s} \]

### Step 4: Calculate the total distance covered when crossing each other
The total distance covered when both trains cross each other is the sum of their lengths.

Let the length of Train A be \( 2L \) and the length of Train B be \( L \). Thus, the total distance is:
\[ \text{Distance} = 2L + L = 3L \]

### Step 5: Use the time taken to cross each other to find L
The time taken to cross each other is 12 seconds.

Using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
we can write:
\[ 3L = 25 \text{ m/s} \times 12 \text{ s} \]
\[ 3L = 300 \text{ m} \]
\[ L = 100 \text{ m} \]

### Step 6: Calculate the length of Train A
Since the length of Train A is \( 2L \):
\[ \text{Length of Train A} = 2 \times 100 = 200 \text{ m} \]

### Step 7: Calculate the length of the platform
Now, we know the length of Train A (200 m). The train also crosses a platform in 45 seconds.

Using the speed of Train A (in m/s):
\[ \text{Speed of Train A} = 13.33 \text{ m/s} \]

The distance covered while crossing the platform is:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 13.33 \text{ m/s} \times 45 \text{ s} = 600 \text{ m} \]

This distance includes the length of Train A and the length of the platform:
\[ \text{Distance} = \text{Length of Train A} + \text{Length of Platform} \]
\[ 600 = 200 + \text{Length of Platform} \]
\[ \text{Length of Platform} = 600 - 200 = 400 \text{ m} \]

### Final Answer:
The length of the platform is **400 meters**.

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