A train travelling at 48 kmph completely crosses another train having half of its length and travelling in opposite direction at 42 kmph , in 12 seconds. It also passes a railway platform in 45 seconds. The length of the platform is :

Correct option is A

Given:

1. The speed of the first train = 48 km/h.
2. The speed of the second train = 42 km/h.
3. The length of the second train = half the length of the first train.
4. The first train completely crosses the second train in 12 seconds.
5. The first train passes a platform in 45 seconds.
6. We need to find the length of the platform.

Formula Used:

1. Speed Conversion:

$$\text{Speed (in m/s)} = \text{Speed (in km/h)} \times \frac{5}{18}$$​​

2. Time Relation:

$$\text{Relative Distance} = \text{Relative Speed} \times \text{Time}$$​​

3. Passing a Platform:

$$\text{Length of Train} + \text{Length of Platform} = \text{Speed of Train} \times \text{Time}$$​​

Solution:

1. Convert speeds to m/s:
- Speed of the first train:

$$\text{Speed of Train 1} = 48 \times \frac{5}{18} = 13.33 \, \text{m/s}$$​​

- Speed of the second train:

$$\text{Speed of Train 2} = 42 \times \frac{5}{18} = 11.67 \, \text{m/s}$$​​

2. Relative speed of the two trains (opposite directions):

$$\text{Relative Speed} = 13.33 + 11.67 = 25 \, \text{m/s}$$​​

3. Relative distance when crossing each other:
Let the length of the first train be L meters. Then, the length of the second train is $$\frac{L}{2}$$​ .
Total distance = $$L + \frac{L}{2} = \frac{3L}{2}$$​​

Using the formula:

$$\frac{3L}{2} = 25 \times 12$$​​

Simplify:

$$\frac{3L}{2} = 300 \implies L = \frac{300 \times 2}{3} = 200 \, \text{m}$$​​

Thus, the length of the first train is 200 m

4. Passing the platform:
The total distance covered when passing the platform = Length of Train + Length of Platform.
Using the formula:

$$L + \text{Platform Length} = \text{Speed of Train 1} \times \text{Time}$$​​

Substitute values:

$$200 + \text{Platform Length} = 13.33 \times 45$$​​

Simplify:

$$200 + \text{Platform Length} = 600$$​​

​$$\text{Platform Length} = 600 - 200 = 400 \, \text{m}$$​​

Final Answer:

The length of the platform is 400 meters
**Option A: 400 m**