A man standing on a railway platform observes that a train going in one direction takes 4 s to pass him. Another train of same length going in the opposite direction takes 5 s to pass him. The time taken (in s) by the two trains to cross each other is

Options

A. 40
B. 40/9
C. 22500
D. 36

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Problem
We have two trains of the same length (let's denote the length of each train as \( L \)). The first train takes 4 seconds to pass a man standing on a platform, and the second train takes 5 seconds to pass him while moving in the opposite direction.

### Step 2: Calculate the Speed of Each Train
1. **For the first train:**
- Time taken to pass the man = 4 seconds
- Speed of the first train \( V_1 = \frac{L}{4} \) (since speed = distance/time)

2. **For the second train:**
- Time taken to pass the man = 5 seconds
- Speed of the second train \( V_2 = \frac{L}{5} \)

### Step 3: Determine the Relative Speed When Both Trains Cross Each Other
When the two trains are moving towards each other, their speeds add up. Therefore, the relative speed \( V_{relative} \) when both trains cross each other is:
\[ V_{relative} = V_1 + V_2 = \frac{L}{4} + \frac{L}{5} \]

### Step 4: Find a Common Denominator and Simplify
To add the speeds, we need a common denominator:
- The least common multiple of 4 and 5 is 20.
- Rewrite the speeds:
\[ V_1 = \frac{L}{4} = \frac{5L}{20} \]
\[ V_2 = \frac{L}{5} = \frac{4L}{20} \]
Now, add them:
\[ V_{relative} = \frac{5L}{20} + \frac{4L}{20} = \frac{9L}{20} \]

### Step 5: Calculate the Time Taken to Cross Each Other
When the two trains cross each other, the total distance they need to cover is the sum of their lengths, which is \( 2L \). The time \( T \) taken to cross each other can be calculated using the formula:
\[ T = \frac{\text{Total Distance}}{\text{Relative Speed}} = \frac{2L}{V_{relative}} = \frac{2L}{\frac{9L}{20}} = \frac{2L \times 20}{9L} = \frac{40}{9} \]

### Step 6: Final Answer
The time taken by the two trains to cross each other is:
\[ T = \frac{40}{9} \text{ seconds} \]