A train `100 m` long travelling at `40 ms^(-1)` starts overtaking another train `200 m` long travelling at `30 ms^(-1)`. The time taken by the forst train to pass the second train comletely is .

Options

A. ` 30s`
B. ` 40 s`
C. ` 50 s`
D. ` 60 s`

Explanation

To solve the problem of how long it takes for the first train to completely overtake the second train, we can follow these steps:

### Step 1: Understand the situation
We have two trains:
- Train A (100 m long) traveling at 40 m/s.
- Train B (200 m long) traveling at 30 m/s.

### Step 2: Determine the relative speed
Since the trains are moving in the same direction, we need to calculate the relative speed of Train A with respect to Train B. The relative speed \( v_{relative} \) is given by:

\[ v_{relative} = v_A - v_B \]

Substituting the values:

\[ v_{relative} = 40 \, \text{m/s} - 30 \, \text{m/s} = 10 \, \text{m/s} \]

### Step 3: Calculate the total distance to be covered
For Train A to completely overtake Train B, it needs to cover the length of both trains. Therefore, the total distance \( d \) to be covered is:

\[ d = \text{Length of Train A} + \text{Length of Train B} = 100 \, \text{m} + 200 \, \text{m} = 300 \, \text{m} \]

### Step 4: Use the formula for time
We can use the formula relating distance, speed, and time:

\[ d = v_{relative} \times t \]

Rearranging for time \( t \):

\[ t = \frac{d}{v_{relative}} \]

Substituting the known values:

\[ t = \frac{300 \, \text{m}}{10 \, \text{m/s}} = 30 \, \text{s} \]

### Conclusion
The time taken by the first train to pass the second train completely is **30 seconds**.

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