A train of length 200 m traveling at `30 ms ^(-1)` overtakes another train of length 300 m traveling at `20 ms ^(-1).` The time taken by the first train to pass the second is

Options

A. 10 s
B. 30 s
C. 40 s
D. 50 s

Answer

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

### Step 1: Understand the Problem
We have two trains:
- Train A (the first train): Length = 200 m, Speed = 30 m/s
- Train B (the second train): Length = 300 m, Speed = 20 m/s

We need to find the time taken by Train A to completely pass Train B.

### Step 2: Calculate the Total Distance to be Covered
When Train A overtakes Train B, it needs to cover the entire length of Train B plus its own length. Therefore, the total distance \( D \) that Train A needs to travel to completely pass Train B is:
\[ D = \text{Length of Train A} + \text{Length of Train B} = 200 \, \text{m} + 300 \, \text{m} = 500 \, \text{m} \]

### Step 3: Determine the Relative Speed
Since both trains are moving in the same direction, the relative speed \( v_{rel} \) of Train A with respect to Train B is given by:
\[ v_{rel} = \text{Speed of Train A} - \text{Speed of Train B} = 30 \, \text{m/s} - 20 \, \text{m/s} = 10 \, \text{m/s} \]

### Step 4: Calculate the Time Taken to Overtake
Using the formula for time, which is given by:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
we can substitute the values we have:
\[ \text{Time} = \frac{D}{v_{rel}} = \frac{500 \, \text{m}}{10 \, \text{m/s}} = 50 \, \text{seconds} \]

### Conclusion
The time taken by the first train to completely pass the second train is **50 seconds**.

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