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-by-Step Solution:

1. **Identify the lengths of the trains and their speeds:**
- Length of the first train (Train 1), \( L_1 = 200 \, m \)
- Length of the second train (Train 2), \( L_2 = 300 \, m \)
- Speed of Train 1, \( V_1 = 30 \, m/s \)
- Speed of Train 2, \( V_2 = 20 \, m/s \)

2. **Calculate the relative velocity:**
- Since Train 1 is overtaking Train 2, we need to find the relative velocity of Train 1 with respect to Train 2.
- The formula for relative velocity when two objects are moving in the same direction is:
\[ V_{relative} = V_1 - V_2 \]
- Substituting the values:
\[ V_{relative} = 30 \, m/s - 20 \, m/s = 10 \, m/s \]

3. **Calculate the total distance to be covered:**
- The total distance that Train 1 needs to cover to completely overtake Train 2 is the sum of the lengths of both trains:
\[ \text{Total Distance} = L_1 + L_2 = 200 \, m + 300 \, m = 500 \, m \]

4. **Calculate the time taken to overtake:**
- The time taken to overtake can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Velocity}} \]
- Substituting the values:
\[ \text{Time} = \frac{500 \, m}{10 \, m/s} = 50 \, s \]

5. **Conclusion:**
- Therefore, the time taken by the first train to pass the second train is \( 50 \, seconds \).