A 180m. long train crosses another train of length 270 m in 10.8 seconds by running towards each other. If T the ratio of speed of the first train to second train is 2:3. Then find the time taken by `2^("nd")` train to cross first train if both run in the same direction.

Options

A. A)47
B. B)26
C. C)54
D. D)50

Answer

To solve the problem step by step, we will analyze the situation where two trains cross each other both when they are moving towards each other and when they are moving in the same direction.

### Step 1: Understand the problem
We have two trains:
- Train A: Length = 180 m
- Train B: Length = 270 m

They cross each other in 10.8 seconds when moving towards each other. The ratio of their speeds is given as 2:3.

### Step 2: Calculate the total distance when trains cross each other
When the two trains cross each other while moving towards each other, the total distance covered is the sum of their lengths:
\[ \text{Total Distance} = \text{Length of Train A} + \text{Length of Train B} = 180 \, \text{m} + 270 \, \text{m} = 450 \, \text{m} \]

**Hint:** Remember that when two objects move towards each other, their speeds add up.

### Step 3: Calculate the combined speed of the trains
Let the speed of Train A be \( 2x \) and the speed of Train B be \( 3x \). Therefore, the combined speed when they are moving towards each other is:
\[ \text{Combined Speed} = 2x + 3x = 5x \]

### Step 4: Use the time taken to find the speed
We know the time taken to cross each other is 10.8 seconds. Using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
we can write:
\[ 450 = 5x \times 10.8 \]

### Step 5: Solve for \( x \)
Now, rearranging the equation to solve for \( x \):
\[ 5x = \frac{450}{10.8} \]
\[ x = \frac{450}{5 \times 10.8} \]
\[ x = \frac{450}{54} \]
\[ x = 8.33 \, \text{m/s} \]

### Step 6: Calculate the speeds of each train
Now we can find the speeds of each train:
- Speed of Train A: \( 2x = 2 \times 8.33 = 16.66 \, \text{m/s} \)
- Speed of Train B: \( 3x = 3 \times 8.33 = 25 \, \text{m/s} \)

### Step 7: Calculate the time taken for Train B to cross Train A when moving in the same direction
When the trains are moving in the same direction, the relative speed is the difference of their speeds:
\[ \text{Relative Speed} = 25 - 16.66 = 8.34 \, \text{m/s} \]

The total distance to be covered when Train B crosses Train A is the length of Train B:
\[ \text{Distance} = 270 \, \text{m} \]

Using the formula again:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} \]
\[ \text{Time} = \frac{270}{8.34} \]

### Step 8: Calculate the final time
Calculating the time:
\[ \text{Time} \approx 32.4 \, \text{seconds} \]

### Conclusion
Thus, the time taken by the second train to cross the first train when both run in the same direction is approximately **32.4 seconds**.

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