Two train running at 45 kmph and 54 kmph cross each other in 12 s when they run in opposite directions. When they run in the same direction , a person in the faster train observes that he crosses the other train in 32 s . Find the lengths of the two trains

Answer

To solve the problem of finding the lengths of the two trains, we can break it down into several steps:

### Step 1: Convert Speeds from km/h to m/s
The speeds of the two trains are given as 45 km/h and 54 km/h. To convert these speeds into meters per second (m/s), we use the conversion factor \( \frac{5}{18} \).

- Speed of Train 1 (L1):
\[ 45 \text{ km/h} = 45 \times \frac{5}{18} = 12.5 \text{ m/s} \]

- Speed of Train 2 (L2):
\[ 54 \text{ km/h} = 54 \times \frac{5}{18} = 15 \text{ m/s} \]

### Step 2: Calculate the Combined Speed When Trains Cross Each Other in Opposite Directions
When the trains cross each other in opposite directions, their speeds add up.

- Combined Speed = Speed of Train 1 + Speed of Train 2
\[ \text{Combined Speed} = 12.5 + 15 = 27.5 \text{ m/s} \]

### Step 3: Calculate the Total Length of the Trains
The total length of the two trains can be calculated using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Given that they cross each other in 12 seconds:
\[ \text{Total Length} (L1 + L2) = 27.5 \text{ m/s} \times 12 \text{ s} = 330 \text{ m} \]

### Step 4: Calculate the Relative Speed When Trains Run in the Same Direction
When the trains run in the same direction, the relative speed is the difference between their speeds.

- Relative Speed = Speed of Train 2 - Speed of Train 1
\[ \text{Relative Speed} = 15 - 12.5 = 2.5 \text{ m/s} \]

### Step 5: Calculate the Length of the Slower Train (L2)
The time taken for the faster train to cross the slower train is given as 32 seconds. The distance covered by the faster train in this time is equal to the length of the slower train (L2).

Using the formula:
\[ \text{Distance} = \text{Relative Speed} \times \text{Time} \]
\[ L2 = 2.5 \text{ m/s} \times 32 \text{ s} = 80 \text{ m} \]

### Step 6: Calculate the Length of the Faster Train (L1)
Now that we have the total length of both trains and the length of the slower train, we can find the length of the faster train.

Using the equation from Step 3:
\[ L1 + L2 = 330 \text{ m} \]
Substituting the value of L2:
\[ L1 + 80 = 330 \]
\[ L1 = 330 - 80 = 250 \text{ m} \]

### Final Answer
- Length of Train 1 (L1) = 250 m
- Length of Train 2 (L2) = 80 m