Two trains whose respective lengths are 200 m and 250 m cross each other in 18 s, when they are travelling in opposite direction and in 1 min, when they are travelling in the same direction. What is the speed of the faster train (in km/h)?

Options

A. 38.5
B. 48.5
C. 54
D. 58.5

Answer

To find the speed of the faster train, we can follow these steps:

### Step 1: Define the lengths of the trains
Let the length of Train 1 (L1) be 200 m and the length of Train 2 (L2) be 250 m.

### Step 2: Calculate the relative speed when trains are moving in opposite directions
When the trains cross each other in opposite directions, the relative speed (V1 + V2) can be calculated using the formula:
\[ \text{Relative Speed} = \frac{\text{Total Length}}{\text{Time taken to cross}} \]
Here, the total length is \( L1 + L2 = 200 + 250 = 450 \) m, and the time taken to cross is 18 seconds.
\[ V1 + V2 = \frac{450 \text{ m}}{18 \text{ s}} = 25 \text{ m/s} \]
This gives us our first equation:
\[ (1) \quad V1 + V2 = 25 \text{ m/s} \]

### Step 3: Calculate the relative speed when trains are moving in the same direction
When the trains cross each other in the same direction, the relative speed (V1 - V2) can be calculated similarly. The time taken to cross is 1 minute, which is 60 seconds.
\[ V1 - V2 = \frac{450 \text{ m}}{60 \text{ s}} = 7.5 \text{ m/s} \]
This gives us our second equation:
\[ (2) \quad V1 - V2 = 7.5 \text{ m/s} \]

### Step 4: Solve the equations simultaneously
Now we have two equations:
1. \( V1 + V2 = 25 \)
2. \( V1 - V2 = 7.5 \)

We can add these two equations to eliminate \( V2 \):
\[ (V1 + V2) + (V1 - V2) = 25 + 7.5 \]
This simplifies to:
\[ 2V1 = 32.5 \]
So,
\[ V1 = \frac{32.5}{2} = 16.25 \text{ m/s} \]

### Step 5: Find \( V2 \)
Now, substitute \( V1 \) back into one of the equations to find \( V2 \):
Using equation (1):
\[ 16.25 + V2 = 25 \]
Thus,
\[ V2 = 25 - 16.25 = 8.75 \text{ m/s} \]

### Step 6: Convert the speed of the faster train to km/h
The faster train is \( V1 \). To convert from m/s to km/h, we use the conversion factor \( \frac{18}{5} \):
\[ V1 = 16.25 \times \frac{18}{5} = 16.25 \times 3.6 = 58.5 \text{ km/h} \]

### Final Answer
The speed of the faster train is **58.5 km/h**.

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