Two trains 130 m and 110 m long are going in the same direction. The faster train takes one minute to pass the other completely. If they are moving in opposite directions, they pass each other completely in 3 seconds. Find the speed of each train?

Options

A. 38 m/s, 36 m/s
B. 42 m/s, 38 m/s
C. 36 m/s, 42 m/s
D. 40 m/s, 36 m/s

Answer

To solve the problem of finding the speeds of two trains, we can follow these steps:

### Step 1: Define Variables
Let the speed of the faster train be \( x \) meters per second and the speed of the slower train be \( y \) meters per second.

### Step 2: Analyze the Scenario When Trains Are in the Same Direction
When the two trains are moving in the same direction, the relative speed is given by:
\[ \text{Relative Speed} = x - y \]
The total distance to be covered when the faster train passes the slower train is the sum of their lengths:
\[ \text{Distance} = 130 \, \text{m} + 110 \, \text{m} = 240 \, \text{m} \]
The time taken to pass completely is 1 minute, which is 60 seconds. Using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
we can write:
\[ 60 = \frac{240}{x - y} \]
Rearranging gives:
\[ x - y = \frac{240}{60} = 4 \quad \text{(Equation 1)} \]

### Step 3: Analyze the Scenario When Trains Are in Opposite Directions
When the two trains are moving in opposite directions, the relative speed is given by:
\[ \text{Relative Speed} = x + y \]
Using the same total distance of 240 m, and knowing they pass each other in 3 seconds, we have:
\[ 3 = \frac{240}{x + y} \]
Rearranging gives:
\[ x + y = \frac{240}{3} = 80 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( x - y = 4 \)
2. \( x + y = 80 \)

We can add these two equations:
\[ (x - y) + (x + y) = 4 + 80 \]
This simplifies to:
\[ 2x = 84 \implies x = \frac{84}{2} = 42 \, \text{m/s} \]

### Step 5: Find the Speed of the Slower Train
Now, substitute \( x = 42 \) into Equation 1:
\[ 42 - y = 4 \]
Rearranging gives:
\[ y = 42 - 4 = 38 \, \text{m/s} \]

### Conclusion
The speeds of the trains are:
- Speed of the faster train \( x = 42 \, \text{m/s} \)
- Speed of the slower train \( y = 38 \, \text{m/s} \)