Two trains each of length 50m. are running with constant speeds on parallel tracks. While moving in same direction one over takes the other in 40 seconds and while moving in opposite direction on ecrosses the other in 20 seonds. The speeds of trains will be:-

Explanation

To solve the problem of two trains overtaking each other, we will follow these steps:

### Step 1: Understand the problem
We have two trains, each of length 50 meters. They are moving on parallel tracks. We need to find their speeds based on the time taken to overtake each other when moving in the same and opposite directions.

### Step 2: Define variables
Let:
- \( v_1 \) = speed of the first train (m/s)
- \( v_2 \) = speed of the second train (m/s)

### Step 3: Analyze the overtaking scenario in the same direction
When the trains are moving in the same direction, the relative speed is given by:
\[ v_{relative} = v_1 - v_2 \]
The total distance to be covered when one train overtakes the other is the sum of their lengths:
\[ \text{Distance} = 50 \, \text{m} + 50 \, \text{m} = 100 \, \text{m} \]
The time taken to overtake is 40 seconds. Therefore, we can write:
\[ 100 = (v_1 - v_2) \times 40 \]
From this, we can derive:
\[ v_1 - v_2 = \frac{100}{40} = 2.5 \, \text{m/s} \quad \text{(Equation 1)} \]

### Step 4: Analyze the overtaking scenario in the opposite direction
When the trains are moving in opposite directions, the relative speed is given by:
\[ v_{relative} = v_1 + v_2 \]
Again, the distance to be covered is 100 m, and the time taken is 20 seconds. Therefore, we can write:
\[ 100 = (v_1 + v_2) \times 20 \]
From this, we can derive:
\[ v_1 + v_2 = \frac{100}{20} = 5 \, \text{m/s} \quad \text{(Equation 2)} \]

### Step 5: Solve the equations
Now we have two equations:
1. \( v_1 - v_2 = 2.5 \)
2. \( v_1 + v_2 = 5 \)

We can solve these equations simultaneously.

Adding both equations:
\[ (v_1 - v_2) + (v_1 + v_2) = 2.5 + 5 \]
\[ 2v_1 = 7.5 \]
\[ v_1 = \frac{7.5}{2} = 3.75 \, \text{m/s} \]

Now, substituting \( v_1 \) back into Equation 2 to find \( v_2 \):
\[ 3.75 + v_2 = 5 \]
\[ v_2 = 5 - 3.75 = 1.25 \, \text{m/s} \]

### Final Answer
The speeds of the trains are:
- Speed of the first train \( v_1 = 3.75 \, \text{m/s} \)
- Speed of the second train \( v_2 = 1.25 \, \text{m/s} \)

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