Solution:

1. Part 1: Overtaking a train moving in the same direction

• Speed of the first train = $96 \, \text{km/h}$
• Speed of the second train = $72 \, \text{km/h}$
• Length of the second train = $144 \, \text{m}$
• Time taken to overtake the second train = $30 \, \text{seconds}$

To find the length of the first train, we first determine the relative speed between the two trains (since they are moving in the same direction).

Step 1: Calculate relative speed

• Relative speed = Speed of first train – Speed of second train
• Convert the speeds into meters per second:
  • Speed of first train:
    $96 \, \text{km/h} = \frac{96 \times 1000}{3600} = 26.67 \, \text{m/s}$
  • Speed of second train:
    $72 \, \text{km/h} = \frac{72 \times 1000}{3600} = 20 \, \text{m/s}$
• Relative speed:
  $26.67 \, \text{m/s} - 20 \, \text{m/s} = 6.67 \, \text{m/s}$

Step 2: Calculate the distance covered during overtaking

• The total distance covered by the first train to overtake the second train is the length of the second train plus the length of the first train. Let the length of the first train be $L$.
• Distance covered = $L + 144 \, \text{m}$
• Using the formula $\text{Distance} = \text{Relative speed} \times \text{Time}$: $L + 144 = 6.67 \times 30$ $L + 144 = 200.1$ $L = 200.1 - 144 = 56.1 \, \text{m}$ Thus, the length of the first train is approximately 56.1 m.

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2. Part 2: Passing a train in the opposite direction

Now, the first train is to pass another train of the same length, but this time the trains are moving in opposite directions. The speed of the second train is $48 \, \text{km/h}$.

Step 3: Calculate the relative speed

• Speed of the second train:
  $48 \, \text{km/h} = \frac{48 \times 1000}{3600} = 13.33 \, \text{m/s}$
• Since the trains are moving in opposite directions, we add their speeds: $\text{Relative speed} = 26.67 \, \text{m/s} + 13.33 \, \text{m/s} = 40 \, \text{m/s}$

Step 4: Calculate the time taken to pass the second train

• The total distance to be covered is the length of both trains:
  $\text{Total distance} = 56.1 + 56.1 = 112.2 \, \text{m}$
• Using the formula $\text{Time} = \frac{\text{Distance}}{\text{Relative speed}}$: $\text{Time} = \frac{112.2}{40} = 2.805 \, \text{seconds}$

Thus, the first train will pass the second train in approximately 2.81 seconds.

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Would you like more details or explanations? Here are five related questions for further exploration:

1. How would the result change if both trains were running at the same speed?
2. How is the time affected if the second train was longer or shorter?
3. What is the concept of relative speed and how is it applied in such problems?
4. Can you generalize a formula for overtaking when two trains have different speeds and lengths?
5. What would happen if both trains were moving in the same direction but the second train was much faster?

Tip: When trains move in opposite directions, their relative speed is the sum of their individual speeds.