To solve this, we need to determine the time taken by the slower train to pass the driver of the faster train. The formula for time is:

$\text{Time} = \frac{\text{Distance}}{\text{Relative Speed}}$

Step 1: Convert speeds from km/h to m/s

We know that: $1 \, \text{km/h} = \frac{1000}{3600} = \frac{5}{18} \, \text{m/s}$

So:

• Speed of the faster train = $45 \, \text{km/h} = 45 \times \frac{5}{18} = 12.5 \, \text{m/s}$
• Speed of the slower train = $30 \, \text{km/h} = 30 \times \frac{5}{18} = 8.33 \, \text{m/s}$

Step 2: Determine the relative speed

Since the trains are moving in opposite directions, their relative speed is the sum of their individual speeds: $\text{Relative Speed} = 12.5 \, \text{m/s} + 8.33 \, \text{m/s} = 20.83 \, \text{m/s}$

Step 3: Determine the distance to be covered

The total distance to be covered by the slower train to pass the driver of the faster train is the sum of the lengths of both trains: $\text{Distance} = 500 \, \text{m} + 500 \, \text{m} = 1000 \, \text{m}$

Step 4: Calculate the time taken

Now, using the formula for time: $\text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{1000 \, \text{m}}{20.83 \, \text{m/s}} \approx 48 \, \text{seconds}$

Thus, the time taken by the slower train to pass the driver of the faster train is approximately 48 seconds.

Would you like any further clarification or details on any part of the solution?

Here are some related questions:

1. How do you convert km/h to m/s?
2. What is the formula for relative speed when two objects move in opposite directions?
3. How do you calculate the time taken to pass an object when two objects are moving?
4. What is the total distance to be covered if two objects of equal length are moving in opposite directions?
5. How would the result change if the trains were moving in the same direction?

Tip: When dealing with problems involving relative motion, always remember to determine whether the objects are moving towards each other (sum of speeds) or away from each other (difference of speeds).