The two trains are approaching each other on parallel tracks with speeds of 75 km/h and 100 km/h, and the initial distance between them is 100 km. We need to find the time $T$ when the distance between them becomes 100 km again after passing each other.

Step 1: Combined Speed of Trains

When two objects move towards each other, their relative speed is the sum of their individual speeds. Therefore, the relative speed of the two trains is:

$\text{Relative speed} = 75 \, \text{km/h} + 100 \, \text{km/h} = 175 \, \text{km/h}$

Step 2: Total Distance Traveled

After passing each other, the total distance traveled by both trains when the distance between them becomes 100 km again is:

• Initially, the distance between them is 100 km.
• After crossing, the trains must travel enough distance so that their separation becomes 100 km again.

This means the total distance traveled by the two trains is: $\text{Total distance} = 100 \, \text{km} + 100 \, \text{km} = 200 \, \text{km}$

Step 3: Time Calculation

Since the two trains are moving towards each other with a relative speed of 175 km/h, the time $T$ required to cover 200 km at this speed is:

$T = \frac{\text{Total distance}}{\text{Relative speed}} = \frac{200 \, \text{km}}{175 \, \text{km/h}} = \frac{200}{175} = \frac{8}{7} \, \text{hours}$

Final Answer:

The time $T$ is $\frac{8}{7}$ hours, or approximately 1.14 hours.

Would you like more details or have any questions?

Here are some related questions for you to consider:

1. How would the solution change if both trains had the same speed?
2. What if the initial distance between the trains was different?
3. How would the problem differ if one train was stationary?
4. Can you calculate how far each train has traveled by the time they meet?
5. How would the solution change if the two trains were traveling in the same direction?

Tip: When objects move towards each other, their relative speed is the sum of their individual speeds!