Let’s calculate when the two trains meet.

Given:

1. Distance between stations $A$ and $B$: $560 \, \text{km}$.
2. Train 1 (from $A$):
   • Starts at $7:00 \, \text{a.m.}$,
   • Speed: $80 \, \text{km/h}$.
3. Train 2 (from $B$):
   • Starts at $9:00 \, \text{a.m.}$,
   • Speed: $100 \, \text{km/h}$.

Step-by-Step Solution:

1. Distance traveled by Train 1 by 9:00 a.m.

Train 1 travels for 2 hours before Train 2 starts.
$\text{Distance traveled by Train 1} = \text{Speed} \times \text{Time} = 80 \times 2 = 160 \, \text{km}.$ So, at 9:00 a.m., Train 1 is $560 - 160 = 400 \, \text{km}$ away from $B$.

2. Relative speed of the two trains:

Both trains are traveling towards each other.
$\text{Relative speed} = 80 + 100 = 180 \, \text{km/h}.$

3. Time taken for the trains to meet:

The distance between the two trains at 9:00 a.m. is $400 \, \text{km}$. Using the relative speed: $\text{Time to meet} = \frac{\text{Distance}}{\text{Relative speed}} = \frac{400}{180} \approx 2.22 \, \text{hours}.$

4. Meeting time:

Since Train 2 starts at $9:00 \, \text{a.m.}$ and it takes approximately $2.22 \, \text{hours}$ to meet: $\text{Meeting time} = 9:00 + 2.22 \, \text{hours} \approx 11:13 \, \text{a.m.}.$

Final Answer:

The two trains will meet at approximately 11:13 a.m..

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Do you want detailed diagrams or further clarifications?
Here are 5 related questions you might explore:

1. What would happen if one train's speed increased by 20 km/h?
2. How would the meeting time change if Train 1 started earlier or later?
3. Can you calculate the distance each train travels before meeting?
4. How does the solution change if both trains started at the same time?
5. What if the distance between $A$ and $B$ was 700 km instead?

Tip: When solving relative motion problems, always break them into individual components (speed, time, distance).