We can solve this problem step by step.

Step 1: Calculate the distance covered by the first train before the second train starts.

• Train A starts at 7 a.m. and travels at 80 km/h.
• Train B starts at 9 a.m., meaning Train A has 2 hours of head start.

Distance covered by Train A in 2 hours: $\text{Distance} = \text{Speed} \times \text{Time} = 80 \, \text{km/h} \times 2 \, \text{hours} = 160 \, \text{km}.$

So, Train A is 560 - 160 = 400 km away from B when Train B starts at 9 a.m.

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Step 2: Relative speed of the trains.

• Train A travels towards B at 80 km/h.
• Train B travels towards A at 120 km/h.
• The relative speed of the trains (since they are moving towards each other) is: $\text{Relative Speed} = 80 \, \text{km/h} + 120 \, \text{km/h} = 200 \, \text{km/h}.$

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Step 3: Time taken for the trains to meet.

The trains now need to cover the 400 km gap between them at a combined speed of 200 km/h.

Time to meet: $\text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{400 \, \text{km}}{200 \, \text{km/h}} = 2 \, \text{hours}.$

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Step 4: Find the meeting time.

• Train B starts at 9 a.m., and they meet 2 hours later.
• Therefore, they will meet at: $9 \, \text{a.m.} + 2 \, \text{hours} = 11 \, \text{a.m.}.$

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Final Answer:

The two trains will meet at 11 a.m..

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Would you like a detailed explanation of the concepts used here, or do you have any questions? Here are 5 related problems you can explore:

1. What if the trains started at the same time but with their respective speeds?
2. How far from Station A will they meet?
3. What would happen if Train B traveled at 100 km/h instead of 120 km/h?
4. How long would it take if Train A's speed was reduced to 60 km/h?
5. What is the total distance covered by both trains before meeting?

Tip: When solving problems involving relative speed, always determine whether the objects are moving towards or away from each other, as it affects the combined speed.