To find the average speed of the train, we need to use the formula for average speed when the distances are different and the speeds vary. The formula is:

$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$

Step 1: Calculate the total distance

The total distance the train travels is the sum of the two segments:

$\text{Total Distance} = 80 \, \text{km} + 120 \, \text{km} = 200 \, \text{km}$

Step 2: Calculate the time for each segment

We know the time for each segment is given by the formula:

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

For the first segment (80 km at 40 km/h):

$\text{Time}_1 = \frac{80 \, \text{km}}{40 \, \text{km/h}} = 2 \, \text{hours}$

For the second segment (120 km at 60 km/h):

$\text{Time}_2 = \frac{120 \, \text{km}}{60 \, \text{km/h}} = 2 \, \text{hours}$

Step 3: Calculate the total time

The total time is the sum of the times for each segment:

$\text{Total Time} = 2 \, \text{hours} + 2 \, \text{hours} = 4 \, \text{hours}$

Step 4: Calculate the average speed

Now we can calculate the average speed:

$\text{Average Speed} = \frac{200 \, \text{km}}{4 \, \text{hours}} = 50 \, \text{km/h}$

Final Answer:

The average speed of the train is 50 km/h.

Would you like a more detailed explanation or have any questions about this solution? Here are some related questions to explore:

1. How would the average speed change if the speeds were different over the same distances?
2. How do you calculate the total time if the speed for each segment is known?
3. What is the difference between average speed and average velocity?
4. How would the result change if the train had an additional segment with a different speed?
5. How can you calculate average speed when the time taken for each part of the journey is known?

Tip: The key to calculating average speed is to use total distance divided by total time, rather than averaging the speeds directly.