Two trains started at 7 AM from the same point. The first train travelled north at a speed of 80 km/h and the second train travelled south at a speed of 100 km/h. The time at which they were 540 km apart is ______ AM.

Calculating Train Separation Time

This problem involves calculating the time it takes for two trains, moving in opposite directions, to reach a certain distance apart. We need to determine when they will be 540 km apart, given their speeds and starting time.

Understanding Relative Speed

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. This relative speed represents how quickly the distance between them is increasing.

- Train 1 Speed ($S_1$): $80$ km/h (North)
- Train 2 Speed ($S_2$): $100$ km/h (South)

Calculating Relative Speed

The relative speed ($S_{rel}$) of the two trains is calculated as:

$$S_{rel}=S_1+S_2$$

Substituting the values:

$$S_{rel}=80\text{ km/h}+100\text{ km/h}=180\text{ km/h}$$

This means the distance between the two trains increases by 180 km every hour.

Calculating Time to Reach Separation Distance

We need to find the time ($t$) it takes for the trains to be 540 km apart. We can use the formula:

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

In this case:

$$t=\frac{\text{Total Separation Distance}}{\text{Relative Speed}}$$

$$t=\frac{540\text{ km}}{180\text{ km/h}}$$

$$t=3\text{ hours}$$

Determining the Final Time

The trains started at 7 AM. We need to add the calculated time ($t$) to the starting time:

$$\text{Final Time}=\text{Start Time}+t$$

$$\text{Final Time}=7\text{ AM}+3\text{ hours}$$

$$\text{Final Time}=10\text{ AM}$$

Therefore, the trains will be 540 km apart at 10 AM.