To determine how long it will take for the two trains to meet, we need to calculate the combined distance they cover per hour and then use that to find the time it takes for them to cover the total distance between the stations.

Let the distance between the two stations be $D = 392$ miles.

The speed of the first train is $v_1 = 75$ miles per hour. The speed of the second train is $v_2 = 65$ miles per hour.

The combined speed of both trains traveling toward each other is: $v_{\text{combined}} = v_1 + v_2 = 75 + 65 = 140 \text{ miles per hour}$

To find the time $t$ it takes for the two trains to meet, we use the formula: $t = \frac{D}{v_{\text{combined}}}$

Substitute the values for $D$ and $v_{\text{combined}}$: $t = \frac{392 \text{ miles}}{140 \text{ miles per hour}}$ $t = \frac{392}{140} = 2.8 \text{ hours}$

Therefore, it will take 2.8 hours for the two trains to meet.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. How far will each train have traveled when they meet?
2. What would be the meeting time if one of the trains had a different speed?
3. How does the distance between stations affect the meeting time?
4. What if the two trains started at different times?
5. How would the calculation change if the trains traveled away from each other instead?
6. What if one train stopped for a certain period during the journey?
7. How can you use relative speed to solve similar problems?
8. What is the importance of understanding relative motion in physics?

Tip: When solving problems involving relative motion, it is often helpful to consider the combined speed of the objects moving toward or away from each other.