To solve this problem, let's use the relative speed concept and some basic equations of motion.

Problem Details:

• Distance between cities $A$ and $B$: $500 \, \text{km}$
• Train from $A$ to $B$:
  • Speed: $70 \, \text{km/hr}$
  • Start time: $8:00 \, \text{am}$
• Train from $B$ to $A$:
  • Speed: $110 \, \text{km/hr}$
  • Start time: $10:00 \, \text{am}$

We need to find the time at which the two trains meet.

Solution:

1. Calculate the distance covered by Train 1 (from A to B) by 10:00 am:

   • Time elapsed: $10:00 \, \text{am} - 8:00 \, \text{am} = 2 \, \text{hours}$
   • Distance covered: $70 \, \text{km/hr} \times 2 \, \text{hours} = 140 \, \text{km}$

   Remaining distance between the trains at 10:00 am:
   $500 \, \text{km} - 140 \, \text{km} = 360 \, \text{km}$

2. Relative speed of the two trains: When two objects move toward each other, their relative speed is the sum of their individual speeds: $70 \, \text{km/hr} + 110 \, \text{km/hr} = 180 \, \text{km/hr}$

3. Time taken to meet: Time $t$ is given by: $t = \frac{\text{Remaining distance}}{\text{Relative speed}} = \frac{360 \, \text{km}}{180 \, \text{km/hr}} = 2 \, \text{hours}$

   So, the trains meet $2 \, \text{hours}$ after $10:00 \, \text{am}$, i.e., at 12:00 pm.

Final Answer:

The two trains meet at 12:00 pm. The correct option is (2) 12 pm.

Would you like a more detailed explanation or additional examples?

Related Questions:

1. How does the relative speed concept apply to trains moving in the same direction?
2. What is the effect of changing the speed of one of the trains on their meeting time?
3. How would the solution change if one train started later than 10:00 am?
4. What if the trains were traveling in opposite directions (away from each other)?
5. How do you calculate the distance covered by each train individually before they meet?

Tip:

When solving relative motion problems, always start by determining whether the objects are moving toward or away from each other to choose the correct relative speed formula.