Train Meeting Time Calculation

This problem involves calculating the time when two trains moving towards each other will meet. We need to account for their different starting times and speeds.

Calculate Head Start Distance

Train A starts at 7 a.m. and Train B starts at 8 a.m. In the one hour between 7 a.m. and 8 a.m., Train A travels a certain distance.

• Speed of Train A: $35 \text{ km/h}$
• Time duration: $8 \text{ a.m.} - 7 \text{ a.m.} = 1 \text{ hour}$
• Distance covered by Train A in 1 hour: Speed $\times$ Time = $35 \text{ km/h} \times 1 \text{ h} = 35 \text{ km}$

Determine Remaining Distance and Relative Speed

At 8 a.m., when Train B starts, the distance between the two trains is reduced.

• Total distance between A and B: $210 \text{ km}$
• Distance covered by Train A by 8 a.m.: $35 \text{ km}$
• Remaining distance between trains at 8 a.m.: $210 \text{ km} - 35 \text{ km} = 175 \text{ km}$
• Speed of Train A: $35 \text{ km/h}$
• Speed of Train B: $40 \text{ km/h}$
• Since the trains are moving towards each other, their relative speed is the sum of their speeds:
• Relative speed = $35 \text{ km/h} + 40 \text{ km/h} = 75 \text{ km/h}$

Calculate Time to Meet

Now, we calculate the time it takes for the trains to cover the remaining distance at their relative speed.

• Time = $\frac{\text{Remaining Distance}}{\text{Relative Speed}}$
• Time = $\frac{175 \text{ km}}{75 \text{ km/h}} = \frac{175}{75} \text{ hours}$
• Simplifying the fraction: Time = $\frac{7}{3} \text{ hours}$

Convert the time into hours and minutes:

• $\frac{7}{3} \text{ hours} = 2 \frac{1}{3} \text{ hours}$
• $2 \frac{1}{3} \text{ hours} = 2 \text{ hours} + \frac{1}{3} \times 60 \text{ minutes} = 2 \text{ hours} + 20 \text{ minutes}$

Determine the Meeting Time

The trains will meet 2 hours and 20 minutes after 8 a.m.

• Meeting Time = $8:00 \text{ a.m.} + 2 \text{ hours } 20 \text{ minutes}$
• Meeting Time = $10:20 \text{ a.m.}$