Calculating the Meeting Time of Two Trains

This problem involves calculating the time when two trains, starting from different stations and at different times, will meet. We need to consider the distance between the stations, the speeds of the trains, and their respective start times.

Understanding the Problem Parameters

• Total distance between Station A and Station B: 428 km.
• Train 1 starts from Station A at 6:00 a.m.
• Speed of Train 1: 48 km/h.
• Train 2 starts from Station B at 6:20 a.m.
• Speed of Train 2: 55 km/h.
• Both trains move towards each other.

Step 1: Calculate the Distance Covered by the First Train Before the Second Train Starts

Train 1 starts 20 minutes earlier than Train 2. We first calculate how far Train 1 travels in this 20-minute interval.

• Time difference = 6:20 a.m. - 6:00 a.m. = 20 minutes.
• Convert minutes to hours: 20 minutes = $ \frac{20}{60} $ hours = $ \frac{1}{3} $ hour.
• Distance covered by Train 1 in $ \frac{1}{3} $ hour = Speed × Time
• Distance = $ 48 \, \text{km/h} \times \frac{1}{3} \, \text{h} = 16 \, \text{km} $.

So, by 6:20 a.m., Train 1 has already covered 16 km from Station A.

Step 2: Calculate the Remaining Distance Between the Trains at 6:20 a.m.

When Train 2 starts its journey at 6:20 a.m., Train 1 has already moved 16 km towards Station B.

• Initial distance between stations A and B = 428 km.
• Distance covered by Train 1 = 16 km.
• Remaining distance between the two trains at 6:20 a.m. = Total Distance - Distance covered by Train 1
• Remaining distance = $ 428 \, \text{km} - 16 \, \text{km} = 412 \, \text{km} $.

Step 3: Calculate the Relative Speed of the Trains

Since the trains are moving towards each other, their speeds add up. This is their relative speed, which determines how quickly the distance between them decreases.

• Speed of Train 1 = 48 km/h.
• Speed of Train 2 = 55 km/h.
• Relative speed = Speed of Train 1 + Speed of Train 2
• Relative speed = $ 48 \, \text{km/h} + 55 \, \text{km/h} = 103 \, \text{km/h} $.

Step 4: Calculate the Time Taken for the Trains to Meet

Now, we calculate the time it will take for the trains to cover the remaining distance (412 km) at their relative speed (103 km/h).

• Time = Remaining Distance / Relative Speed
• Time = $ \frac{412 \, \text{km}}{103 \, \text{km/h}} $.
• Time = 4 hours.

This means the trains will meet 4 hours after 6:20 a.m.

Step 5: Determine the Final Meeting Time

To find the exact time they meet, we add the calculated time (4 hours) to the time when both trains were moving simultaneously (6:20 a.m.).

• Start time for relative motion calculation = 6:20 a.m.
• Time taken to meet = 4 hours.
• Meeting Time = 6:20 a.m. + 4 hours = 10:20 a.m.

Conclusion

The two trains will meet at 10:20 a.m.