Train Meeting Time Calculation Explained

This problem involves two trains traveling towards each other from different stations at different speeds and starting times. To find out when they meet, we need to consider the distance covered by the first train before the second train starts and then calculate the time it takes for them to cover the remaining distance together using their relative speed.

Understanding the Train Journey

• Station A and B are 110 km apart.
• Train 1 starts from A at 7 a.m. towards B at 20 km/h.
• Train 2 starts from B at 8 a.m. towards A at 25 km/h.

Step-by-Step Solution for Meeting Time

Let's break down the problem:

Step 1: Calculate the distance covered by Train 1 before Train 2 starts.

Train 1 starts at 7 a.m., and Train 2 starts at 8 a.m. This means Train 1 travels alone for 1 hour (from 7 a.m. to 8 a.m.).

Distance covered by Train 1 in 1 hour = Speed of Train 1 $\times$ Time

Distance = $20 \text{ km/h} \times 1 \text{ hour} = 20 \text{ km}$

Step 2: Calculate the remaining distance between the trains when Train 2 starts.

The total distance between A and B is 110 km. After Train 1 has covered 20 km, the remaining distance is:

Remaining distance = Total distance - Distance covered by Train 1

Remaining distance = $110 \text{ km} - 20 \text{ km} = 90 \text{ km}$

At 8 a.m., the two trains are 90 km apart and are moving towards each other.

Step 3: Calculate the relative speed of the two trains.

Since the trains are moving towards each other, their speeds add up to determine how quickly the distance between them is decreasing. This is called their relative speed.

Relative speed = Speed of Train 1 + Speed of Train 2

Relative speed = $20 \text{ km/h} + 25 \text{ km/h} = 45 \text{ km/h}$

Step 4: Calculate the time it takes for the trains to meet after 8 a.m.

They need to cover the remaining 90 km distance at a combined relative speed of 45 km/h.

Time taken to meet = Remaining distance / Relative speed

Time taken = $90 \text{ km} / 45 \text{ km/h} = 2 \text{ hours}$

Step 5: Determine the actual meeting time.

The trains started moving towards each other from the remaining distance calculation point (when Train 2 started) at 8 a.m. They will meet 2 hours after 8 a.m.

Meeting time = Start time of Train 2 + Time taken to meet

Meeting time = 8:00 a.m. + 2 hours = 10:00 a.m.

Thus, the two trains will meet at 10 a.m.

Checking the Meeting Time

Let's verify the distances covered by each train until 10 a.m.

• Train 1 starts at 7 a.m. and meets at 10 a.m., so it travels for 3 hours. Distance covered by Train 1 = $20 \text{ km/h} \times 3 \text{ hours} = 60 \text{ km}$.
• Train 2 starts at 8 a.m. and meets at 10 a.m., so it travels for 2 hours. Distance covered by Train 2 = $25 \text{ km/h} \times 2 \text{ hours} = 50 \text{ km}$.
• Total distance covered by both trains = $60 \text{ km} + 50 \text{ km} = 110 \text{ km}$. This matches the total distance between A and B, confirming the meeting time is correct.

Revision Table: Key Concepts for Train Problems

Additional Information: Relative Speed in Train Problems

Relative speed is a crucial concept in problems involving objects moving relative to each other. When two objects are moving:

• Towards each other: The relative speed is the sum of their individual speeds. This is because the distance between them is reducing at a rate equal to the sum of how much each object covers per unit of time.
• In the same direction: The relative speed is the absolute difference between their individual speeds (speed of the faster object minus the speed of the slower object). This is because the distance between them is changing at a rate equal to the difference in how much each object covers per unit of time.

Understanding relative speed simplifies problems involving meeting points or overtakes.