Solving Train Meeting Time Problems

This problem involves two trains moving towards each other on the same track between two stations. To find when they meet, we need to consider their speeds and the time difference in their departures.

Understanding the Train Journey Details

• Station A and Station B are 455 km apart.
• Train 1 starts from A at 9 am, speed = 55 kmph (towards B).
• Train 2 starts from B at 10 am, speed = 45 kmph (towards A).

Train 1 starts one hour earlier than Train 2. In that first hour (from 9 am to 10 am), Train 1 covers some distance. We must account for this head start.

Calculating Initial Distance Covered by Train 1

Train 1 travels from 9 am to 10 am, which is a duration of 1 hour.

\(\text{Distance} = \text{Speed} \times \text{Time}\)

Distance covered by Train 1 in the first hour = \(55 \text{ kmph} \times 1 \text{ hour} = 55 \text{ km}\).

Determining the Remaining Distance at 10 am

At 10 am, Train 1 has already covered 55 km of the total 455 km distance.

Remaining distance between Train 1 and Train 2 at 10 am = Total distance - Distance covered by Train 1

Remaining distance = \(455 \text{ km} - 55 \text{ km} = 400 \text{ km}\).

Calculating the Relative Speed of the Trains

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

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

Relative speed = \(55 \text{ kmph} + 45 \text{ kmph} = 100 \text{ kmph}\).

Finding the Time to Meet After 10 am

Now, we need to find how long it takes for the trains to cover the remaining 400 km distance at their relative speed of 100 kmph.

\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)

Time taken to meet after 10 am = \(\frac{400 \text{ km}}{100 \text{ kmph}} = 4 \text{ hours}\).

Determining the Final Meeting Time

The trains start closing the 400 km gap at 10 am. It takes them 4 hours to meet from that point.

Meeting time = Start time (for relative motion) + Time taken to meet

Meeting time = 10 am + 4 hours = 2 pm.

Based on the calculations, the trains meet at 2 pm.

Evaluating the Options

Let's look at the given options:

1. 2 pm
2. 2:15 pm
3. 2:30 pm
4. 2:45 pm

Our calculation shows the meeting time is 2 pm.

Revision Table: Key Concepts

Additional Information: Train Problems

Train problems are common in aptitude tests and often involve concepts of speed, distance, and time. Key variations include:

• Trains moving in opposite directions: When two trains move towards each other or away from each other, their relative speed is the sum of their individual speeds.
• Trains moving in the same direction: When one train overtakes another moving in the same direction, their relative speed is the difference between their speeds.
• Train crossing a pole or a person: The distance covered is equal to the length of the train itself.
• Train crossing a platform, bridge, or another train: The distance covered is the sum of the length of the train and the length of the other object (platform, bridge, or second train).

Always ensure the units (km, meters, hours, seconds) are consistent before performing calculations.