Understanding the Train Meeting Problem

This problem involves two trains traveling towards each other from different locations, A and B, at different speeds. We are given their speeds and the difference in the distance they cover until they meet. Our goal is to find the total distance between A and B.

Analyzing the Given Information

• Speed of the faster train (starting from A, let's assume) = 60 km/h.
• Speed of the slower train (starting from B) = 50 km/h.
• The faster train travels 110 km more than the slower train by the time they meet.
• They start at the same time and travel for the same duration until they meet.

Setting up the Equations

Let \(t\) be the time in hours until the two trains meet. Since they start simultaneously and travel towards each other until they meet, the time taken by both trains will be the same.

• Distance covered by the faster train = Speed \(\times\) Time = \(60t\) km.
• Distance covered by the slower train = Speed \(\times\) Time = \(50t\) km.

We are told that the faster train travels 110 km more than the slower train. We can write this as an equation:

Distance covered by faster train - Distance covered by slower train = 110 km

\(60t - 50t = 110\)

Solving for Time Taken to Meet

Now, let's solve the equation for \(t\):

\( (60 - 50)t = 110 \)

\( 10t = 110 \)

\( t = \frac{110}{10} \)

\( t = 11 \) hours

So, the trains meet after 11 hours.

Calculating the Distances Covered by Each Train

Now that we know the time \(t = 11\) hours, we can calculate the distance covered by each train:

• Distance covered by faster train = \(60 \times 11 = 660\) km.
• Distance covered by slower train = \(50 \times 11 = 550\) km.

Finding the Total Distance Between A and B

When the two trains meet, the sum of the distances they have covered is equal to the total distance between A and B.

Total distance = Distance covered by faster train + Distance covered by slower train

Total distance = \(660 \text{ km} + 550 \text{ km}\)

Total distance = \(1210 \text{ km}\)

Let's verify the difference in distance: \(660 \text{ km} - 550 \text{ km} = 110 \text{ km}\), which matches the information given in the problem.

Therefore, the distance between A and B is 1210 km.

Revision Table: Key Concepts

Additional Information: Relative Speed Concept

While we solved this problem using the difference in distances, it can also be conceptualized using relative speed. When two objects move towards each other, their relative speed is the sum of their individual speeds. The relative speed tells us how quickly the distance between them is decreasing.

• Relative speed = Speed of train A + Speed of train B
• Relative speed = \(60 \text{ km/h} + 50 \text{ km/h} = 110 \text{ km/h}\).

The difference in distance covered (110 km) is because the faster train travels for 11 hours, and in each hour, it covers \(60 - 50 = 10\) km more than the slower train. Over 11 hours, this difference accumulates to \(10 \times 11 = 110\) km.

The total distance can also be found by calculating the time it takes for the 110 km "gap" created by the speed difference to result in a 110 km lead for the faster train. This time is \(110 \text{ km} / 10 \text{ km/h} = 11\) hours. Then, multiply this time by the relative speed to get the total distance: \(110 \text{ km/h} \times 11 \text{ hours} = 1210 \text{ km}\).