Calculating Train Meeting Time: Delhi to Patna Distance Problem

This solution explains how to find the time when two trains traveling towards each other will meet. We are given the total distance between two cities, the speeds of the trains, and the fact that one train starts later than the other. We need to calculate the time elapsed since the first train started.

Understanding the Problem Parameters

Let's break down the information provided:

• Total distance between Delhi and Patna: $D = 480$ km.
• Speed of the first train (starting from Delhi): $S_1 = 60$ kmph.
• Speed of the second train (starting from Patna): $S_2 = 80$ kmph.
• The second train starts $1$ hour after the first train.
• We need to find the time ($t$) after the *first* train started when they meet.

Step-by-Step Solution

We can solve this problem by considering the head start of the first train.

Step 1: Calculate the Distance Covered by the First Train in the Head Start

The first train travels for 1 hour before the second train starts. The distance it covers in this hour is:

Distance = Speed $\times$ Time

$d_{headstart} = S_1 \times 1 \text{ hour} = 60 \text{ kmph} \times 1 \text{ hour} = 60$ km.

Step 2: Determine the Remaining Distance

When the second train starts its journey, the first train has already covered 60 km. The distance remaining between the two trains is:

$D_{remaining} = \text{Total Distance} - d_{headstart}$

$D_{remaining} = 480 \text{ km} - 60 \text{ km} = 420$ km.

Step 3: Calculate the Relative Speed

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

$S_{relative} = S_1 + S_2$

$S_{relative} = 60 \text{ kmph} + 80 \text{ kmph} = 140$ kmph.

Step 4: Calculate the Time to Meet After the Second Train Starts

Now, we can find the time it takes for the trains to cover the remaining distance ($420$ km) at their relative speed ($140$ kmph). Let's call this time $t_{relative}$:

Time = Distance / Speed

$t_{relative} = \frac{D_{remaining}}{S_{relative}} = \frac{420 \text{ km}}{140 \text{ kmph}} = 3$ hours.

This means the trains will meet 3 hours *after* the second train started.

Step 5: Calculate the Total Time from the First Train's Start

The question asks for the time after the *first* train started. The second train started 1 hour after the first train, and they meet 3 hours after the second train started. Therefore, the total time is:

Total Time $T = (\text{Head start time}) + t_{relative}$

$T = 1 \text{ hour} + 3 \text{ hours} = 4$ hours.

Alternative Calculation Approach

We can also solve this using algebra. Let $t$ be the time in hours since the first train left Delhi.

• The first train travels for $t$ hours. Distance covered: $d_1 = 60t$.
• The second train starts 1 hour later, so it travels for $(t-1)$ hours. Distance covered: $d_2 = 80(t-1)$. (This assumes $t \ge 1$).
• When they meet, the sum of the distances covered equals the total distance: $d_1 + d_2 = 480$.

Setting up the equation:

$60t + 80(t-1) = 480$

Now, solve for $t$:

$60t + 80t - 80 = 480$

$140t = 480 + 80$

$140t = 560$

$t = \frac{560}{140}$

$t = 4$ hours.

Conclusion

Both methods show that the two trains will meet $4$ hours after the first train started its journey from Delhi.