A 260 m long train is running at 120 km/h and crosses another train coming from the opposite direction at 80 km/h in 9 seconds. What is the length (in metres) of the second train?

Introduction / Context:
This is a classic aptitude question on trains and relative speed. The goal is to use the concept of relative speed in opposite directions to find the unknown length of the second train, given the crossing time and the speed of both trains.

Given Data / Assumptions:

• Length of first train = 260 m.
• Speed of first train = 120 km/h.
• Speed of second train = 80 km/h.
• They move in opposite directions.
• Time taken to completely cross each other = 9 seconds.
• We assume both trains move at constant speeds in a straight line.

Concept / Approach:
When two objects move towards each other on a straight track, their relative speed is the sum of their individual speeds. The total distance covered during the crossing is the sum of their lengths. Using speed = distance / time, we first compute the relative speed in metres per second, then multiply by the time to get the total distance, and finally subtract the known length of the first train to obtain the length of the second train.

Step-by-Step Solution:
Step 1: Relative speed in km/h = 120 + 80 = 200 km/h. Step 2: Convert 200 km/h to m/s using 1 km/h = 5/18 m/s. Step 3: Relative speed = 200 * 5 / 18 = 1000 / 18 m/s. Step 4: Total distance covered during crossing = relative speed * time. Step 5: Distance = (1000 / 18) * 9 = 500 m. Step 6: Sum of lengths of both trains = 500 m. Step 7: Length of second train = 500 - 260 = 240 m.

Verification / Alternative check:
We can verify by reversing the process. If the second train is 240 m, total length becomes 260 + 240 = 500 m. With a relative speed of 1000 / 18 m/s, the time is distance / speed = 500 / (1000 / 18) = 500 * 18 / 1000 = 9 seconds, which matches the given condition. So the answer is consistent.

Why Other Options Are Wrong:

• 250 m: This gives total length 510 m, which would require slightly more than 9 seconds, so it does not fit.
• 260 m: This gives total length 520 m and would also increase the time above 9 seconds.
• 270 m: This gives total length 530 m, making the required time even larger than 9 seconds.

Common Pitfalls:
Students often subtract speeds instead of adding them for opposite directions. Another common mistake is to forget to convert km/h into m/s, which leads to incorrect distances. Sometimes learners wrongly assume that only one train length is involved when two trains are crossing, but in such problems the distance covered during crossing is always the sum of both lengths.

Final Answer:
Therefore, the length of the second train is 240 m.