Two trains are moving in opposite directions at speeds of 43 km/h and 51 km/h respectively. The slower train takes 9 seconds to completely cross a man sitting in the faster train. What is the length (in metres) of the slower train?

Introduction / Context:
This problem involves two trains moving in opposite directions. We focus on the perspective of a man sitting in the faster train while the slower train passes him. The key idea is that the relative speed of the trains determines how quickly the slower train passes the man, and that the distance covered in this time equals the length of the slower train.

Given Data / Assumptions:

• Speed of slower train = 43 km/h.
• Speed of faster train = 51 km/h.
• Trains move in opposite directions.
• Time taken by slower train to pass the man in the faster train = 9 seconds.
• We assume straight track and constant speeds.

Concept / Approach:
From the viewpoint of the man on the faster train, the slower train appears to approach with a relative speed that is the sum of both speeds, because they move in opposite directions. Once we compute this relative speed in metres per second, we multiply it by the given time to get the distance covered, which is simply the length of the slower train since only that train is passing the man.

Step-by-Step Solution:
Step 1: Relative speed in km/h = 43 + 51 = 94 km/h. Step 2: Convert 94 km/h to m/s using 1 km/h = 5/18 m/s. Step 3: Relative speed = 94 * 5 / 18 m/s. Step 4: Calculate: 94 * 5 / 18 = 470 / 18 m/s which simplifies to approximately 26.11 m/s. Step 5: Time taken for slower train to pass the man = 9 seconds. Step 6: Distance covered relative to the man (equal to length of slower train) = relative speed * time. Step 7: Length of slower train = (470 / 18) * 9 = 470 / 2 = 235 m.

Verification / Alternative check:
We can verify by simplifying carefully. Since (470 / 18) * 9 equals 470 * 9 / 18, we can cancel 9 and 18 to get 470 / 2, which is 235 m. The units are metres, and this length is realistic for a train. With a relative speed of roughly 26.11 m/s, in 9 seconds the slower train would indeed cover about 235 m, confirming the result.

Why Other Options Are Wrong:

• 338.4 m: This would require a longer time or higher relative speed than given.
• 470 m and 940 m: These are multiples of the relative speed factor but would correspond to 18 or 36 seconds, not 9 seconds, for the given relative speed.

Common Pitfalls:
Learners sometimes subtract speeds instead of adding them in opposite direction problems. Another typical error is to confuse which train length is involved. In this situation, only the slower train is passing the man, so we use its length, not the combined length of both trains. Incorrect unit conversion from km/h to m/s can also lead to wrong answers, so it is essential to handle the factor 5/18 accurately.

Final Answer:
The length of the slower train is 235 m.