Two trains are running on parallel tracks at 60 km/h and 20 km/h respectively in the same direction. The faster train completely passes a man sitting in the slower train in 6 seconds. Approximately what is the length of the faster train in metres?

Introduction / Context:
This problem extends the concept of relative speed to a situation where a fast train overtakes a man who is sitting in a slower moving train. Since both are moving in the same direction, the effective speed of the fast train relative to the man is the difference of their speeds. The question aims to find the length of the faster train using the time taken to completely pass the man.

Given Data / Assumptions:

• Speed of faster train = 60 km/h.
• Speed of slower train (where the man is sitting) = 20 km/h.
• Time taken for faster train to completely pass the man = 6 s.
• Both trains move in the same direction on parallel tracks with constant speeds.
• The man is treated as a point on the slower train.

Concept / Approach:
When two bodies move in the same direction, relative speed = faster speed − slower speed. The distance that must be covered for the faster train to completely pass the man is just the length of the fast train. Once the relative speed is converted to m/s, distance (train length) can be calculated using distance = speed * time.

Step-by-Step Solution:
Relative speed in km/h = 60 − 20 = 40 km/h. Convert relative speed to m/s: 40 * (5 / 18) ≈ 11.11 m/s. Time taken to pass the man = 6 s. Length of the faster train = relative speed * time = 11.11 * 6 ≈ 66.67 m.

Verification / Alternative check:
We can check the answer by converting 66.67 m back into time. At a relative speed of about 11.11 m/s, the time taken to cover 66.67 m is 66.67 / 11.11 ≈ 6 s, which matches the data in the question. Therefore, the calculation is consistent and the length is correct to two decimal places.

Why Other Options Are Wrong:
A length of 54 m or 60 m would result in a passing time smaller than 6 s at the same relative speed. A length of 72 m would require more than 6 s to pass. Since the question specifies 6 s, only a length close to 66.67 m satisfies the given conditions.

Common Pitfalls:
One frequent error is to add the speeds instead of subtracting them, which is only correct for opposite directions. Another mistake is to think that the slower train length is also involved, but for a man sitting in that train only the length of the faster train needs to pass him. Also, learners sometimes forget to convert km/h to m/s, which leads to numerical errors.

Final Answer:
Thus, the approximate length of the faster train is 66.67 m.