A train is moving along a straight track and crosses a man who is running on a platform at a speed of 8 km/h in the same direction as the train. The train takes 9 seconds to completely pass the man. If the speed of the train is 74 km/h, what is the length of the train in metres?

Introduction / Context:
This problem deals with relative speed when a fast object such as a train overtakes a slower moving person on a platform. It examines your ability to find the relative speed when both move in the same direction and then use that relative speed to compute the length of the train from the time taken to overtake.

Given Data / Assumptions:

• Speed of the train = 74 km/h.
• Speed of the man running on the platform = 8 km/h.
• The man and the train move in the same direction.
• The train takes 9 seconds to completely pass the man.
• We need to determine the length of the train in metres.

Concept / Approach:
When two objects move in the same direction, the relative speed is the difference between their speeds. The length of the train is the distance covered by the train relative to the man during the time it takes to overtake him. We first compute the relative speed in km/h, convert it into metres per second, and then multiply by the time in seconds to obtain the length of the train in metres.

Step-by-Step Solution:
Step 1: Relative speed when moving in the same direction = speed of train - speed of man = 74 - 8 = 66 km/h.Step 2: Convert 66 km/h into metres per second. Use 1 km/h = 5 / 18 metres per second.Step 3: Relative speed in metres per second = 66 * 5 / 18.Step 4: Simplify: 66 / 18 = 11 / 3, so relative speed = (11 / 3) * 5 = 55 / 3 metres per second.Step 5: Time taken to overtake = 9 seconds, so distance covered relative to the man (which equals the train length) = relative speed * time = (55 / 3) * 9 = 55 * 3 = 165 metres.

Verification / Alternative check:
If the train length is 165 metres and relative speed is 55 / 3 metres per second, then time = distance / speed = 165 / (55 / 3) = 165 * 3 / 55 = 9 seconds.This matches the time given in the question, confirming that the computed length is correct.

Why Other Options Are Wrong:
Values such as 156 metres, 188 metres or 202 metres arise if you approximate relative speed incorrectly or mishandle unit conversions. The value 140 metres is too small and would imply a smaller overtaking time than 9 seconds given the same speeds. Only 165 metres is consistent with the data and formulae.

Common Pitfalls:
A common mistake is to add the speeds instead of subtracting them when both objects move in the same direction. Learners may also forget to convert km/h to metres per second before multiplying by time in seconds, which leads to a result in mixed units. Keeping units consistent and remembering that relative speed is the difference in the same direction prevent these errors.

Final Answer:
The length of the train is 165 metres.