A jogger running at 9 km/h along a railway track is 260 metres ahead of the engine of a 140 metre long train that is running at 45 km/h in the same direction. In how many seconds will the train completely pass the jogger?

Introduction / Context:
This problem involves relative speed between a train and a jogger moving in the same direction along a railway track. The jogger is initially ahead of the train by a certain distance. For the train to completely pass the jogger, the engine must cover the gap plus the full length of the train. Using relative speed and distance ideas, we can find the time taken.

Given Data / Assumptions:

• Jogger speed = 9 km/h.
• Train speed = 45 km/h.
• Jogger is 260 m ahead of the engine initially.
• Train length = 140 m.
• Both move in the same direction at constant speeds.

Concept / Approach:
When two objects move in the same direction, their relative speed is the difference of their speeds. For the train to completely pass the jogger, the engine must first close the 260 metre gap and then the entire 140 metre length of the train must pass in front of the jogger. Hence, the total distance that must be covered relative to the jogger is 260 + 140 metres. We compute the relative speed in m/s and then apply time = distance / speed.

Step-by-Step Solution:
Step 1: Total relative distance to be covered = 260 + 140 = 400 m. Step 2: Convert speeds from km/h to m/s. Step 3: Train speed = 45 km/h = 45 * 5/18 = 12.5 m/s. Step 4: Jogger speed = 9 km/h = 9 * 5/18 = 2.5 m/s. Step 5: Relative speed = 12.5 - 2.5 = 10 m/s. Step 6: Time taken to pass = distance / relative speed = 400 / 10 = 40 seconds.

Verification / Alternative check:
In 40 seconds, the relative distance covered at 10 m/s is exactly 400 m, equal to the initial head start plus the train length. At that moment, the rear end of the train has just passed the jogger. Any smaller or larger time would either not fully clear the jogger or overshoot, which confirms that 40 seconds is the correct exact answer.

Why Other Options Are Wrong:
Times like 25, 32, or 38 seconds correspond to smaller distances than 400 m at 10 m/s, so the train would not have fully passed the jogger. The distractor 30 seconds gives only 300 m of relative displacement, again insufficient. Therefore, only 40 seconds satisfies the exact distance requirement.

Common Pitfalls:
Learners may forget to add the train length to the initial separation or may incorrectly add the two speeds instead of subtracting them for same direction motion. Mixing up metres and kilometres or failing to convert speeds properly can also lead to errors. Keeping careful track of relative speed and total distance to be covered solves the problem correctly.

Final Answer:
The train will completely pass the jogger in 40 sec.