A train 110 m long is running at a speed of 60 km/h. In what time (in seconds) will it pass a man who is running at 6 km/h in the opposite direction to the train?

Introduction / Context:
This is a standard question on relative speed where a train passes a man running in the opposite direction. It tests the understanding that when two bodies move towards each other, their relative speed is the sum of their individual speeds, and that the distance to be covered during crossing is the train length only.

Given Data / Assumptions:

• Train length = 110 m.
• Speed of train = 60 km/h.
• Speed of man = 6 km/h.
• They move in opposite directions along the same line.
• Speeds are constant and the track is straight.

Concept / Approach:
When two objects move towards each other, their relative speed is the sum of their speeds. To find how long the train takes to pass the man completely, we calculate this relative speed in metres per second, then divide the train length by the relative speed. This gives the crossing time in seconds.

Step-by-Step Solution:
Step 1: Relative speed in km/h = 60 + 6 = 66 km/h. Step 2: Convert 66 km/h to m/s using 1 km/h = 5/18 m/s. Step 3: Relative speed = 66 * 5 / 18 = 330 / 18 m/s, which simplifies to 55 / 3 m/s. Step 4: Distance to be covered relative to the man = length of train = 110 m. Step 5: Use time = distance / speed, so time = 110 / (55 / 3) seconds. Step 6: Simplify: time = 110 * 3 / 55 = 330 / 55 = 6 seconds.

Verification / Alternative check:
We can check by quick mental math. Relative speed 55 / 3 is about 18.33 m/s. In 6 seconds, distance = 18.33 * 6 ≈ 110 m, which matches the train length. This confirms that 6 seconds is the correct crossing time for the train to pass the man completely.

Why Other Options Are Wrong:

• 5 sec: At this time the train would only cover about 91.67 m, which is less than its length.
• 7 sec: This would give over 128 m, more than the required 110 m, which does not match the crossing condition.
• 10 sec: This is too long and would involve the train covering much more distance than its length relative to the man.

Common Pitfalls:
Common mistakes include subtracting the man speed from the train speed even though they move in opposite directions, or using only the man speed by accident. Another typical error is forgetting to convert km/h to m/s, which leads to time values with wrong units. Remember that conversion and the idea that crossing a point object involves only train length are crucial to solving these problems correctly.

Final Answer:
The train will pass the man completely in 6 sec.