A man is sitting in a train that is moving at 50 km/h. He observes that a goods train, 280 meters long, passes him completely in 9 seconds while moving in the opposite direction. What is the speed of the goods train in km/h?

Introduction / Context:
This is a classic relative speed question involving two trains moving in opposite directions. One train carries an observer and the other is a goods train of known length. The goal is to find the speed of the goods train using the time taken to pass the observer.

Given Data / Assumptions:
- Speed of the passenger train with the man inside is 50 km/h.
- The goods train is 280 meters long.
- The goods train passes the man completely in 9 seconds.
- The two trains move in opposite directions in a straight line.
- We must find the speed of the goods train in km/h.

Concept / Approach:
When two objects move in opposite directions, their relative speed is the sum of their individual speeds. The time taken for the goods train to pass the man equals the time it takes for its entire length to cover the relative distance at the relative speed. We convert everything to consistent units, compute the relative speed, and then find the unknown speed of the goods train.

Step-by-Step Solution:
Step 1: Let the speed of the goods train be v km/h.Step 2: Since the trains are moving in opposite directions, relative speed = 50 + v km/h.Step 3: Convert the relative speed into meters per second by multiplying by 5 / 18.Step 4: The time taken to pass completely is 9 seconds and the distance covered relative to the man is the length of the goods train, which is 280 meters.Step 5: Relative speed in m/s = distance / time = 280 / 9 m/s.Step 6: Equate the two expressions for relative speed: (50 + v) * 5 / 18 = 280 / 9.Step 7: Multiply both sides by 18 to get 5(50 + v) = 560.Step 8: Simplify: 5(50 + v) = 560 leads to 50 + v = 112, so v = 62 km/h.

Verification / Alternative check:
Compute the relative speed in km/h from 280 / 9 m/s. Multiply 280 / 9 by 18 / 5 to convert, which gives (280 * 18) / (9 * 5) = 5040 / 45 = 112 km/h. Since the passenger train moves at 50 km/h, the goods train must move at 112 - 50 = 62 km/h. This matches the earlier result.

Why Other Options Are Wrong:
Options 60, 64, 65, and 58 km/h are values that do not produce the correct relative speed. For example, if the goods train had speed 60 km/h, relative speed would be 110 km/h, which in meters per second is about 30.56 and would give a different passing time. Only 62 km/h gives a relative speed of 112 km/h that matches the observed time and length.

Common Pitfalls:
Students often forget to add the speeds for opposite directions or they subtract them instead. Another common error is to skip unit conversion and mix km/h with meters and seconds. Correctly applying the conversion factor 5 / 18 and carefully equating the relative speed to distance over time is crucial.

Final Answer:
The speed of the goods train is 62 km/h.