Three hours after a goods train passes a station, an express train travelling at 88 km/h in the same direction passes the station and overtakes the goods train 8 hours after passing the station. What is the speed of the goods train in km/h?

Introduction / Context:
This is a classic relative speed problem where a faster vehicle starts later, then catches up with a slower one. It checks understanding of how to equate distances travelled by two objects when one is trying to overtake the other, using basic time and distance formulas.

Given Data / Assumptions:

• Goods train speed = v km/h (unknown).
• Goods train passes the station first.
• Express train passes the same station 3 hours later at 88 km/h.
• Express train overtakes the goods train 8 hours after it passes the station.
• Both trains move in the same direction on straight track with constant speeds.

Concept / Approach:
At the instant of overtaking, both trains have travelled the same distance from the station. The goods train has been moving for longer time, while the express train has a higher speed. We set up an equation based on equal distances and solve for the unknown speed v of the goods train.

Step-by-Step Solution:
Let v be the speed of the goods train in km/h. When overtaking happens, the express train has been travelling for 8 hours. During the same period from the station, the goods train has been travelling for 3 + 8 = 11 hours. Distance travelled by express train = 88 * 8 km. Distance travelled by goods train = v * 11 km. At overtaking, distances are equal: 88 * 8 = 11 * v. Compute: 88 * 8 = 704, so 704 = 11v. Therefore v = 704 / 11 = 64 km/h.
Verification / Alternative check:
Goods train distance from station at overtaking: 64 * 11 = 704 km. Express train distance from station: 88 * 8 = 704 km. Both match, so the calculation is consistent with the scenario.
Why Other Options Are Wrong:
76.8 km/h would give distance 76.8 * 11 = 844.8 km, which does not match express distance 704 km. 96 km/h gives goods distance 96 * 11 = 1056 km, far larger than the express train distance. 51.2 km/h gives 51.2 * 11 = 563.2 km, less than the express train distance.
Common Pitfalls:
A common error is to subtract times incorrectly and set 88 * 5 equal to 11v by confusing the time difference. Another mistake is to work with relative speed directly without carefully tracking the starting time difference, which leads to wrong equations. Always align both motions with a common reference point, here the station, and equalize distances at the point of overtaking.
Final Answer:
The speed of the goods train is 64 km/h.