Four hours after a goods train passes a station, another train travelling in the same direction at a speed of 72 km/hr passes through the same station. If this faster train overtakes the goods train 8 hours after passing the station, what is the speed of the goods train in km/hr?

Introduction / Context:
This is another catch-up problem involving a goods train and a faster passenger train moving in the same direction. The goods train leaves first, giving it a head start, and the second train starts from the same station later but at a higher speed. We use distances travelled over different times to find the unknown speed of the goods train.

Given Data / Assumptions:
- Goods train passes a station at time t = 0 with speed v km/hr (unknown).
- Four hours later, a second train passes the station at 72 km/hr in the same direction.
- The second train overtakes the goods train 8 hours after it passes the station.
- Both trains move on straight tracks at constant speeds.

Concept / Approach:
The logic is similar to the previous overtaking problem. The goods train has a 4 hour head start. At the catch-up moment, the distance of each train from the station is equal. We express these distances in terms of their speeds and times, then solve the resulting equation for v, the speed of the goods train.

Step-by-Step Solution:
Step 1: Let v be the goods train's speed in km/hr.Step 2: Time travelled by the goods train at the overtaking moment = 4 hours (head start) + 8 hours (while second train moves) = 12 hours.Step 3: Distance covered by the goods train = v * 12 km.Step 4: Time travelled by the faster train at that moment = 8 hours, so its distance from the station = 72 * 8 = 576 km.Step 5: At the overtaking moment, distances are equal: v * 12 = 576.Step 6: Solve for v: v = 576 / 12 = 48 km/hr.

Verification / Alternative check:
Check using relative speed and head start. The speed difference is 72 - 48 = 24 km/hr. The goods train's head start distance after 4 hours is 48 * 4 = 192 km. Time needed to close a 192 km gap at 24 km/hr is 192 / 24 = 8 hours, which matches the given time that the faster train takes to catch up after passing the station. This confirms v = 48 km/hr.

Why Other Options Are Wrong:
57.6 km/hr would give a head start of 57.6 * 4 = 230.4 km and a speed difference of 14.4 km/hr, so the catch-up time would not be 8 hours. 72 km/hr would mean the trains have the same speed and the faster train would never catch up. 38.4 km/hr yields a different head start and catch-up time inconsistent with 8 hours. 42 km/hr similarly fails the distance–time equality.

Common Pitfalls:
Some students incorrectly add the times instead of considering how long each train has been in motion at the catch-up moment. Others forget the 4 hour head start and simply set 8v equal to 72 * 8. Always track timelines carefully and ensure that distances from the station at the catch-up time match for both trains.

Final Answer:
The speed of the goods train is 48 km/hr.