Seven hours after a goods train passes a station, another train travelling in the same direction at a speed of 54 km/hr passes that station. If this faster train overtakes the goods train 11 hours after passing the station, what is the speed of the goods train in km/hr?

Introduction / Context:
This question examines relative motion in the same direction, focusing on how a faster train catches up with a slower goods train that has departed earlier. By translating the situation into distances travelled over time, we can set the positions equal at the catch-up moment and solve for the unknown speed of the goods train.

Given Data / Assumptions:
- A goods train passes a station at time t = 0 with speed v km/hr (unknown).
- Seven hours later, a faster train passes the same station at 54 km/hr in the same direction.
- The faster train overtakes the goods train 11 hours after it passes the station.
- Both trains move at constant speeds on a straight track.

Concept / Approach:
We work with the positions of both trains relative to the station. The goods train has a 7 hour head start. When the faster train has been running for 11 hours, the goods train has been running for 7 + 11 = 18 hours. At the moment of overtaking, both trains are at the same distance from the station. Setting the distances equal gives a simple linear equation in v.

Step-by-Step Solution:
Step 1: Let v be the speed of the goods train in km/hr.Step 2: Time traveled by the goods train when it is overtaken = 7 hours (lead) + 11 hours (while the faster train moves) = 18 hours.Step 3: Distance covered by the goods train by then = v * 18 km.Step 4: Time traveled by the faster train = 11 hours, so distance covered by it = 54 * 11 = 594 km.Step 5: At the overtaking point, both distances are equal: v * 18 = 594.Step 6: Solve for v: v = 594 / 18.Step 7: Compute 594 / 18: since 18 * 30 = 540 and 18 * 33 = 594, v = 33 km/hr.

Verification / Alternative check:
Check relative speed and head start. The speed difference between the faster train and the goods train is 54 - 33 = 21 km/hr. The goods train's head start distance after 7 hours is 33 * 7 = 231 km. Time needed to close a 231 km gap at 21 km/hr is 231 / 21 = 11 hours, which matches the given 11 hours after the faster train passes the station. This confirms the correctness of v = 33 km/hr.

Why Other Options Are Wrong:
39.6 km/hr, 49.5 km/hr, 26.4 km/hr, and 42 km/hr all fail when substituted back. For example, if v = 39.6 km/hr, the head start distance is 39.6 * 7 = 277.2 km, and the speed difference is 54 - 39.6 = 14.4 km/hr, giving a catch-up time of 277.2 / 14.4 ≈ 19.25 hours, not 11 hours. Similar inconsistencies occur with the other values.

Common Pitfalls:
Some learners mistakenly add the distances instead of equating them, or think the 11 hours is the total time for the goods train instead of the faster train. Another common error is to forget the 7 hour head start and set 18v equal to 54 * 11 without understanding where 18 comes from. Carefully tracking how long each train has been moving is crucial.

Final Answer:
The speed of the goods train is 33 km/hr.