A goods train passes a station, and 5 hours later another train travelling at 70 km/h in the same direction passes the same station and then overtakes the goods train in 9 hours. What is the speed (in km/h) of the goods train?

Introduction / Context:
This train problem focuses on relative speed in the same direction and the idea of a head start. A goods train leaves first, and a faster train follows later, eventually overtaking it. The question asks for the speed of the slower, goods train based on the given timings.

Given Data / Assumptions:

• The goods train passes the station first.
• Second train speed = 70 km/h.
• The second train passes the station 5 hours after the goods train.
• From the moment it passes the station, the faster train takes 9 hours to overtake the goods train.
• Both trains move in the same direction at constant speeds.

Concept / Approach:
Let the speed of the goods train be v km/h. When the faster train starts from the station, the goods train already has a time lead and is ahead by a certain distance. The faster train catches up because its speed is higher. The head start distance equals speed of goods train multiplied by 5 hours. The closing speed is 70 - v. Using time = distance / relative speed, we form an equation and solve for v.

Step-by-Step Solution:
Step 1: Let speed of goods train be v km/h. Step 2: In 5 hours, goods train travels a head start distance = 5 * v km. Step 3: From the moment the second train passes the station, it takes 9 hours to catch the goods train. Step 4: Relative speed (closing speed) = 70 - v km/h. Step 5: Distance to be closed = head start distance = 5v. Step 6: Use formula: distance = relative speed * time, so 5v = (70 - v) * 9. Step 7: Simplify: 5v = 630 - 9v, so 14v = 630 and v = 630 / 14 = 45 km/h.

Verification / Alternative check:
At 45 km/h, in 5 hours the goods train goes 225 km ahead. Relative speed of the faster train is 70 - 45 = 25 km/h. Time to cover 225 km at 25 km/h is 225 / 25 = 9 hours, which agrees with the given data. Thus the value is correct.

Why Other Options Are Wrong:

• 54 km/h: This would give head start 270 km and relative speed 16 km/h, requiring 270 / 16 hours, which is not 9.
• 67.5 km/h: This is too close to 70 km/h, giving very small relative speed and far more than 9 hours.
• 36 km/h: This gives head start 180 km and relative speed 34 km/h, and 180 / 34 is not equal to 9.

Common Pitfalls:
Learners sometimes forget that the goods train has moved for 5 extra hours before the second train starts. Another mistake is incorrectly using 70 + v instead of 70 - v for the relative speed because the trains move in the same direction. It is also easy to mix up which time interval corresponds to which portion of motion, so handling time references carefully is important.

Final Answer:
The speed of the goods train is 45 km/h.