A passenger train running at 80 km/h leaves a railway station 6 hours after a goods train has departed and overtakes the goods train in 4 hours. What is the speed (in km/h) of the goods train?

Introduction / Context:
This problem involves two trains moving in the same direction on a straight railway track. A slower goods train starts first, followed by a faster passenger train that eventually overtakes it. We use the idea of relative speed and head start distance to calculate the speed of the goods train.

Given Data / Assumptions:

• Speed of passenger train = 80 km/h.
• Goods train leaves first.
• Passenger train leaves the station 6 hours later.
• Passenger train overtakes the goods train in 4 hours after its own departure.
• Both trains travel at constant speeds along the same direction.

Concept / Approach:
Let the speed of the goods train be v km/h. The goods train travels for 10 hours by the time they meet, while the passenger train travels for 4 hours. Since they meet at the same point, the distances covered by both trains must be equal. This leads to a linear equation in v, which we solve to obtain the required speed.

Step-by-Step Solution:
Step 1: Let v be the speed of the goods train in km/h. Step 2: Time travelled by goods train until overtaken = 6 + 4 = 10 hours. Step 3: Time travelled by passenger train until the meeting point = 4 hours. Step 4: Distance covered by goods train = v * 10 km. Step 5: Distance covered by passenger train = 80 * 4 = 320 km. Step 6: Because both reach the same point, v * 10 = 320. Step 7: Solve for v, giving v = 320 / 10 = 32 km/h.

Verification / Alternative check:
Check with distances. In 10 hours at 32 km/h, the goods train travels 320 km. The passenger train, in 4 hours at 80 km/h, also travels 320 km. Since both distances are equal, they indeed meet at the same position after the stated times, confirming that 32 km/h is the correct speed for the goods train.

Why Other Options Are Wrong:

• 50 km/h: This would give 500 km for the goods train and only 320 km for the passenger train, so they would never meet in 4 hours.
• 45 km/h: Gives 450 km for the goods train and 320 km for the passenger, again not equal.
• 64 km/h: Leads to 640 km for the goods train and 320 km for the passenger train, violating the meeting condition.

Common Pitfalls:
Some learners incorrectly use relative speed methods without accounting for the different start times, or they forget that the total time the goods train travels is larger than the time for the passenger train. Others mistakenly try to add distances or use different reference frames. Writing a clear equation based on equal distances at the meeting point helps avoid these issues.

Final Answer:
The speed of the goods train is 32 km/h.