Two trains of lengths 400 m and 600 m are running in the same direction on parallel tracks. The faster train completely overtakes the slower train in 180 seconds. If the speed of the slower train is 48 km/h, what is the speed of the faster train?

Introduction / Context:
This question covers overtaking between two trains of different lengths moving in the same direction. One train is faster, and the time needed to overtake completely is given, along with the speed of the slower train. The task is to determine the speed of the faster train, using the total distance involved and relative speed.

Given Data / Assumptions:

• Length of faster train = 400 m.
• Length of slower train = 600 m.
• Time taken for the faster train to completely overtake the slower train = 180 s.
• Speed of slower train = 48 km/h.
• Both trains move in the same direction on parallel tracks at constant speeds.

Concept / Approach:
In an overtaking scenario, the distance that the faster train must cover relative to the slower train is the sum of both train lengths. The relative speed in the same direction is the difference between their speeds. We can first compute the relative speed in m/s from the given distance and time, convert it to km/h, and then add it to the speed of the slower train to find the speed of the faster train.

Step-by-Step Solution:
Total distance for complete overtaking = 400 m + 600 m = 1000 m. Time taken = 180 s. Relative speed in m/s = distance / time = 1000 / 180 = 50 / 9 m/s. Convert relative speed to km/h: (50 / 9) * (18 / 5) = (50 * 18) / (9 * 5) = 1000 / 45 = 20 km/h. Let speed of faster train = V km/h. Relative speed = V − 48 = 20, so V = 48 + 20 = 68 km/h.

Verification / Alternative check:
Check by direct calculation. If slower train speed is 48 km/h and faster train speed is 68 km/h, relative speed = 20 km/h. In m/s, 20 km/h = 20 * 5 / 18 = 50 / 9 m/s. Time needed to cover 1000 m at 50 / 9 m/s is 1000 / (50 / 9) = 1000 * 9 / 50 = 180 s, which matches the given time.

Why Other Options Are Wrong:
If the faster train moved at 50 km/h or 52 km/h, relative speed would be too small and the overtaking would take much longer than 180 s. If it moved at 76 km/h, relative speed would be 28 km/h, making the overtaking time shorter than 180 s. Only 68 km/h leads to the correct overtaking time for the given distance.

Common Pitfalls:
Errors often arise from forgetting that total distance is the sum of both train lengths, incorrectly converting km/h to m/s, or adding speeds instead of subtracting them in a same direction overtaking scenario. Carefully identifying relative speed and distance ensures a correct solution.

Final Answer:
The speed of the faster train is 68 km/h.