A 200 m long train passes a motorcyclist running in the same direction at 12 km/h in 15 seconds and also passes a jeep travelling in the same direction in 20 seconds. Assuming the lengths of both the motorcyclist and the jeep are negligible, at what speed is the jeep travelling (in km/h)?

Introduction / Context:
This question involves a train overtaking two different vehicles (a motorcyclist and a jeep) in the same direction. From the time taken to pass each, we infer relative speeds. First, we use the time taken to pass the motorcyclist to find the speed of the train. Then, we use the passing time with the jeep to determine the jeep's speed. This type of problem tests your understanding of relative motion and proper handling of units.

Given Data / Assumptions:
- Length of the train = 200 m.
- The train and both vehicles move in the same direction in a straight line.
- Speed of the motorcyclist = 12 km/h.
- The train completely passes the motorcyclist in 15 seconds.
- The train completely passes the jeep in 20 seconds.
- Lengths of the motorcyclist and the jeep are negligible compared to the train, so only the train length is used as the passing distance.

Concept / Approach:
When the train overtakes another moving object in the same direction, the relevant speed is the relative speed, which is the difference between the train's speed and the other object's speed. Time taken to pass is distance divided by relative speed. We express speeds in m/s when working with distance in metres and time in seconds. First, we use the motorcyclist data to find the train speed. Then, using the jeep data and the known train speed, we find the jeep speed. Finally, we convert the jeep speed back into km/h and compare with the options.

Step-by-Step Solution:
Step 1: Find the relative speed between the train and the motorcyclist.Distance to be covered for passing = 200 m.Time taken = 15 seconds.Relative speed (train minus motorcyclist) in m/s = 200 / 15 ≈ 13.33 m/s.Step 2: Convert relative speed to km/h.Speed in km/h = 13.33 * (18/5) ≈ 48 km/h.So train speed - motorcyclist speed = 48 km/h.Given motorcyclist speed = 12 km/h, so train speed = 48 + 12 = 60 km/h.Step 3: Use the jeep data to find jeep speed.Again, passing distance = 200 m.Time taken to pass jeep = 20 seconds.Relative speed between train and jeep in m/s = 200 / 20 = 10 m/s.Convert 10 m/s to km/h: 10 * (18/5) = 36 km/h.Thus, train speed - jeep speed = 36 km/h.We already know train speed = 60 km/h, so jeep speed = 60 - 36 = 24 km/h.

Verification / Alternative check:
Check both passing times with the derived speeds. With train speed 60 km/h and motorcyclist speed 12 km/h, relative speed = 48 km/h. In m/s, that is 48 * (5/18) ≈ 13.33 m/s. Time to cover 200 m at 13.33 m/s is 200 / 13.33 ≈ 15 seconds, which matches the first condition. With jeep speed 24 km/h, relative speed with train is 60 - 24 = 36 km/h = 36 * (5/18) = 10 m/s, so time to cover 200 m is 200 / 10 = 20 seconds, matching the second condition.

Why Other Options Are Wrong:
- 36 km/h, 32 km/h, and 29 km/h do not fit both passing-time conditions when you recompute the relative speeds and times.
- Only 24 km/h makes the 15 second and 20 second overtaking times both consistent with a single train speed of 60 km/h.

Common Pitfalls:
One common mistake is to forget that overtaking distance is just the length of the train when the other object's length is negligible. Another is to add speeds instead of subtracting them when vehicles move in the same direction. Also, mixing km/h and m/s without proper conversion often leads to wrong answers. Always convert to m/s when working with metres and seconds and back to km/h at the end if needed.

Final Answer:
The jeep is travelling at 24 km/h.