Two trains start simultaneously from the same place but move along straight tracks at right angles to each other. Their speeds are 36 km/h and 48 km/h respectively. What will be the distance between them (in metres) after 30 seconds?

Introduction / Context:
This problem combines time and distance concepts with basic geometry. Two trains move away from the same starting point in perpendicular directions. The question asks for the distance between them after a certain time interval. This requires converting speeds, computing distances travelled along each track, and then using the Pythagorean theorem to find the straight line distance between the two trains.

Given Data / Assumptions:
- Speed of the first train = 36 km/h.
- Speed of the second train = 48 km/h.
- They start at the same time from the same point and move along tracks at right angles (forming a 90 degree angle between their paths).
- Time elapsed = 30 seconds.
- We need the distance between the trains after 30 seconds, in metres.

Concept / Approach:
We first convert the speeds from km/h to m/s, because the time is in seconds and the answer is required in metres. Then we calculate how far each train travels in 30 seconds. Since the trains move along perpendicular directions, the distance between them is the hypotenuse of a right-angled triangle whose legs are the distances travelled by each train. We then apply the Pythagorean theorem: hypotenuse^2 = leg1^2 + leg2^2.

Step-by-Step Solution:
Step 1: Convert speeds to m/s.First train: 36 km/h = 36 * (5/18) = 10 m/s.Second train: 48 km/h = 48 * (5/18) = 40/3 m/s ≈ 13.33 m/s.Step 2: Compute distances in 30 seconds.First train distance = 10 m/s * 30 s = 300 m.Second train distance = (40/3) m/s * 30 s = 40 * 10 = 400 m.Step 3: Use the Pythagorean theorem for the right-angled triangle.Let d be the distance between the trains.Then d^2 = 300^2 + 400^2 = 90000 + 160000 = 250000.So d = square root of 250000 = 500 m.

Verification / Alternative check:
The 3-4-5 triangle pattern appears here: the legs are in the ratio 3 : 4 (300 m and 400 m), and the hypotenuse is 5 parts, which gives 500 m. This is a well-known Pythagorean triple, so the calculation is clearly consistent and provides a quick mental check that the answer is correct.

Why Other Options Are Wrong:
- 300 m and 400 m are the individual distances travelled by each train but not the distance between them.
- 900 m is much larger and would require both trains to travel much farther than they actually do in 30 seconds at the given speeds.
- 250 m is too small, incorrect under the Pythagorean relation for 300 m and 400 m legs.

Common Pitfalls:
A frequent mistake is to add or subtract the distances directly instead of using the Pythagorean theorem when the paths are at right angles. Another common error is forgetting to convert speeds to m/s, which leads to unit inconsistencies. Always be careful with units and geometry when dealing with motion in two perpendicular directions.

Final Answer:
The distance between the two trains after 30 seconds is 500 m.