Two runners start simultaneously from the same point and run in opposite directions around a circular track of length 300 metres. Their constant speeds are 15 km/h and 25 km/h respectively. After how many seconds will they first meet each other again anywhere on the track?

Introduction / Context:
This problem checks understanding of relative speed on a circular track when two objects move in opposite directions. We must find the first time the runners meet again after the start, expressed in seconds, using their linear speeds and the length of the track.

Given Data / Assumptions:

• Length of the circular track = 300 metres.
• Runner 1 speed = 15 km/h.
• Runner 2 speed = 25 km/h.
• They start at the same time from the same point and run in opposite directions.
• Speeds remain constant and motion is along the track.

Concept / Approach:
When two objects move in opposite directions along a circular path, their relative speed equals the sum of their individual speeds. The first meeting after the start occurs when the distance between them along the track equals one full circumference. Therefore, the time to meet is track length divided by relative speed, with all units converted consistently to metres and seconds.

Step-by-Step Solution:
First convert speeds from km/h to m/s. 15 km/h = 15 * 1000 / 3600 = 15000 / 3600 = 25 / 6 m/s. 25 km/h = 25 * 1000 / 3600 = 25000 / 3600 = 125 / 18 m/s. Relative speed when moving in opposite directions = 25 / 6 + 125 / 18. Find a common denominator: 25 / 6 = 75 / 18, so total = (75 + 125) / 18 = 200 / 18 = 100 / 9 m/s. Track length L = 300 m, so time to first meeting t = L / relative speed = 300 / (100 / 9) seconds. Compute t = 300 * 9 / 100 = 2700 / 100 = 27 seconds.

Verification / Alternative check:
In 27 seconds runner 1 covers distance d1 = (25 / 6) * 27 = 25 * 4.5 = 112.5 m. Runner 2 covers d2 = (125 / 18) * 27 = 125 * 1.5 = 187.5 m. The sum d1 + d2 = 112.5 + 187.5 = 300 m, exactly one lap of the track, confirming that they meet somewhere on the track after 27 seconds.

Why Other Options Are Wrong:
If we try 31, 23 or 29 seconds, the total distance covered by both runners is not equal to 300 metres. For example, with 31 seconds the sum of distances exceeds 300 metres, meaning they would have met earlier. With 23 or 29 seconds, the sum is less than 300 metres, so they have not yet completed one full relative lap and therefore cannot have met again.

Common Pitfalls:
Learners often forget to convert speeds from km/h to m/s before using metres in the distance. Another common mistake is to use the difference of speeds instead of the sum when motion is in opposite directions. Remember that for opposite directions on a circular track, relative speed is always the sum of the two speeds.

Final Answer:
The two runners will first meet again after 27 seconds.