Introduction
Problems on a circular track with same starting point ask about meeting times, overtakes, number of rounds, and relative positions when two or more runners start together (same or opposite directions).
This pattern is important because it combines relative speed ideas with the concept of laps (track length) and helps solve many contest-style questions quickly using simple formulas.
Pattern: Circular Track / Same Starting Point
Pattern
Key concept: Convert motion on a circle into linear distance (laps) and use relative speed: when two people meet or overtake, they cover integer laps difference.
- Same direction: Relative speed = (faster - slower). Time to first overtake = Track length ÷ relative speed.
- Opposite directions: Relative speed = (sum of speeds). Time to first meeting = Track length ÷ relative speed.
- Number of meetings in time T: If opposite directions → meetings = ⌊(relative speed × T) ÷ track length⌋ (count integer meetings). If same direction → overtakes = ⌊(relative speed × T) ÷ track length⌋.
- When asked about positions after time T: Compute distance each covers (speed × T), reduce modulo track length to find position on track.
- For integer lap questions: If faster completes n more laps than slower in time T, (faster - slower) × T = n × track length.
Step-by-Step Example
Question
Two runners start at the same point on a circular track of length 400 m and run in opposite directions. Their speeds are 6 m/s and 4 m/s. When will they meet first?
Solution
-
Step 1: Identify relative speed
Opposite directions → relative speed = 6 + 4 = 10 m/s. -
Step 2: Time to meet
Time = Track length ÷ Relative speed = 400 ÷ 10 = 40 seconds. -
Final Answer:
They meet first after 40 s. -
Quick Check:
In 40 s, A covers 240 m and B covers 160 m; together 400 m → they meet ✅
Quick Variations
1. How many times they meet in T seconds: use (relative speed × T) ÷ track length and take integer part.
2. When will they be together after k meetings: set (relative speed × time) = k × track length and solve for time.
3. Positions after time T: compute (speed × T) mod track length for each runner to get exact location.
4. Opposite-direction multiple meetings: meetings are evenly spaced by time = track length ÷ (sum of speeds).
5. When counting overtakes: faster can overtake multiple times; each overtake corresponds to one extra lap covered relative to the slower runner.
Trick to Always Use
- Step 1: Convert circular motion into linear laps: treat each lap as length L.
- Step 2: Use relative speed: add for opposite, subtract for same direction.
- Step 3: For meetings/overtakes set (relative speed × time) = n × L for integer n (number of meetings/overtakes).
- Step 4: For final positions compute distance modulo L to find the place on the track.
Summary
Summary
For circular track problems with the same starting point:
- Use relative speed (add if opposite, subtract if same direction).
- Time to meet/overcome = track length ÷ relative speed (for first event).
- Number of meetings/overtakes in time T = ⌊(relative speed × T) ÷ track length⌋.
- Find exact location after time T by taking each runner’s distance (speed × T) modulo track length.
- Always keep units consistent and use the actionable steps to structure your solution.
Hint: For opposite directions, add speeds and divide track length by sum.
Common Mistakes: Subtracting speeds instead of adding for opposite direction.