Quiz: Circular Track Problems
This quiz tests the core ideas of relative speed on circular tracks from Lesson 8: Circular Tracks: Meeting and Passing. You will apply the key formulas for objects moving in opposite directions and in the same direction, find the time to meet for the first time, count the number of meetings in a given time, and determine when runners meet again at the starting point.
Key formulas to keep in mind
- Opposite directions: relative speed = ##a + b##; time to first meet = ##\dfrac{L}{a+b}##
- Same direction: relative speed = ##a – b##; time to first meet = ##\dfrac{L}{a-b}##
- Meetings in time ##T##: number of meetings = ##\left\lfloor \dfrac{T}{\text{time per meeting}} \right\rfloor##
- Meeting at the starting point: time = LCM of the individual lap times
Work carefully with units and always identify whether the motion is in the same or opposite direction. Good luck!
Q1. Two runners move on a circular track of length ##400## m in opposite directions with speeds ##6## m/s and ##4## m/s. What is their relative speed?
When moving in opposite directions, relative speed is the sum of the speeds: ##6+4=10## m/s.
Q2. Two runners move on a circular track of length ##400## m in the same direction with speeds ##6## m/s and ##4## m/s. What is their relative speed?
When moving in the same direction, relative speed is the difference of the speeds: ##6-4=2## m/s.
Q3. Two runners move in opposite directions on a ##400## m circular track with speeds ##6## m/s and ##4## m/s. How long do they take to meet for the first time?
Opposite directions: ##time = \dfrac{L}{a+b}=\dfrac{400}{10}=40## s.
Q4. Two runners move in the same direction on a ##400## m circular track with speeds ##6## m/s and ##4## m/s. How long do they take to meet for the first time?
Same direction: ##time = \dfrac{L}{a-b}=\dfrac{400}{2}=200## s.
Q5. Two runners move in opposite directions on a ##600## m circular track with speeds ##8## m/s and ##4## m/s. How many times do they meet in ##5## minutes?
Time per ##meeting = \dfrac{600}{8+4}=50## s. In ##300## s, ##meetings = \dfrac{300}{50}=6##.
Q6. Two runners move in the same direction on a ##400## m circular track with speeds ##10## m/s and ##6## m/s. How many times does the faster runner overtake the slower one in ##10## minutes?
Time per ##meeting = \dfrac{400}{10-6}=100## s. In ##600## s, ##meetings = \dfrac{600}{100}=6##.
Q7. A and B run on a circular track in opposite directions with speeds ##5## m/s and ##3## m/s. The track length is ##240## m. After how many seconds do they meet for the first time?
Opposite directions: ##time = \dfrac{240}{5+3}=\dfrac{240}{8}=30## s.
Q8. A and B run on a circular track in the same direction with speeds ##5## m/s and ##3## m/s. The track length is ##240## m. After how many seconds does A first overtake B?
Same direction: ##time = \dfrac{240}{5-3}=\dfrac{240}{2}=120## s.
Q9. Two runners complete one lap of a circular track in ##4## minutes and ##6## minutes respectively. If they start together from the same point in the same direction, after how many minutes will they meet again at the starting point?
They meet at the starting point after the LCM of the lap times: LCM of ##4## and ##6## is ##12## minutes.
Q10. Two runners complete one lap of a circular track in ##3## minutes and ##5## minutes respectively. Starting together from the same point, after how many minutes will they next meet at the starting point?
Meeting at the starting point occurs after the LCM of the lap times: LCM of ##3## and ##5## is ##15## minutes.
Q11. Two runners move in opposite directions on a circular track with speeds ##7## m/s and ##3## m/s. The track length is ##500## m. How many times do they meet in ##100## seconds?
Time per ##meeting = \dfrac{500}{7+3}=50## s. In ##100## s, ##meetings = \dfrac{100}{50}=2##.
Q12. On a circular track, two runners move in the same direction with speeds ##9## m/s and ##5## m/s. The track length is ##800## m. How long does the faster runner take to gain one full lap on the slower runner?
Gaining one full lap means meeting once: ##time = \dfrac{800}{9-5}=\dfrac{800}{4}=200## s.
Q13. Two cyclists ride in opposite directions on a circular track of length ##1## km with speeds ##12## km/h and ##8## km/h. How long do they take to meet for the first time?
Relative ##speed = 12+8=20## km/h. ##Time = \dfrac{1}{20}## ##h = 3## minutes.
Q14. Two cyclists ride in the same direction on a circular track of length ##1## km with speeds ##12## km/h and ##8## km/h. How long does the faster cyclist take to first overtake the slower one?
Relative ##speed = 12-8=4## km/h. ##Time = \dfrac{1}{4}## ##h = 15## minutes.
Q15. Three runners complete one lap of a circular track in ##2##, ##3## and ##4## minutes respectively. Starting together from the same point, after how many minutes will all three next meet at the starting point?
They meet at the starting point after the LCM of the lap times: LCM of ##2,3,4## is ##12## minutes.
Q16. Two runners move in opposite directions on a circular track of length ##300## m with speeds ##4## m/s and ##2## m/s. How many times do they meet in ##5## minutes?
Time per ##meeting = \dfrac{300}{4+2}=50## s. In ##300## s, ##meetings = \dfrac{300}{50}=6##.
Q17. Two runners move in the same direction on a circular track of length ##600## m with speeds ##8## m/s and ##5## m/s. How many times does the faster runner overtake the slower one in ##10## minutes?
Time per ##meeting = \dfrac{600}{8-5}=200## s. In ##600## s, ##meetings = \dfrac{600}{200}=3##.
Q18. Two runners start together from the same point on a circular track and run in opposite directions with speeds ##6## m/s and ##4## m/s. The track length is ##500## m. How far has the faster runner travelled when they first meet?
Time to ##meet = \dfrac{500}{10}=50## s. Distance by faster ##runner = 6\times 50=300## m.
Q19. Two runners start together from the same point on a circular track and run in the same direction with speeds ##7## m/s and ##5## m/s. The track length is ##400## m. How far has the faster runner travelled when he first overtakes the slower one?
Time to ##overtake = \dfrac{400}{7-5}=200## s. Distance by faster ##runner = 7\times 200=1400## m.
Q20. Two runners complete one lap of a circular track in ##6## minutes and ##9## minutes respectively. Starting together from the same point in the same direction, after how many minutes will they next meet at the starting point?
They meet at the starting point after the LCM of the lap times: LCM of ##6## and ##9## is ##18## minutes.