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Circular Track / Same Starting Point

Introduction

circular track with same starting point पर होने वाले प्रश्न मिलन के समय, ओवरटेक, รอบों की संख्या, और सापेक्ष स्थितियों के बारे में पूछते हैं जब दो या अधिक धावक साथ में start करते हैं (same या opposite directions)।

यह pattern महत्वपूर्ण है क्योंकि यह relative speed के विचार को laps (track length) की समझ के साथ जोड़ता है और सरल सूत्रों से कई contest-style सवाल जल्दी हल करने में मदद करता है।

Pattern: Circular Track / Same Starting Point

Pattern

Key concept: वृत्ताकार गति को linear distance (laps) में बदलें और relative speed का उपयोग करें: जब दो लोग मिलते या ओवरटेक करते हैं, तो वे integer laps का अंतर कवर करते हैं।

  • 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: अगर opposite directions → meetings = ⌊(relative speed × T) ÷ track length⌋ (integer meetings को count करें). अगर same direction → overtakes = ⌊(relative speed × T) ÷ track length⌋.
  • When asked about positions after time T: प्रत्येक का distance निकालें (speed × T), और position पाने के लिए उसे track length से modulo करें।
  • For integer lap questions: अगर faster ने slower से time T में n अधिक lap पूरे किए हों, तो (faster - slower) × T = n × track length।

Step-by-Step Example

Question

दो धावक एक 400 m लंबी circular track पर same point से start करते हैं और opposite directions में दौड़ते हैं। उनकी speeds 6 m/s और 4 m/s हैं। वे पहली बार कब मिलेंगे?

Solution

  1. Step 1: Identify relative speed

    Opposite directions → relative speed = 6 + 4 = 10 m/s.

  2. Step 2: Time to meet

    Time = Track length ÷ Relative speed = 400 ÷ 10 = 40 seconds.

  3. Final Answer:

    वे पहली बार 40 s के बाद मिलते हैं।

  4. Quick Check:

    40 s में, A 240 m और B 160 m कवर करता है; कुल मिलाकर 400 m → वे मिलते हैं ✅

Quick Variations

1. How many times they meet in T seconds: (relative speed × T) ÷ track length का integer part लें।

2. When will they be together after k meetings: (relative speed × time) = k × track length रखकर time निकाले।

3. Positions after time T: हर धावक का (speed × T) mod track length निकालें ताकि exact location मिले।

4. Opposite-direction multiple meetings: meetings का अंतर समय = track length ÷ (sum of speeds) होता है।

5. When counting overtakes: faster कई बार overtake कर सकता है; हर overtake slower के सापेक्ष एक extra lap पूरा करने के बराबर होता है।

Trick to Always Use

  • Step 1: Circular motion को linear laps में बदलें: हर lap को length L मानें।
  • Step 2: Relative speed यूज़ करें: opposite में जोड़ें, same direction में घटा दें।
  • Step 3: Meetings/overtakes के लिए set करें (relative speed × time) = n × L जहाँ n integer (meetings/overtakes) हो।
  • Step 4: अंतिम स्थिति के लिए distance modulo L लें ताकि track पर position मिले।

Summary

Summary

Same starting point वाले circular track के प्रश्नों के लिए:

  • Use relative speed (opposite में add, same direction में subtract)।
  • पहली meeting/overcome का समय = track length ÷ relative speed (for first event)।
  • T समय में meetings/overtakes की संख्या = ⌊(relative speed × T) ÷ track length⌋।
  • समय T के बाद exact location पाने के लिए प्रत्येक की distance (speed × T) को track length से modulo करें।
  • Units consistent रखें और समाधान को structure करने के लिए दिए गए actionable steps का पालन करें।

Practice

(1/5)
1. Two runners start from the same point on a circular track of length 300 m and run in opposite directions with speeds 5 m/s and 7 m/s. After how many seconds will they meet for the first time?
easy
A. 25 s
B. 20 s
C. 22 s
D. 30 s

Solution

  1. Step 1: Identify motion type

    Opposite directions → relative speed = 5 + 7 = 12 m/s.

  2. Step 2: Compute time

    Time = Track length ÷ Relative speed = 300 ÷ 12 = 25 s.

  3. Final Answer:

    They meet after 25 s → Option A.

  4. Quick Check:

    Together they cover 12 × 25 = 300 m → one full lap ✅
Hint: For opposite directions, add speeds and divide track length by sum.
Common Mistakes: Subtracting speeds instead of adding for opposite direction.
2. Two boys A and B start from the same point on a circular track of 400 m and run in the same direction at 6 m/s and 4 m/s. After how many seconds will A overtake B for the first time?
easy
A. 100 s
B. 200 s
C. 150 s
D. 250 s

Solution

  1. Step 1: Motion type

    Same direction → relative speed = 6 - 4 = 2 m/s.

  2. Step 2: Compute time

    Time = Track length ÷ Relative speed = 400 ÷ 2 = 200 s.

  3. Final Answer:

    A overtakes B after 200 s → Option B.

  4. Quick Check:

    In 200 s, A covers 6×200 = 1200 m and B 4×200 = 800 m → difference = 400 m (one lap) ✅
Hint: For same direction, subtract speeds and divide track length by the difference.
Common Mistakes: Using total speeds instead of difference for same direction.
3. Two cyclists start together on a circular track of 360 m in opposite directions. Their speeds are 8 m/s and 4 m/s. How long will they take to meet for the second time?
easy
A. 30 s
B. 40 s
C. 60 s
D. 45 s

Solution

  1. Step 1: Relative speed

    Opposite directions → relative speed = 8 + 4 = 12 m/s.

  2. Step 2: Time for first meeting

    Time₁ = 360 ÷ 12 = 30 s.

  3. Step 3: Second meeting

    The second meeting occurs after 2 × Time₁ = 60 s.

  4. Final Answer:

    They meet the second time after 60 s → Option C.

  5. Quick Check:

    After 60 s they cover 12×60 = 720 m combined → two laps (2×360) ✅
Hint: Second meeting = 2 × (track ÷ relative speed) for opposite-direction starters.
Common Mistakes: Counting the starting instant as a meeting or using only first-meeting time.
4. On a circular track of 500 m, A runs at 10 m/s and B at 6 m/s in the same direction. After how many seconds will A be 1 lap ahead of B?
medium
A. 125 s
B. 120 s
C. 130 s
D. 150 s

Solution

  1. Step 1: Relative speed

    Same direction → relative speed = 10 - 6 = 4 m/s.

  2. Step 2: Time to gain one lap

    Time = Track length ÷ Relative speed = 500 ÷ 4 = 125 s.

  3. Final Answer:

    A will be one lap ahead after 125 s → Option A.

  4. Quick Check:

    In 125 s A covers 10×125 = 1250 m, B covers 6×125 = 750 m → difference = 500 m (one lap) ✅
Hint: One-lap lead → track length ÷ (faster - slower).
Common Mistakes: Confusing lap distance with total distance run.
5. Two runners start simultaneously from the same point on a circular track of 600 m in opposite directions. Speeds are 5 m/s and 7 m/s. How many times will they meet in 10 minutes?
medium
A. 4 times
B. 5 times
C. 6 times
D. 12 times

Solution

  1. Step 1: Relative speed

    Opposite directions → relative speed = 5 + 7 = 12 m/s.

  2. Step 2: Time per meeting

    Time per meeting = Track length ÷ Relative speed = 600 ÷ 12 = 50 s.

  3. Step 3: Meetings in 10 minutes

    Total time = 10 × 60 = 600 s → Meetings = 600 ÷ 50 = 12 times.

  4. Final Answer:

    They meet 12 times in 10 minutes → Option D.

  5. Quick Check:

    Every 50 s they meet → 600/50 = 12 meetings (first meeting at 50 s, last at 600 s) ✅
Hint: Number of meetings = (relative speed × total time) ÷ track length.
Common Mistakes: Forgetting to convert minutes to seconds or excluding meetings at exact endpoints incorrectly.

Mock Test

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