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Meeting and Overtaking Problems

Introduction

Meeting और overtaking problems Time, Speed & Distance का एक मुख्य हिस्सा हैं। ये पूछते हैं कि दो moving objects कब मिलेंगे (एक-दूसरे की ओर आते हुए) या कब एक объект दूसरे को पकड़कर पास कर देगा (same direction)। इन सवालों को हल करना आसान हो जाता है जब आप scenario को relative speed और उस specific distance में बदल दें जिसे close करना है।

यह pattern important है क्योंकि यह competitive exams और रोज़मर्रा की reasoning में अक्सर आता है - trains, cars, runners - और हर बार इन्हें एक ही छोटे set of steps से हल किया जा सकता है।

Pattern: Meeting and Overtaking Problems

Pattern

Key concept: दो-body problem को एक-body में reduce कर दें, relative speed और उस खास distance का उपयोग करके।

  • Meeting (एक-दूसरे की ओर चल रहे): Relative speed = v₁ + v₂. Time = Initial separation ÷ (v₁ + v₂).
  • Overtaking (same direction): Relative speed = |v₁ - v₂| (faster - slower). Time = Distance to be covered ÷ (v_faster - v_slower).
  • Distance to be covered: meeting के लिए = initial gap; overtaking के लिए = object की length + कोई initial gap (यदि लागू)।
  • Unit consistency: km/h के साथ hours या m/s के साथ seconds उपयोग करें - compute करने से पहले convert कर लें।

Step-by-Step Example

Question

दो cars 150 km दूर हैं और एक-दूसरे की ओर 70 km/h और 50 km/h की speeds से चल रही हैं। वे कब मिलेंगी?

Solution

  1. Step 1: दिए गए values पहचानें

    Distance = 150 km; speeds v₁ = 70 km/h, v₂ = 50 km/h.

  2. Step 2: केस तय करें और relative speed निकालें

    वे एक-दूसरे की ओर चल रहे हैं → Relative speed = v₁ + v₂ = 70 + 50 = 120 km/h.

  3. Step 3: Time formula लागू करें

    Time = Distance ÷ Relative speed = 150 ÷ 120 = 1.25 hours = 1 hour 15 minutes.

  4. Final Answer:

    वे 1 hour 15 minutes में मिलेंगे।

  5. Quick Check:

    1.25 घंटे में पहला 87.5 km और दूसरा 62.5 km चलता है → कुल 150 km ✅

Quick Variations

1. Two objects different start times: meeting के लिए covered distances निकाले (relative speed का उपयोग करें और head-start time adjust करें)।

2. दो लंबी वस्तुओं का overtaking: cover की जाने वाली distance = दोनों lengths का sum (यदि एक को पूरी तरह clear करना हो)।

3. Circular track पर meeting: relative speed इस्तेमाल करें; repeated meetings के लिए lap-times का LCM या circumference ÷ relative speed लें।

4. Mixed units: पहले units convert करें (जैसे km/h और m/s) और फिर compute करें।

Trick to Always Use

  • Step 1: पूछें: "क्या वे एक-दूसरे की ओर (meeting) हैं या same direction में (overtaking)?"
  • Step 2: Meeting → relative speed = v₁ + v₂. Overtaking → relative speed = v_faster - v_slower.
  • Step 3: सही distance तय करें - initial separation, object की length, या combined lengths।
  • Step 4: Units match हों यह सुनिश्चित करें। समय को seconds/minutes में चाहिए तो appropriate conversion करें।
  • Step 5: Time = Distance ÷ Relative speed। आवश्यकता होने पर answer को minutes/seconds में दिखाएँ।

Summary

Summary

  • पहचानें कि situation meeting है या overtaking - यही सही relative speed तय करेगा।
  • Relative speed निकालने के लिए meeting में जोड़ें, overtaking में घटाएं।
  • Overtaking में distance हमेशा सही तरह से लें - initial gap, length या दोनों।
  • Units consistent रहें (e.g., km/h के साथ hours या m/s के साथ seconds)।

याद रखने का formula:
“Meeting → add speeds, Overtaking → subtract speeds, फिर Time = Distance ÷ Relative Speed।”

Practice

(1/5)
1. Two cars start from opposite points 200 km apart and move towards each other at speeds of 60 km/h and 40 km/h. How long will they take to meet?
easy
A. 2 hours
B. 2.5 hours
C. 3 hours
D. 4 hours

Solution

  1. Step 1: Identify the Case

    They are moving towards each other → meeting problem.

  2. Step 2: Compute Relative Speed

    Relative speed = 60 + 40 = 100 km/h.

  3. Step 3: Compute Time

    Time = Distance ÷ Relative speed = 200 ÷ 100 = 2 hours.

  4. Final Answer:

    They will meet in 2 hours → Option A.

  5. Quick Check:

    60×2 + 40×2 = 120 + 80 = 200 ✅
Hint: Meeting → Add speeds to get relative speed.
Common Mistakes: Subtracting speeds instead of adding for opposite directions.
2. Two runners start from the same point in the same direction. The first runs at 8 km/h and the second at 10 km/h. How long will it take for the faster runner to be 2 km ahead?
easy
A. 0.5 hours
B. 1 hour
C. 1.5 hours
D. 2 hours

Solution

  1. Step 1: Identify the Case

    Same direction → overtaking type (faster gains on slower).

  2. Step 2: Compute Relative Speed

    Relative speed = 10 - 8 = 2 km/h.

  3. Step 3: Compute Time

    Time = Distance ÷ Relative speed = 2 ÷ 2 = 1 hour.

  4. Final Answer:

    The faster runner will be 2 km ahead in 1 hour → Option B.

  5. Quick Check:

    In 1 h: faster = 10 km, slower = 8 km → gap = 2 km ✅
Hint: Same direction → Subtract speeds (faster - slower).
Common Mistakes: Adding speeds instead of subtracting for overtaking cases.
3. Two trains, each 100 m long, run on parallel tracks in opposite directions at 54 km/h and 36 km/h. How long will they take to cross each other completely?
easy
A. 6 s
B. 7 s
C. 8 s
D. 9 s

Solution

  1. Step 1: Convert Speeds to m/s

    54×5/18 = 15 m/s; 36×5/18 = 10 m/s.

  2. Step 2: Relative Speed

    Opposite directions → add speeds: 15 + 10 = 25 m/s.

  3. Step 3: Distance to Cover

    Total distance = 100 + 100 = 200 m. Time = 200 ÷ 25 = 8 s.

  4. Final Answer:

    They cross each other in 8 seconds → Option C.

  5. Quick Check:

    25 × 8 = 200 m ✅
Hint: Opposite direction → Add speeds, convert to m/s when lengths are in metres.
Common Mistakes: Using difference of speeds for opposite-direction crossing.
4. A train 180 m long travels at 54 km/h and overtakes another train 120 m long running at 36 km/h in the same direction. Find the time to overtake completely.
medium
A. 20 s
B. 30 s
C. 40 s
D. 60 s

Solution

  1. Step 1: Convert Speeds to m/s

    54×5/18 = 15 m/s; 36×5/18 = 10 m/s.

  2. Step 2: Relative Speed

    Same direction → relative speed = 15 - 10 = 5 m/s.

  3. Step 3: Distance to Cover

    Distance = 180 + 120 = 300 m (faster must cover both lengths to pass completely).

  4. Step 4: Compute Time

    Time = 300 ÷ 5 = 60 s.

  5. Final Answer:

    They overtake in 60 seconds → Option D.

  6. Quick Check:

    5 × 60 = 300 m ✅
Hint: Same direction overtaking → subtract speeds (in m/s) and add lengths.
Common Mistakes: Forgetting to convert units or to add both train lengths.
5. Two cyclists start 60 km apart and travel towards each other at 15 km/h and 25 km/h. After how much time will they meet?
medium
A. 1.5 hours
B. 2 hours
C. 2.5 hours
D. 3 hours

Solution

  1. Step 1: Identify the Case

    They move towards each other → meeting problem.

  2. Step 2: Compute Relative Speed

    Relative speed = 15 + 25 = 40 km/h.

  3. Step 3: Compute Time

    Time = Distance ÷ Relative speed = 60 ÷ 40 = 1.5 hours.

  4. Final Answer:

    They will meet in 1.5 hours → Option A.

  5. Quick Check:

    15×1.5 + 25×1.5 = 22.5 + 37.5 = 60 ✅
Hint: Meeting → Add speeds then divide the separation by total speed.
Common Mistakes: Subtracting speeds when objects move toward each other.

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