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Relative Speed (Same or Opposite Direction)

Introduction

कई motion problems में दो objects एक-दूसरे के relative move करते हैं - जैसे दो cars, दो trains, या एक व्यक्ति और train। Relative speed यह बताती है कि उनके बीच की दूरी कितनी तेजी से बदल रही है, जिससे meeting और overtaking वाले सवाल बहुत आसान हो जाते हैं।

Relative speed में mastery होने से आप जल्दी से meeting time, overtaking time और closing/separating rate निकाल सकते हैं - चाहे दोनों same direction में हों या opposite direction में।

Pattern: Relative Speed (Same or Opposite Direction)

Pattern

Key concept: दो objects move कर रहे हों तो एक object को stationary मानकर दूसरे की relative speed से calculation करो।

  • Same direction (दोनों एक ही दिशा में): Relative speed = |v₁ - v₂| (speeds का difference)।
  • Opposite direction (एक-दूसरे की ओर): Relative speed = v₁ + v₂ (speeds का sum)।

Units हमेशा same रखें (km/h → hours के लिए, m/s → seconds के लिए)। Overtaking type में distance = cover की जाने वाली पूरी लंबाई → Time = Distance ÷ Relative speed.

Step-by-Step Example

Question

दो trains 300 km दूर हैं और एक-दूसरे की ओर 80 km/h और 70 km/h की speed से चल रही हैं। वे कितने समय बाद मिलेंगी?

Solution

  1. Step 1: Given values पहचानें

    Distance = 300 km; v₁ = 80 km/h; v₂ = 70 km/h.

  2. Step 2: Relative speed निकालें

    Opposite direction → Relative speed = 80 + 70 = 150 km/h.

  3. Step 3: Meeting time निकालें

    Time = 300 ÷ 150 = 2 hours.

  4. Final Answer:

    वे 2 hours में मिलेंगी।

  5. Quick Check:

    2 hours में: first train = 160 km, second = 140 km → total = 300 km ✔️

Quick Variations

1. Train overtaking person/another train → distance = length of train + initial gap।

2. Circular track पर repeated meetings → direction + relative speed + laps/LCM logic।

3. Mixed units → speeds को पहले km/h ↔ m/s में convert करें और फिर apply करें।

4. One object stationary हो (v₂ = 0) → relative speed = moving object की speed।

Trick to Always Use

  • Step 1: सबसे पहले direction check करें - same या opposite?
  • Step 2: Opposite → speeds add; Same → speeds subtract (v₁ - v₂)।
  • Step 3: Formula: Time = Distance ÷ Relative speed (units हमेशा match रखें)।
  • Step 4: Overtaking में distance = दूसरे object की length + कोई भी initial gap।

Summary

Summary

Key takeaways:

  • Relative speed = opposite direction → sum; same direction → difference।
  • Units हमेशा consistent रखें (km/h + hours या m/s + seconds)।
  • Overtaking में distance हमेशा full length cover करना होता है।
  • Confusion हो तो दोनों objects द्वारा same time में तय की गई distances compare करके answer verify करें।

Practice

(1/5)
1. Two trains are 300 km apart and move towards each other at speeds of 80 km/h and 70 km/h. How long will they take to meet?
easy
A. 2 hours
B. 1.5 hours
C. 2.5 hours
D. 3 hours

Solution

  1. Step 1: Identify the Case

    They move towards each other → opposite directions.

  2. Step 2: Determine Relative Speed

    Relative speed = 80 + 70 = 150 km/h.

  3. Step 3: Compute Time

    Time = Distance ÷ Relative speed = 300 ÷ 150 = 2 hours.

  4. Final Answer:

    They meet in 2 hours → Option A.

  5. Quick Check:

    In 2 h, first covers 160 km and second 140 km → total 300 km ✅
Hint: Opposite direction → add speeds (v1 + v2).
Common Mistakes: Subtracting speeds instead of adding when objects move toward each other.
2. Car A travels at 90 km/h and Car B at 70 km/h in the same direction. If Car A is 2 km behind Car B, how long will it take to overtake?
easy
A. 5 min
B. 6 min
C. 8 min
D. 10 min

Solution

  1. Step 1: Identify the Case

    Same direction → use relative speed = difference of speeds.

  2. Step 2: Determine Relative Speed

    Relative speed = 90 - 70 = 20 km/h = 20 km per hour.

  3. Step 3: Compute Time

    Time (hours) = Distance ÷ Relative speed = 2 ÷ 20 = 0.1 h = 6 minutes.

  4. Final Answer:

    Car A will overtake in 6 minutes → Option B.

  5. Quick Check:

    In 0.1 h, Car A goes 9 km, Car B 7 km → gap closed by 2 km ✅
Hint: Same direction → subtract speeds (v1 - v2).
Common Mistakes: Adding speeds instead of subtracting for same-direction problems.
3. Two cyclists move in opposite directions at 15 km/h and 25 km/h. How far apart will they be after 2 hours?
easy
A. 70 km
B. 60 km
C. 80 km
D. 90 km

Solution

  1. Step 1: Identify the Case

    Opposite directions → relative speed = sum of speeds.

  2. Step 2: Calculate Relative Speed

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

  3. Step 3: Compute Separation After 2 Hours

    Distance apart = Relative speed × Time = 40 × 2 = 80 km.

  4. Final Answer:

    They will be 80 km apart → Option C.

  5. Quick Check:

    Each covers 30 km and 50 km respectively → total 80 km ✅
Hint: Opposite directions → add speeds then multiply by time.
Common Mistakes: Using difference of speeds instead of sum.
4. Two trains, each 120 m long, run in opposite directions at 54 km/h and 36 km/h. How long will they take to cross each other completely?
medium
A. 9.6 s
B. 8.6 s
C. 10.4 s
D. 12.4 s

Solution

  1. Step 1: Convert Units

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

  2. Step 2: Find Relative Speed

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

  3. Step 3: Calculate Time to Cross

    Total length = 120 + 120 = 240 m. Time = 240 ÷ 25 = 9.6 s.

  4. Final Answer:

    They cross each other in 9.6 seconds → Option A.

  5. Quick Check:

    25 × 9.6 = 240 m ✅
Hint: Convert km/h → m/s first (×5/18), add speeds for opposite direction, then use Distance ÷ Speed.
Common Mistakes: Forgetting to convert units or omitting one train's length from distance.
5. A train 200 m long overtakes a man walking at 6 km/h in the same direction. The train speed is 54 km/h. Find the time taken to pass the man.
medium
A. 12 s
B. 14 s
C. 16 s
D. 15 s

Solution

  1. Step 1: Convert Speeds to m/s

    Train: 54×5/18 = 15 m/s; Man: 6×5/18 = 1.666... m/s.

  2. Step 2: Determine Relative Speed

    Same direction → relative speed = 15 - 1.666... = 13.333... m/s.

  3. Step 3: Compute Time

    Distance to cover = length of train = 200 m. Time = 200 ÷ 13.333... = 15 seconds.

  4. Final Answer:

    Time taken ≈ 15 seconds → Option D.

  5. Quick Check:

    13.333... × 15 = 200 m ✅
Hint: Same direction → convert to m/s and subtract speeds before Distance ÷ Relative speed.
Common Mistakes: Using km/h directly with seconds or forgetting to subtract the walker's speed.

Mock Test

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