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Complex Motion / Variable Speeds

Introduction

Complex Motion / Variable Speeds उन समस्याओं को कवर करता है जहाँ कोई चलती चीज़ अपनी यात्रा के दौरान गति बदलती है - उदाहरण के लिए, अलग-अलग हिस्सों में अलग speeds, रुकना, या acceleration/deceleration phases। ये प्रश्न कई time-distance segments को जोड़ते हैं और हर segment की distance, time और effective speed का सावधानीपूर्वक हिसाब माँगते हैं।

यह pattern महत्वपूर्ण है क्योंकि वास्तविक दुनिया की गति अक्सर एक निरंतर speed पर नहीं रहती; segment-wise गणनाएँ और reductions (जैसे averages, total time, net distance) में महारत आपको परतदार aptitude प्रश्नों को आत्मविश्वास से हल करने देती है।

Pattern: Complex Motion / Variable Speeds

Pattern

Key concept: यात्रा को स्पष्ट segments में बाँटें, हर segment के लिए distance/time निकालें, फिर उन्हें मिलाकर काम करें। Total Distance = Σ distances और Total Time = Σ times से overall average speed = Total Distance ÷ Total Time निकालें।

  • Segment approach: हर भाग को उसकी खुद की speed/time/distance मानकर रखें।
  • Average speed over whole trip: यह speeds का arithmetic mean नहीं होता - कुल distance ÷ कुल time निकालें।
  • When distances are equal but speeds differ: दो segments के लिए harmonic mean उपयोग करें: 2ab/(a+b) (ज़्यादा segments के लिए total distance/total time से generalize करें)।
  • When times are equal but speeds differ: equal-time segments के लिए speeds का arithmetic mean लें।
  • Include rests or pauses: pause का time total time में जोड़ें (distance अपरिवर्तित) उसके बाद average speed निकालें।

Step-by-Step Example

Question

एक cyclist 30 km की दूरी 15 km/h से और उसके बाद 20 km 10 km/h से चलता है। पूरे trip के लिए cyclist की average speed क्या होगी?

Solution

  1. Step 1: प्रत्येक segment का time निकालें

    Segment 1 time = Distance ÷ Speed = 30 ÷ 15 = 2 hours.
    Segment 2 time = 20 ÷ 10 = 2 hours.

  2. Step 2: कुल distance और कुल time

    Total distance = 30 + 20 = 50 km.
    Total time = 2 + 2 = 4 hours.

  3. Step 3: Average speed

    Average speed = Total distance ÷ Total time = 50 ÷ 4 = 12.5 km/h.

  4. Final Answer:

    Average speed = 12.5 km/h.

  5. Quick Check:

    अगर average (15+10)/2 = 12.5 होता तो भी सही रहता (क्योंकि times बराबर हैं)। जाँच: 12.5 × 4 = 50 ✅

Quick Variations

1. Segments with pauses: pause time को total time में जोड़ें (distance वही रहती है) और फिर average speed निकालें।

2. Variable speeds over unequal distances: हर segment का time निकालें, times जोड़ें, फिर total distance ÷ total time करें।

3. Piecewise acceleration: अगर किसी segment में acceleration uniform है, तो उस segment का average speed = (u + v)/2 लेकर उस segment के time/distance निकालें।

4. Multiple equal-distance segments: generalized harmonic mean उपयोग करें: Total distance ÷ (Σ (distance_i / speed_i)).

5. Return trips with different speeds: outward और return को दो segments मानें - कुल time महत्वपूर्ण है।

Trick to Always Use

  • Step 1: यात्रा को साफ़ segments में बाँटें और हर एक को label करें (distance, speed, time)।
  • Step 2: प्रत्येक segment का time निकालें: time_i = distance_i ÷ speed_i (या distance = speed × time)।
  • Step 3: distances और times जोड़ें: Total distance = Σ distance_i; Total time = Σ time_i (rests शामिल करें)।
  • Step 4: Overall average speed = Total distance ÷ Total time. Speeds को सीधे average मत करें (जब तक times equal न हों) - totals पर सोचें।
  • Optional: distances बराबर हों तो harmonic mean, times बराबर हों तो arithmetic mean उपयोग करें।

Summary

Summary

Complex motion / variable speeds के लिए:

  • हमेशा trip को segments में बाँटें और हर segment का time स्पष्ट रूप से निकालें।
  • पूरे trip का average speed = (total distance) ÷ (total time) - सिम्पल mean केवल special cases में सही होता है।
  • रुकने का समय total time में जोड़ें; इससे average speed घट जाएगी जबकि distance नहीं बदलती।
  • बराबर-distance segments के लिए harmonic mean; बराबर-time segments के लिए arithmetic mean उपयोग करें।
  • Quick checks (totals फिर से गिनना या alternate formulas) सामान्य गलतियों को जल्दी पकड़ लेते हैं।

Practice

(1/5)
1. A car travels 40 km at 20 km/h and the next 40 km at 60 km/h. Find its average speed for the whole journey.
easy
A. 30 km/h
B. 35 km/h
C. 40 km/h
D. 45 km/h

Solution

  1. Step 1: Recognise equal distances

    Both segments are 40 km each → use harmonic mean for equal-distance segments.

  2. Step 2: Apply harmonic mean

    Average speed = (2ab)/(a + b) = (2 × 20 × 60) ÷ (20 + 60) = 2400 ÷ 80 = 30 km/h.

  3. Final Answer:

    Average speed = 30 km/h → Option A.

  4. Quick Check:

    Time1 = 40/20 = 2 h; Time2 = 40/60 = 0.6667 h; total time = 2.6667 h; total distance = 80 km → 80 ÷ 2.6667 = 30 ✅
Hint: Equal distances → harmonic mean (2ab)/(a+b).
Common Mistakes: Taking arithmetic mean of speeds instead of harmonic mean.
2. A bus covers the first 50 km at 25 km/h and the next 100 km at 50 km/h. What is its average speed for the total trip?
easy
A. 33.33 km/h
B. 35 km/h
C. 37.5 km/h
D. 40 km/h

Solution

  1. Step 1: Compute times for segments

    Time₁ = 50 ÷ 25 = 2 h; Time₂ = 100 ÷ 50 = 2 h.

  2. Step 2: Total distance & time

    Total distance = 150 km; Total time = 4 h.

  3. Step 3: Average speed

    Average speed = 150 ÷ 4 = 37.5 km/h.

  4. Final Answer:

    Average speed = 37.5 km/h → Option C.

  5. Quick Check:

    Equal times → arithmetic mean (25 + 50)/2 = 37.5 ✅
Hint: Equal-time segments → arithmetic mean of speeds.
Common Mistakes: Using harmonic mean when times are equal.
3. A car travels from A to B at 30 km/h and returns at 45 km/h. What is its average speed for the round trip?
easy
A. 36 km/h
B. 37.5 km/h
C. 38 km/h
D. 40 km/h

Solution

  1. Step 1: Round trip equal distances

    To-and-fro are equal-distance segments → use harmonic mean.

  2. Step 2: Apply formula

    Average speed = (2ab)/(a + b) = (2 × 30 × 45) ÷ (30 + 45) = 2700 ÷ 75 = 36 km/h.

  3. Final Answer:

    Average speed = 36 km/h → Option A.

  4. Quick Check:

    If distance one way = D, total time = D/30 + D/45 = D(1/30 + 1/45) = D(1/18) → average = 2D ÷ (D/18) = 36 ✅
Hint: Round trip → harmonic mean of the two speeds.
Common Mistakes: Using arithmetic mean instead of harmonic mean for round trips.
4. A man walks 10 km at 5 km/h, rests for 1 hour, then walks another 15 km at 3 km/h. Find his average speed for the entire journey.
medium
A. 3.00 km/h
B. 3.125 km/h
C. 3.75 km/h
D. 4.00 km/h

Solution

  1. Step 1: Compute times of motion and rest

    Time₁ = 10 ÷ 5 = 2 h; Rest = 1 h; Time₂ = 15 ÷ 3 = 5 h.

  2. Step 2: Total distance & total time

    Total distance = 10 + 15 = 25 km.
    Total time = 2 + 1 + 5 = 8 h.

  3. Step 3: Average speed

    Average speed = 25 ÷ 8 = 3.125 km/h.

  4. Final Answer:

    Average speed = 3.125 km/h → Option B.

  5. Quick Check:

    Including rest increases total time (8 h) → 25/8 = 3.125 ✅
Hint: Include rest time in total time; average = total distance ÷ total time.
Common Mistakes: Ignoring rest or rounding too early.
5. A truck covers half the distance at 40 km/h and the other half at 60 km/h. Find its average speed.
medium
A. 45 km/h
B. 46 km/h
C. 50 km/h
D. 48 km/h

Solution

  1. Step 1: Recognise equal half-distances

    Each half is equal distance → harmonic mean of speeds applies.

  2. Step 2: Apply harmonic mean

    Average speed = (2ab)/(a + b) = (2 × 40 × 60) ÷ (40 + 60) = 4800 ÷ 100 = 48 km/h.

  3. Final Answer:

    Average speed = 48 km/h → Option D.

  4. Quick Check:

    If total distance = 2D, time = D/40 + D/60 = D(1/40 + 1/60) = D(1/24) → average = 2D ÷ (D/24) = 48 ✅
Hint: Half-distance segments → harmonic mean (2ab)/(a+b).
Common Mistakes: Averaging speeds directly instead of computing total time.

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