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Complex Motion / Variable Speeds

Introduction

Complex Motion / Variable Speeds covers problems where a moving object changes speed during its journey - for example, different speeds on different segments, pauses, or acceleration/deceleration phases. These questions combine multiple time-distance segments and require careful bookkeeping of each segment’s distance, time and effective speed.

This pattern is important because real-world motion rarely stays at one constant speed; mastering segment-wise calculations and reductions (averages, total time, net distance) lets you solve layered aptitude questions confidently.

Pattern: Complex Motion / Variable Speeds

Pattern

Key concept: Break the journey into clear segments, compute distance/time for each segment, then combine. Use Total Distance = Σ distances and Total Time = Σ times to get overall average speed = Total Distance ÷ Total Time.

  • Segment approach: Treat each part with its own speed/time/distance values.
  • Average speed over whole trip: Not the arithmetic mean of speeds - compute total distance ÷ total time.
  • When distances are equal but speeds differ: Use harmonic mean for two segments: 2ab/(a+b) (generalize for more segments by total distance/total time).
  • When times are equal but speeds differ: Use arithmetic mean of speeds for equal-time segments.
  • Include rests or pauses: Add pause time to total time (distance unchanged) before computing average speed.

Step-by-Step Example

Question

A cyclist rides 30 km at 15 km/h, then continues 20 km at 10 km/h. What is the cyclist’s average speed for the whole trip?

Solution

  1. Step 1: Compute time for each segment

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

  2. Step 2: Total distance and total 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:

    If average were (15+10)/2 = 12.5 (works here because times equal). Verify: 12.5 × 4 = 50 ✅

Quick Variations

1. Segments with pauses: Add pause time to total time (distance unchanged) before computing average speed.

2. Variable speeds over unequal distances: Compute time per segment, sum times, then divide total distance by total time.

3. Piecewise acceleration: If acceleration is uniform in a segment, use average speed in that segment = (u + v)/2 for that segment’s time/distance.

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

5. Return trips with different speeds: Treat outward and return as two segments - total time matters.

Trick to Always Use

  • Step 1: Split the journey into clear segments and label each (distance, speed, time).
  • Step 2: Compute time for each segment: time_i = distance_i ÷ speed_i (or distance = speed × time).
  • Step 3: Sum distances and times: Total distance = Σ distance_i; Total time = Σ time_i (include rests).
  • Step 4: Overall average speed = Total distance ÷ Total time. Don't average speeds directly (unless times are equal) - think totals.
  • Optional: Use harmonic mean only when distances are equal; use arithmetic mean only when times are equal.

Summary

Summary

For complex motion / variable speeds:

  • Always segment the trip and compute each segment’s time explicitly.
  • Average speed over whole trip = (total distance) ÷ (total time) - not the simple mean of segment speeds except in special equal-time/equal-distance cases.
  • Include pauses in total time; they lower average speed though distance unchanged.
  • For equal-distance segments, use harmonic mean; for equal-time segments, use arithmetic mean of speeds.
  • Quick checks (recomputing totals or using alternate formulas) catch common mistakes early.

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.

Mock Test

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