Elementary Math Word Problems · Angular Velocity & Clock Angles
Clock Angle & Catch-up Chase Simulator
Key pedagogical takeaway: Minute hand travels at 6°/min, hour hand at 0.5°/min. Relative chasing speed is strictly 5.5°/min (11/2°/min). They overlap 22 times per 24 hours, NOT 24.
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- Live Derivation
- Projector Mode
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3D Simulation Stage● 60 FPS
Parameter Controls
points
min
Live Derivations & Metrics
Current Hand Angle θ
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Next Exact Overlap
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Relative Angular Speed
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Total 24h Overlaps
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Real-Time Formula Derivation
Governing Formula:
--Substitution:
--Regime: --
Worked Examples & Exam Question Deep Dive
Worked Problem
Problem: Between 4:00 and 5:00, at what exact time do the clock hands first overlap?
Step-by-Step Derivation:
- At 4:00, separation is 4*30° = 120°. Relative chase speed = 5.5°/min. Time = 120 / 5.5 = 240/11 = 21 9/11 min. Exact time: 4:21 9/11.
Worked Problem
Problem: What is the angle between clock hands at 8:30?
Step-by-Step Derivation:
- Formula θ = |30H - 5.5M|: |30*8 - 5.5*30| = |240 - 165| = 75°. (2.5 hour divisions * 30° = 75°).
Student Inquiry Worksheet & Practice Problems
Q1. Variant 1: Between 2:00 and 3:00, when are hands perpendicular (90°)? How many times?
Answer Key: At 2:00 hands are 60° apart. Chase to 90° after passing requires 60° + 90° = 150°. t = 150/5.5 = 27 3/11 min. Exactly 1 time (2:27 3/11).
Q2. Variant 2: Clock loses 2 min per hour. Set at 8:00 AM. When clock reads 3:00 PM, what is actual time?
Answer Key: Clock speed ratio = 58/60. Clock elapses 7h = 420 min. True time = 420 * (60/58) = 434.48 min = 3:14:29 PM.
Q3. Variant 3: How many times do clock hands overlap in 24 hours? Why not 24 times?
Answer Key: Exactly 22 times! In 12h, minute hand completes 12 laps, hour hand completes 1 lap. Net laps = 12 - 1 = 11 overlaps per 12h = 22 overlaps per 24h.
Frequently Asked Questions & Misconceptions
Why is a clock puzzle fundamentally a circular track chase problem?
The dial is a 360° circular track. Minute hand runs at 6°/min; hour hand at 0.5°/min. Lapping the hour hand is an overlap at relative 5.5°/min.
What if the angle formula yields θ > 180°?
Take the minor angle! Two lines form two supplementary angles summing to 360°. If calculated θ > 180°, simply take 360° - θ.
How many times are clock hands perpendicular (90°) and collinear in 24 hours?
Perpendicular: 44 times per day (22 per 12h). Collinear: 44 times per day (22 overlaps + 22 opposite lines).
Does the minute hand tick in steps or move continuously in mechanical watches?
Mechanical escapements advance gear trains 5-8 times per second in smooth continuous rotation, matching our continuous rate model.