MINDMAPS FOR UPSCPET 2026 · VISUAL REVISION
Clock hands and relative angular speed
The 12-hour relative-angle cycle from coincidence to opposition and back
12-Hour Relative-Angle Cycle
Angular speeds
- At 12:00, the hands coincide; their relative separation is 0°.
- In 1 minute, the minute hand moves 360° ÷ 60 = 6° and the hour hand moves 30° ÷ 60 = 0.5°.
- Thus the minute hand gains 6° − 0.5° = 5.5° (or 11/2°) per minute over the hour hand.
Angle calculation
- At h:m, with h in 0–11, raw separation is d = |30h − 5.5m| degrees.
- Reduce d into 0°–360° and use min(d, 360° − d) when the smaller angle between hands is asked.
- In each relative lap, hands reach 90° separation twice: at 90° and at 270° of relative gain.
Opposite hands
- Hands form a straight line in opposite directions when the minute hand gains 180°.
- After 12:00, the first opposition occurs at 180 ÷ 5.5 = 360/11 = 32 8/11 minutes past 12.
- Between 5:00 and 7:00, opposition occurs only once, exactly at 6:00 (not twice).
Relative laps and counts
- A full relative lap of 360° takes 360 ÷ 5.5 = 720/11 = 65 5/11 minutes to return to 0°.
- In 12 hours (720 min), relative gain is 720 × 5.5° = 3960° = 11 × 360°, or 11 full laps.
- Each relative lap has one opposition; count coincident endpoints only once. In 24 h: 22 oppositions.
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