The exact time the hands first overlap after noon

Question

At 12:00 the hour and minute hands are exactly together. The minute hand then races ahead, and sometime just after 1:05 it catches back up to the hour hand.

What is the exact time of that first overlap after 12:00, down to the second?

Accepted Answer

The first overlap after 12:00 happens at 1:05 and 27 and 3/11 seconds, i.e. \frac{720}{11} minutes after 12:00. Set up the speeds (in degrees). Measure both hands as an angle from the 12, clockwise. - The minute hand sweeps 360° per 60 minutes = 6° per minute. - The hour hand sweeps 360° per 12 hours = 0.5° per minute. So the minute hand gains on the hour hand at 6 - 0.5 = 5.5° per minute. Find when it has gained a full lap. At 12:00 both hands are at 0°, together. The next overlap occurs when the minute hand has gained a full 360° on the hour hand: 5.5° 	imes t = 360° \quad\Rightarrow\quad …

The exact time the hands first overlap after noon

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