Coincidence

The time on my watch is exactly 12 o’clock. That is, the minute and hour hands are coincident. At exactly what time will they next be coincident?

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If the minute hand "ticks" into place every minute then:

Every hour the hour hand moves 30 degrees, so every minute it moves 1/2 a degree. As there are 60 minutes in an hour, each minute that passes corresponds to 6 degrees (6*60=360). So, at 1:05 the minute hand will be at 30 degrees while the hour hand is at 32.5 degrees, and at 1:06 the minute hand will be at 36 degrees and the hour hand at 33 degrees. So it passes as it goes from 1:05 to 1:06 (during the "tick&quot😉.

If the minute hand moves smoothly throughout the minute (rather than simply "ticking" into place the moment a minute has passed).

Then, every second that passes corresponds to 1/60 of 6 degrees for the minute hand (1/10 degree) and every second that passes corresponds to 1/60 of 1/2 degree (or 1/120 degree). At 1:05 M=30, H=32.5. So to figure out how many seconds, solve for x:

30+(x/10) = 32.5 +(x/120)
=
2.5=(x/10)-(x/120)
=
2.5=(11x/120)
=
x=27.27272727 etc.
Plug in:
(27.2727 * 1/10)+30=32.727272 degrees for the minute hand
(27.2727 * 1/120)+32.5=32.727272 degrees for the hour hand.

Nemesio

Well done, nemesio.

This event occurs 11 times in 12 hours.
Therefore ir occurs once every 12/11 hours
Hence required time is 60/11 minutes past 1 o'clock. 😏

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not a second later? Because my watch as a second hand. Or if the watch breaks at 12 then it will always be. Or I could just hand crank the hands to coincide any time I want

Originally posted by Otis
not a second later? Because my watch as a second hand. Or if the watch breaks at 12 then it will always be. Or I could just hand crank the hands to coincide any time I want

The question never mentioned a second hand. Clocks and watches don't need second hands.

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