To solve the problem of finding the first time after 12 o'clock when the hour and minute hands overlap, we need to consider the following:

Key facts:

1. The minute hand moves 360 degrees in 60 minutes (i.e., 6 degrees per minute).
2. The hour hand moves 360 degrees in 12 hours (i.e., 0.5 degrees per minute).

Approach:

At 12:00, both the hour and minute hands are aligned. After this, the minute hand moves faster than the hour hand and will eventually catch up. Let's determine when this occurs.

Let the time after 12:00 be $t$ minutes.

• In $t$ minutes, the minute hand will have moved $6t$ degrees (since it moves 6 degrees per minute).
• In $t$ minutes, the hour hand will have moved $0.5t$ degrees from the 12 o’clock position (since it moves 0.5 degrees per minute).

For the hands to overlap, the angle moved by the minute hand must equal the angle moved by the hour hand plus 360 degrees (since the minute hand needs to catch up after one full cycle):

$6t = 0.5t + 360$

Solving for $t$:

$6t - 0.5t = 360$ $5.5t = 360$ $t = \frac{360}{5.5} = 65.45 \text{ minutes}.$

Thus, the first time the hands overlap after 12 o'clock is approximately 65.45 minutes, which is 1 hour and 5 minutes and 27 seconds.

So, to the nearest second, the hands overlap at 1:05:27.

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Would you like more details or have any questions?

Here are 5 related questions to explore:

1. At what time after 1:00 will the hands overlap again?
2. How many times do the hands overlap in a 12-hour period?
3. At what time between 12:00 and 12:30 do the hands overlap?
4. How do you calculate the angle between the hour and minute hands at any given time?
5. What is the next time the hands overlap after 6:00?

Tip: The hands of a clock overlap 11 times in a 12-hour period because the 12th time would be at 12 o'clock again.