Clock Hand Crossing Explained

This problem involves understanding the relative movement and speeds of the minute hand and the second hand of a clock to determine how many times they cross each other within a specific time interval.

Understanding Clock Hands' Movement

Let's first understand the speeds at which the minute hand and second hand move on a clock face:

• The second hand completes one full rotation (360 degrees) in 60 seconds (1 minute).
• The minute hand completes one full rotation (360 degrees) in 60 minutes (1 hour). This means it moves 6 degrees in 1 minute.

Since the second hand is much faster than the minute hand, it will constantly gain on the minute hand and "lap" it.

Time Duration Analysis

The question asks for the number of crossings between 09:15:00 AM and 09:45:00 AM.

• Start time: 09:15:00 AM
• End time: 09:45:00 AM
• Total duration = End time - Start time = 09:45:00 AM - 09:15:00 AM = 30 minutes.

We need to find out how many times the minute hand and second hand cross each other during these 30 minutes.

Crossing Occurrences of Minute and Second Hands

For the minute hand and the second hand, a practical approach for determining crossings is generally followed, especially for questions expecting an integer answer.

• In any given 1-minute interval, the second hand completes one full revolution (360 degrees). During this revolution, it sweeps past all possible positions on the clock face.
• During the same 1-minute interval, the minute hand moves only 6 degrees. This movement is very small compared to the second hand's full revolution.
• Because the second hand completes a full rotation while the minute hand barely moves, the second hand will always "pass over" or "cross" the minute hand exactly once within that 1-minute interval. For example, if at the start of a minute, the second hand is at 12 and the minute hand is at 3, the second hand will move past the 3 o'clock position (where the minute hand is) as it completes its revolution.

This means that for every minute that passes, there is one instance where the second hand and the minute hand cross each other.

Calculating Total Crossings

Given the time duration is 30 minutes, and knowing that they cross each other once per minute, we can calculate the total number of crossings:

Total Crossings = Number of minutes in the interval \( \times \) Crossings per minute

Total Crossings = \(30 \text{ minutes} \times 1 \text{ crossing/minute}\)

\( \text{Total Crossings} = 30 \)

Therefore, the minute hand and second hand of a clock cross each other 30 times between 09:15:00 AM and 09:45:00 AM.