Understanding the Problem
To find the average velocity of the tip of the minute hand from 6 AM to 6:30 PM, it's essential to consider the movement of the minute hand and the distance it travels.
Length of the Minute Hand
- The minute hand is 5 cm long.
Calculating the Circumference
- The tip of the minute hand traces a circular path.
- Circumference \( C \) of the circle is given by the formula:
\[
C = 2 \pi r
\]
where \( r \) is the radius (length of the minute hand).
- Substituting the values:
\[
C = 2 \pi (5 \, \text{cm}) \approx 31.42 \, \text{cm}
\]
Time Duration
- The time from 6 AM to 6:30 PM is 12.5 hours or 750 minutes.
Distance Traveled by the Minute Hand
- In 12 hours, the minute hand completes 12 full rotations.
- Therefore, distance traveled in 12 hours:
\[
\text{Distance} = 12 \times C \approx 12 \times 31.42 \approx 377.04 \, \text{cm}
\]
- From 6 AM to 6:30 PM, the minute hand makes an additional half rotation:
\[
\text{Additional Distance} = \frac{C}{2} \approx \frac{31.42}{2} \approx 15.71 \, \text{cm}
\]
- Total distance traveled:
\[
\text{Total Distance} = 377.04 + 15.71 \approx 392.75 \, \text{cm}
\]
Calculating Average Velocity
- Average velocity \( v \) is given by:
\[
v = \frac{\text{Total Distance}}{\text{Total Time}}
\]
- Total time is 750 minutes, converting to seconds:
\[
750 \, \text{min} = 750 \times 60 = 45000 \, \text{s}
\]
- Therefore, average velocity:
\[
v \approx \frac{392.75 \, \text{cm}}{45000 \, \text{s}} \approx 0.00873 \, \text{cm/s}
\]
Conclusion
The average velocity of the tip of the minute hand between 6 AM to 6:30 PM is approximately 0.00873 cm/s.

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