Calculating the Average Velocity of the Tip of the Minute Hand of a Clock

Given:
- Length of minute hand = 5.0cm
- Time interval = 6:00 a.m. to 6:30 p.m.

Formula:
- The circumference of a circle = 2πr
- Distance covered by minute hand = (θ/360) x 2πr
- Average velocity = Total distance covered / Time taken

Calculations:
- Radius of the clock = length of minute hand = 5.0cm
- Circumference of the clock = 2πr = 2π(5.0) = 31.4cm
- In 12 hours, the minute hand completes one full rotation around the clock, which is 360 degrees.
- In 12 hours, the minute hand rotates 720 degrees (2 rotations).
- In 30 minutes, the minute hand rotates 180 degrees.
- The distance covered by the minute hand in 12 hours = (720/360) x 2π(5.0) = 62.8cm
- The distance covered by the minute hand in 30 minutes = (180/360) x 2π(5.0) = 7.85cm
- Total distance covered in the given time interval = (12 x 62.8) + (0.5 x 7.85) = 769.1cm
- Time taken in the given time interval = 12.5 hours = 45000 seconds
- Average velocity = 769.1 / 45000 = 0.0171 cm/s

Therefore, the average velocity of the tip of the minute hand between 6:00 a.m. to 6:30 p.m. is 0.0171 cm/s.

Explanation:
- The minute hand of a clock rotates around the clock in a circular motion.
- The distance covered by the minute hand in one rotation is equal to the circumference of the clock.
- The distance covered by the minute hand in a given time interval can be calculated by finding the angle rotated by the minute hand and using the formula for arc length.
- The average velocity of the minute hand can be calculated by dividing the total distance covered by the time taken.
- In this case, the minute hand rotates 720 degrees in 12 hours and 180 degrees in 30 minutes, which is used to calculate the distance covered.
- The total distance covered in the given time interval is the sum of the distances covered in each interval.
- The time taken is the difference between the start and end time in seconds.
- Dividing the total distance by the time taken gives the average velocity of the tip of the minute hand.

Average velocity= Net Displacement/ Total Time
Displacement= 2 R
Total Time= 30 min= 1800 sec

A.V= 2R / 1800
A.V = 2×5.0/1800
A.V=10/1800
A.V=1/180 cm/sec