A clock has its second hand `2.0 cm` long. Find the average speed and modulus of average velocity of the tip of the second hand in `15 s`.

Explanation

To solve the problem step by step, we will calculate both the average speed and the modulus of average velocity of the tip of the second hand of a clock.

### Step 1: Understand the motion of the second hand
The second hand of the clock is 2.0 cm long, and in 15 seconds, it completes one-fourth of a full circle (90 degrees).

### Step 2: Calculate the circumference of the circle
The circumference \( C \) of a circle is given by the formula:
\[ C = 2 \pi r \]
where \( r \) is the radius of the circle. Here, \( r = 2.0 \) cm.

Substituting the value of \( r \):
\[ C = 2 \pi \times 2.0 = 4 \pi \text{ cm} \]

### Step 3: Calculate the distance traveled by the tip of the second hand
Since the second hand covers one-fourth of the circle in 15 seconds, the distance \( d \) traveled is:
\[ d = \frac{1}{4} C = \frac{1}{4} \times 4 \pi = \pi \text{ cm} \]

### Step 4: Calculate the average speed
Average speed \( v_{avg} \) is defined as the total distance traveled divided by the total time taken:
\[ v_{avg} = \frac{d}{t} \]
where \( t = 15 \) seconds.

Substituting the values:
\[ v_{avg} = \frac{\pi}{15} \text{ cm/s} \]

### Step 5: Calculate the displacement
The displacement is the straight-line distance from the starting point to the endpoint after 15 seconds. The endpoints can be represented as points A and B, with O being the center of the circle. The displacement forms a right triangle OAB, where OA and OB are both equal to the radius \( r \).

Using the Pythagorean theorem, the length of the hypotenuse AB can be calculated:
\[ AB = \sqrt{OA^2 + OB^2} = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2} \]
Substituting \( r = 2.0 \) cm:
\[ AB = 2\sqrt{2} \text{ cm} \]

### Step 6: Calculate the average velocity
Average velocity \( v_{avg, velocity} \) is defined as the total displacement divided by the total time:
\[ v_{avg, velocity} = \frac{AB}{t} \]
Substituting the values:
\[ v_{avg, velocity} = \frac{2\sqrt{2}}{15} \text{ cm/s} \]

### Final Answers
- Average Speed: \( \frac{\pi}{15} \text{ cm/s} \)
- Modulus of Average Velocity: \( \frac{2\sqrt{2}}{15} \text{ cm/s} \)