The length of a seconds hand in watch is `1 cm.` The change in velocity of its tip in `15 s` is

Options

A. zero
B. `pi/(30 sqrt2) cm//s`
C. `pi/30 cm//s`
D. `(pi(sqrt2))/30 cm//s`

Explanation

To solve the problem of finding the change in velocity of the tip of the second hand of a watch in 15 seconds, we can follow these steps:

### Step 1: Determine the radius of the circular motion
The length of the second hand is given as \( r = 1 \, \text{cm} \).

### Step 2: Calculate the velocity of the tip of the second hand
The tip of the second hand moves in a circular path. The circumference of the circle traced by the tip is given by:
\[ \text{Circumference} = 2\pi r = 2\pi \times 1 \, \text{cm} = 2\pi \, \text{cm} \]
The second hand completes one full revolution (360 degrees) in 60 seconds. Therefore, the speed \( v \) of the tip of the second hand is:
\[ v = \frac{\text{Circumference}}{\text{Time for one revolution}} = \frac{2\pi \, \text{cm}}{60 \, \text{s}} = \frac{\pi}{30} \, \text{cm/s} \]

### Step 3: Determine the angle covered in 15 seconds
In 15 seconds, the second hand covers:
\[ \text{Angle covered} = \frac{15 \, \text{s}}{60 \, \text{s}} \times 360^\circ = 90^\circ \]
This means the velocity vector of the tip of the second hand rotates by 90 degrees in this time.

### Step 4: Calculate the change in velocity
The change in velocity \( \Delta v \) can be calculated using the formula for the magnitude of the change in velocity when the angle between the initial and final velocity vectors is \( \theta \):
\[ \Delta v = \sqrt{v^2 + v^2 - 2vv \cos(\theta)} \]
Here, \( \theta = 90^\circ \) and \( \cos(90^\circ) = 0 \). Thus, the formula simplifies to:
\[ \Delta v = \sqrt{v^2 + v^2} = \sqrt{2v^2} = v\sqrt{2} \]
Substituting the value of \( v \):
\[ \Delta v = \left(\frac{\pi}{30}\right) \sqrt{2} = \frac{\pi \sqrt{2}}{30} \, \text{cm/s} \]

### Final Answer
The change in velocity of the tip of the second hand in 15 seconds is:
\[ \Delta v = \frac{\pi \sqrt{2}}{30} \, \text{cm/s} \]

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