Sure, let's solve this step-by-step.
### Step 1: Calculate the Circumference of the Circle
First, we need to determine the distance traveled by the tip of the minute hand, which moves in a circular path around the clock.
Given:
- The length of the minute hand (which is the radius of the circle) = 10 cm
The circumference [tex]\( C \)[/tex] of the circle is given by the formula:
[tex]\[ C = 2 \pi r \][/tex]
where [tex]\( r \)[/tex] is the radius.
Substituting the given radius:
[tex]\[ C = 2 \pi \times 10 \text{ cm} \][/tex]
[tex]\[ C = 20 \pi \text{ cm} \][/tex]
### Step 2: Calculate the Time Elapsed
Next, we need to determine the time elapsed from 8:00 PM to 8:15 PM.
Given:
- Start time = 8:00 PM
- End time = 8:15 PM
The time elapsed is:
[tex]\[ \text{Elapsed Time} = 8:15 \text{ PM} - 8:00 \text{ PM} \][/tex]
Converting this to a fraction of an hour:
[tex]\[ \text{Elapsed Time} = \frac{15}{60} \text{ hours} = 0.25 \text{ hours} \][/tex]
### Step 3: Calculate the Speed
Speed is defined as the distance traveled divided by the time taken.
Given:
- Distance traveled [tex]\( = C \)[/tex] (circumference of the circle)
- Time taken [tex]\( = 0.25 \text{ hours} \)[/tex]
We already calculated the circumference [tex]\( C \)[/tex]:
[tex]\[ C = 20 \pi \text{ cm} \][/tex]
So, the speed [tex]\( v \)[/tex] is:
[tex]\[ v = \frac{C}{\text{Elapsed Time}} \][/tex]
[tex]\[ v = \frac{20 \pi \text{ cm}}{0.25 \text{ hours}} \][/tex]
### Step 4: Calculate the Magnitude of Velocity
In uniform circular motion, the magnitude of velocity is the same as the speed since velocity is the rate at which an object moves along a path.
Thus, the magnitude of velocity is also:
[tex]\[ v = \frac{20 \pi \text{ cm}}{0.25 \text{ hours}} \][/tex]
### Final Results:
- Circumference [tex]\( C = 62.83185307179586 \text{ cm} \)[/tex]
- Total Time [tex]\( = 0.25 \text{ hours} \)[/tex]
- Speed = Magnitude of Velocity [tex]\( v = 251.32741228718345 \text{ cm/hour} \)[/tex]
So, the speed and the magnitude of the velocity of the terminal point of the minute hand when the time elapsed from 8:00 PM to 8:15 PM is:
[tex]\[ v = 251.32741228718345 \text{ cm/hour} \][/tex]
