Answered

Get the best answers to your questions with the help of IDNLearn.com's experts. Ask anything and receive thorough, reliable answers from our community of experienced professionals.

The length of the minute hand of a clock is 10 cm.

i) Find the speed and the magnitude of the velocity of the terminal point of the minute hand when the time elapses from 8:00 to 8:15 PM.

Sagot :

GinnyAnsweridn
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]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.