High School

The angular speed of a point on Earth is [tex]\frac{\pi}{12}[/tex] radians per hour. The Equator lies on a circle with a radius of approximately 4000 miles.

Find the linear velocity, in miles per hour, of a point on the Equator.

Answer :

To find the linear velocity of a point on the Equator, we can use the relationship between angular speed and linear velocity. Here's how we can solve this problem step by step:

1. Understand the Given Data:
- The angular speed of a point on Earth is given as [tex]\(\frac{\pi}{12}\)[/tex] radians per hour.
- The radius of the Equator is approximately 4000 miles.

2. Recall the Formula for Linear Velocity:
- The formula to find the linear velocity ([tex]\(v\)[/tex]) from angular speed ([tex]\(\omega\)[/tex]) is:
[tex]\[
v = \omega \times r
\][/tex]
where [tex]\(r\)[/tex] is the radius of the circular path.

3. Substitute the Given Values into the Formula:
- Here the angular speed ([tex]\(\omega\)[/tex]) is [tex]\(\frac{\pi}{12}\)[/tex] radians per hour.
- The radius ([tex]\(r\)[/tex]) is 4000 miles.

4. Calculate the Linear Velocity:
- Plug the values into the formula:
[tex]\[
v = \left(\frac{\pi}{12}\right) \times 4000
\][/tex]

5. Compute the Result:
- After performing the multiplication, the linear velocity of a point on the Equator is approximately 1047.20 miles per hour.

Therefore, the linear velocity, in miles per hour, of a point on the Equator is 1047.20 miles per hour.

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