What is the approximate tangential speed of an object orbiting Earth with a radius of [tex]\(1.8 \times 10^8 \, m\)[/tex] and a period of [tex]\(2.2 \times 10^4 \, s\)[/tex]?

A. [tex]\(7.7 \times 10^{-4} \, m/s\)[/tex]
B. [tex]\(5.1 \times 10^4 \, m/s\)[/tex]
C. [tex]\(7.7 \times 10^4 \, m/s\)[/tex]
D. [tex]\(5.1 \times 10^5 \, m/s\)[/tex]



Answer :

Certainly! Let's solve for the approximate tangential speed of an object orbiting Earth given the radius and the period of the orbit.

### Step-by-Step Solution:

1. Given:
- Radius of orbit ([tex]\(r\)[/tex]) = [tex]\(1.8 \times 10^8\)[/tex] meters
- Period of orbit ([tex]\(T\)[/tex]) = [tex]\(2.2 \times 10^4\)[/tex] seconds

2. Calculate the Circumference of the Orbit:

The formula for the circumference [tex]\(C\)[/tex] of a circle (which, in this case, is the orbit) is:

[tex]\[ C = 2 \pi r \][/tex]

Plugging in the given radius:

[tex]\[ C = 2 \pi \times 1.8 \times 10^8 \text{ meters} \][/tex]

This results in:

[tex]\[ C \approx 1,130,973,355.2923255 \text{ meters} \][/tex]

3. Calculate the Tangential Speed ([tex]\(v\)[/tex]):

The tangential speed of an object in circular motion is given by the formula:

[tex]\[ v = \frac{C}{T} \][/tex]

where:
- [tex]\(C\)[/tex] is the circumference of the orbit
- [tex]\(T\)[/tex] is the period of the orbit

Plugging in the values for circumference and period:

[tex]\[ v = \frac{1,130,973,355.2923255 \text{ meters}}{2.2 \times 10^4 \text{ seconds}} \][/tex]

Simplifying this:

[tex]\[ v \approx 51407.8797860148 \text{ meters per second (m/s)} \][/tex]

4. Compare the Calculated Tangential Speed to Given Options:

The given multiple-choice options are:
- [tex]\(7.7 \times 10^{-4} \text{ m/s}\)[/tex]
- [tex]\(5.1 \times 10^4 \text{ m/s}\)[/tex]
- [tex]\(7.7 \times 10^4 \text{ m/s}\)[/tex]
- [tex]\(5.1 \times 10^5 \text{ m/s}\)[/tex]

Our calculated tangential speed ([tex]\(v \approx 51407.8797860148 \text{ m/s}\)[/tex]) can be approximated as [tex]\( 5.1 \times 10^4 \text{ m/s}\)[/tex].

Thus, the approximate tangential speed of the object is:

[tex]\[ \boxed{5.1 \times 10^4 \text{ m/s}} \][/tex]