To find the tangential speed of an object orbiting Earth, we use the formula for tangential speed:
[tex]\[ v = \frac{2 \pi r}{T} \][/tex]
where:
- [tex]\( v \)[/tex] is the tangential speed
- [tex]\( r \)[/tex] is the radius of the orbit
- [tex]\( T \)[/tex] is the period of the orbit
Given:
[tex]\[ r = 1.8 \times 10^8 \text{ meters} \][/tex]
[tex]\[ T = 2.2 \times 10^4 \text{ seconds} \][/tex]
1. First, plug the values into the formula:
[tex]\[ v = \frac{2 \pi \times 1.8 \times 10^8 \text{ m}}{2.2 \times 10^4 \text{ s}} \][/tex]
2. Calculate the numerator [tex]\( 2 \pi \times 1.8 \times 10^8 \)[/tex]:
[tex]\[ 2 \pi \times 1.8 \times 10^8 \approx 11.309733552923255 \times 10^8 \text{ m} \][/tex]
3. Next, divide this result by the period [tex]\( 2.2 \times 10^4 \)[/tex]:
[tex]\[ \frac{11.309733552923255 \times 10^8 \text{ m}}{2.2 \times 10^4 \text{ s}} \approx \frac{11.309733552923255 \times 10^8}{2.2 \times 10^4} \approx 51407.8797860148 \text{ m/s} \][/tex]
Hence, the approximate tangential speed of the object is:
[tex]\[ 51407.88 \text{ m/s} \][/tex]
Therefore, among the provided choices, the closest value is:
[tex]\[ 5.1 \times 10^4 \text{ m/s} \][/tex]
