To find the tangential speed of an object orbiting Earth, we use the formula for tangential speed, which is given by:
[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 values:
- Radius, [tex]\( r = 1.8 \times 10^8 \)[/tex] meters,
- Period, [tex]\( T = 2.2 \times 10^4 \)[/tex] seconds.
Let's calculate the tangential speed step-by-step.
1. Substituting the given values into the formula:
[tex]\[ v = \frac{2 \pi \times 1.8 \times 10^8}{2.2 \times 10^4} \][/tex]
2. Calculate the numerator:
[tex]\[ 2 \pi \times 1.8 \times 10^8 \approx 11.309733552 \times 10^8 \][/tex]
3. Divide the numerator by the period:
[tex]\[ v = \frac{11.309733552 \times 10^8}{2.2 \times 10^4} \approx 51407.8797860148 \, \text{m/s} \][/tex]
So, the approximate tangential speed of the object is [tex]\( 51407.8797860148 \)[/tex] meters per second.
Therefore, among the provided options, the closest value is:
[tex]\[ 5.1 \times 10^4 \, \text{m/s} \][/tex]
So, the correct answer is:
[tex]\[ 5.1 \times 10^4 \, \text{m/s} \][/tex]
