Question

the international space station (iss) orbits earth at an altitude of 4.08 × 10^5 m above the surface of the planet. the radius of the earth is 6.37 × 10^6 m. at what velocity must the iss be moving in order to stay in its orbit? (1 point)

3.12 × 10^4 m/s

7.91 × 10^3 m/s

8.17 × 10^3 m/s

7.66 × 10^3 m/s

Explanation:

Step1: Calculate the distance from the center of the Earth to the ISS

The distance \( r \) from the center of the Earth to the ISS is the sum of the Earth's radius \( R_E\) and the altitude \( h \) of the ISS.
\( r=R_E + h=(6.37\times10^{6})+(4.08\times 10^{5})=6.37\times10^{6}+0.408\times10^{6}=6.778\times10^{6}\text{ m}\)

Step2: Use the formula for the orbital velocity

The formula for the orbital velocity \( v\) of an object in orbit around a planet is \( v = \sqrt{\frac{GM}{r}}\), where \( G = 6.67\times10^{- 11}\text{ N}\cdot\text{m}^2/\text{kg}^2\) is the gravitational constant and \( M = 5.97\times10^{24}\text{ kg}\) is the mass of the Earth.
Substitute \( G = 6.67\times10^{-11}\text{ N}\cdot\text{m}^2/\text{kg}^2\), \( M = 5.97\times10^{24}\text{ kg}\) and \( r = 6.778\times10^{6}\text{ m}\) into the formula:

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Answer:

D. \(7.66\times 10^{3}\text{ m/s}\)