17. The International Space Station (ISS) orbits about $420 \mathrm{~km}$ above Earth's surface. Earth's mass is $M_{\oplus}=5.97 \times 10^{24} \mathrm{~kg}$, and Earth's mean radius is $R_{\oplus}=6.37 \times 10^{6} \mathrm{~m}$, and $G=6.674 \times 10^{-11}$ $\mathrm{m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}$. What is the approximate orbital speed of the ISS?
A. $8180 \mathrm{~m} / \mathrm{s}$
P. $9430 \mathrm{~m} / \mathrm{s}$
C. $7660 \mathrm{~m} / \mathrm{s}$
D. $7910 \mathrm{~m} / \mathrm{s}$

Step 1 ：Given values: \(G = 6.674 \times 10^{-11} \mathrm{m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\), \(M_{\oplus} = 5.97 \times 10^{24} \mathrm{~kg}\), \(R_{\oplus} = 6.37 \times 10^{6} \mathrm{~m}\), and altitude = 420,000 m.

Step 2 ：Calculate the distance r from the center of Earth to the ISS: \(r = R_{\oplus} + \text{altitude} = 6.37 \times 10^{6} \mathrm{~m} + 420,000 \mathrm{~m} = 6.79 \times 10^{6} \mathrm{~m}\)

Step 3 ：Use the formula for orbital speed: \(v = \sqrt{\frac{G \times M_{\oplus}}{r}}\)

Step 4 ：Plug in the values: \(v = \sqrt{\frac{6.674 \times 10^{-11} \mathrm{m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2} \times 5.97 \times 10^{24} \mathrm{~kg}}{6.79 \times 10^{6} \mathrm{~m}}}\)

Step 5 ：Calculate the orbital speed: \(v \approx 7660 \mathrm{~m} / \mathrm{s}\)

Step 6 ：\(\boxed{\text{The approximate orbital speed of the ISS is 7660 m/s (Option C)}}\)