8. Sputnik, the first artificial Earth satellite launched by the USSR on Oct. 4 1957, took 96.0 minutes to complete one circular orbit. What was its approximate height above the Earth's surface?
(a) $5680 \mathrm{~km}$
(b) $56.8 \mathrm{~km}$
(c) $5.68 \mathrm{~km}$
(d) $56800 \mathrm{~km}$
(e) $568 \mathrm{~km}$

Question

8. Sputnik, the first artificial Earth satellite launched by the USSR on Oct. 4 1957, took 96.0 minutes to complete one circular orbit. What was its approximate height above the Earth's surface?
(a) $5680 \mathrm{~km}$
(b) $56.8 \mathrm{~km}$
(c) $5.68 \mathrm{~km}$
(d) $56800 \mathrm{~km}$
(e) $568 \mathrm{~km}$

Accepted Answer

∻Solution∻ ‖ a ⋮ Calculate the orbital period in seconds: $$ T = 96.0 \text{ minutes} \times 60 \frac{\text{seconds}}{\text{minute}} = 5760 \text{ seconds} $$ ‖ ‖ b ⋮ Use the formula for the orbital period of a satellite: $$ T = 2\pi\sqrt{\frac{r^3}{GM}} $$ where $ G $ is the gravitational constant and $ M $ is the mass of the Earth. ‖ ‖ c ⋮ Rearrange the formula to solve for the radius $ r $: $$ r = \sqrt[3]{\frac{GMT^2}{4\pi^2}} $$ ‖ ‖ d ⋮ Substitute the values for $ G $, $ M $, and $ T $ into the equation. Use $ G = 6.674 \times 10^{-11} \text{ N(m/kg)}^2 $ and $ M = 5.972 \times 10^{24} \text{ kg} $: $$ r = \sqrt[3]{\frac{(6.674 \times 10^{-11} \text{ N(m/kg)}^2)(5.972 \times 10^{24} \text{ kg})(5760 \text{ s})^2}{4\pi^2}} $$ ‖ ‖ e ⋮ Calculate the radius $ r $ and subtract the Earth's radius to find the height above the Earth's surface. The Earth's radius $ R_{\text{Earth}} $ is approximately $ 6371 \text{ km} $: $$ h = r - R_{\text{Earth}} $$ ‖ ‖ f ⋮ After performing the calculation, the result falls within the range of the provided options. ‖ ***E*** ∻Key Concept∻ ⚹ The key concept is the orbital mechanics of satellites, specifically the relationship between the orbital period, the radius of the orbit, and the mass of the central body around which the satellite orbits. The formula used is derived from Kepler's third law and Newton's law of universal gravitation. ⚹ ∻Explanation∻ ⚹ By using the given orbital period and known constants for Earth's mass and the gravitational constant, we can calculate the orbital radius of Sputnik. Subtracting Earth's radius from this value gives the height above Earth's surface, which matches one of the provided options. ⚹

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