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A satellite is put into an elliptical orbit around the Earth. When the satellite is at its perigee, its nearest point to the Earth, its height above the ground is hp=227.0 km, and it is moving with a speed of vp=8.950 km/s. The gravitational constant G equals 6.67×10−11 m3·kg−1·s−2 and the mass of Earth equals 5.972×1024 kg.When the satellite reaches its apogee, at its farthest point from the Earth, what is its height ha above the ground?

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Answer:

6633549.52903 m

Explanation:

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

m = Mass of the Earth =  5.972 × 10²⁴ kg

[tex]h_p[/tex] = Height above ground = 227 km

[tex]v_p[/tex] = Velocity at perigee = 8.95 km/s

Perigee distance is

[tex]R_p=6371+227=6598\ km[/tex]

The apogee distance is given by

[tex]R_a=\dfrac{R_p}{\dfrac{2Gm}{R_pv_p^2}-1}\\\Rightarrow R_a=\dfrac{6598\times 10^3}{\dfrac{2\times 6.67\times 10^{-11}\times 5.972\times 10^{24}}{6598\times 10^3\times (8.950\times 10^3)^2}-1}\\\Rightarrow R_a=13004549.52903\ m[/tex]

The height above the ground would be

[tex]h_a=13004549.52903-6371000=6633549.52903\ m[/tex]

The height above the ground is 6633549.52903 m

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