Answer:
First cosmic speed = 5195.74m/s
Second cosmic speed = 7346.05m/s
The raduis of the synchronous 0rbit of satellite is 2.80×10^7m
Explanation:
The first cosmic speed Is determined using the Orbital speed equation which is given by:
V = Sqrt(GM/r)
Where G = gravitational constant = 6.67 ×10^-11
M = Mass of planet
r = radius of the planet
V = Sqrt (6.67×10^-11)(3.42×10^24)/(8450×10^3)
V = Sqrt (2.28×10^14)/(8450×10^3)
V = Sqrt ( 26995739.64)
V = 5195.75m/s
The second cosmic speed is given by :
V = Sqrt(2 × GM)/r
V = Sqrt (2 × (6.67×10^-11)(3.42×10^24)/(8450×10^3)
V = Sqrt( 4.5×10^14)/ (8450×10^3)
V = Sqrt(53964497.04)
V = 7346.05m/s
The raduis of the synchronous orbit if the satellite around the planet is given by:
r = Cuberoot(T^2GM/4 pi r^2where T is the period of rotation of the planet in second
Given :
T = 17.1 hours converting to seconds
T = 17.1 ×60 ×60 = 61560 seconds
Substituting into the equation
r = Cuberoot ([(61560)^2×(6.67×10^-11)(3.42×10^24)/ (4 ×3.142×r^2)]
r = 2.80×10^7m
Answer:
First cosmic speed = 5200 m/s
Second cosmic speed = 7350 m/s
Radius of the synchronous orbit of a satellite = 28,000 km or [tex]2.80\times10^7\ \text{m}[/tex]
Explanation:
The first cosmic speed is given by
[tex]V_1 = \sqrt{\dfrac{GM_p}{R_p}}[/tex]
G is the universal gravitational constant with value [tex]6.674\times10^{-11}\ \text{Nm}^2\text{/kg}^2[/tex]
[tex]M_p[/tex] is the mass of the planet; and
[tex]R_p[/tex] is the radius of the planet.
[tex]V_1 = \sqrt{\dfrac{(6.674\times10^{-11}\ \text{Nm}^2\text{/kg}^2)(3.42\times10^{24}\text{ kg})}{(8.45\times10^6\ \text{m})}} = 5200\ \text{m/s}[/tex]
The second cosmic speed is given by
[tex]V_2 = \sqrt{\dfrac{2GM_p}{R_p}} = \sqrt{2}V_1[/tex]
[tex]V_2 = \sqrt{2}\times 5200\text{ m/s} = 7350\ \text{m/s}[/tex]
The radius of the synchronous orbit of a satellite around the planet is given by
[tex]r = \sqrt[3]{\dfrac{T^2GM_p}{4\pi^2}}[/tex]
where T is the period of rotation of the planet in seconds.
Substituting known values,
[tex]r = \sqrt[3]{\dfrac{(17.1\times60\times60\ \text{s})^2(6.674\times10^{-11}\ \text{Nm}^2\text{/kg}^2)(3.42\times10^{24}\text{ kg})}{4\pi^2}}[/tex]
[tex]r = 2.80\times10^7\ \text{m}[/tex]
