Respuesta :
Answer:
7.4 x 10⁸ J
Explanation:
[tex]m[/tex] = mass of the satellite = 50 kg
[tex]M[/tex] = mass of the earth = 5.98 x 10²⁴ kg
[tex]r_{i}[/tex] = initial radius of the orbit
[tex]T_{i}[/tex] = initial time period of the orbit = 100 min = 100 (60) = 6000 s
Initial time period of the orbit is given as
[tex]T_{i}^{2} =\frac{4\pi^{2} r_{i}^{3}}{GM} \\(6000)^{2} = \frac{4(3.14)^{2} r_{i}^{3}}{(6.67\times10^{-11})(5.98\times10^{24})}\\r_{i} =7.14063\times10^{6} m[/tex]
[tex]r_{f}[/tex] = final radius of the orbit
[tex]T_{f}[/tex] = final time period of the orbit = 310 min = 310 (60) s = 18600 s
Final time period of the orbit is given as
[tex]T_{f}^{2} =\frac{4\pi^{2} r_{f}^{3}}{GM} \\(18600)^{2} = \frac{4(3.14)^{2} r_{f}^{3}}{(6.67\times10^{-11})(5.98\times10^{24})}\\r_{f} =1.51814\times10^{7} m[/tex]
Minimum energy required to change the orbit is given as
[tex]E =(0.5)GMm(\frac{1}{r_{i}} - \frac{1}{r_{f}})\\E =(0.5)(6.67\times10^{-11})(5.98\times10^{24})(50)(\frac{1}{7.14063\times10^{6}} - \frac{1}{1.51814\times10^{7}})\\E = 7.4\times10^{8} J[/tex]
The minimum energy E is required to change the orbit to one with a period of 310 min E=7.4 x 10⁸ J
What is satellite ?
A satellite is a moon, planet or machine that orbits a planet or star. For example, Earth is a satellite because it orbits the sun
m = mass of the satellite = 50 kg
M = mass of the earth = 5.98 x 10²⁴ kg
[tex]r_i[/tex] = initial radius of the orbit
[tex]T_i[/tex] = initial time period of the orbit = 100 min = 100 (60) = 6000 s
Initial time period of the orbit is given as
[tex]T_i^2=\dfrac{4 \pi^2 r^3}{GM}[/tex]
[tex]6000^2=\dfrac{4\times (3.14)^2(r_i)^3}{(6.67\times 10^{-11})(5.98\times 10^{24})}[/tex]
[tex]r_i=7.14\times 10^6\ m[/tex]
[tex]r_f[/tex] = final radius of the orbit
[tex]T_f[/tex] = final time period of the orbit = 310 min = 310 (60) s = 18600 s
Final time period of the orbit is given as
[tex]T_f^2=\dfrac{4\pi ^2r_f^3}{GM}[/tex]
[tex](18000)^2=\dfrac{4(3.14)(r_f^3)}{(6.67\times 10^{-11})(5.98\times 10^{24})}[/tex]
[tex]r_f=1.51814\times 10^ 7\ m[/tex]
Minimum energy required to change the orbit is given as
[tex]E=(0.5)GMm(\dfrac{1}{r_i}-\dfrac{1}{r_f})[/tex]
[tex]E=(0.5)(6.67\times 10^{-11}(5.98\times 10^{24})(\dfrac{1}{7.14\times 10^{6}}-\dfrac{1}{1.51814\times 10^7})[/tex]
[tex]E=7.4\tines 10^8\ J[/tex]
Thus the minimum energy E is required to change the orbit to one with a period of 310 min E=7.4 x 10⁸ J
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