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A 4500-kg spaceship is in a circular orbit 190 km above the surface of Earth. It needs to be moved into a higher circular orbit of 380 km to link up with the space station at that altitude. In this problem you can take the mass of the Earth to be 5.97 × 1024 kg.

I am confused how to solve this problem, please show how this answer is obtained.

Answer :

Answer:

Explanation:

Total energy of a satellite in an orbit , h height away

= -  GMm /2 ( R + h )

When h = 380 km

Total energy of a satellite = [tex]\frac{6.67\times10^{-11}\times5.97\times10^{24}\times 4500}{2\times(6378+380)\times10^3}[/tex]

=  - 13.25 x 10¹⁰ J

When h = 190 km

Total energy of a satellite =

[tex]\frac{6.67\times10^{-11}\times5.97\times10^{24}\times 4500}{2\times(6378+190)\times10^3}[/tex]

=   - 13.63 x 10¹⁰ J

Diff

= 38 x 10⁸ J Energy will be required.

The energy required by an spacecraft to move into a higher orbit will be  [tex]E=38\times10^{8}[/tex]J

What will be the energy required by the space craft to move into a higher orbit?

It is given that,

Mass of the spacecraft M=4500kg

Lower orbit =190lm

Higher orbit=380km

mass of earth  [tex]M_{Earth}=5.97\times10^{24} kg[/tex]

Total energy of satellite in an orbit is given by formula

[tex]E=\dfrac{-GMm}{2(R+h)}[/tex]

Now for hieght  h = 380 km

Energy of the satelite will be

[tex]E_{1} =\dfrac{6.67\times10^{-11}\times5.97\times10^{24}\times4500 }{2(6378+380)\times10^{3} }[/tex]

[tex]E_{1} = -13.25\times10^{10} J[/tex]

Now energy when the height of the satellite h=190km

[tex]E_{2} =\dfrac{6.67\times10^{-11}\times5.97\times10^{24}\times4500 }{2(6378+190)\times10^{3} }[/tex]

 [tex]E_{2} =-13.63\times10^{10}[/tex]

Now the difference between the energies

[tex]E_{1}- E_{2}= 38\times10^{8} J[/tex]

Thus the energy required by an spacecraft to move into a higher orbit will be  [tex]E=38\times10^{8}[/tex]J

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