The minimum energy input necessary to place this satellite in orbit is
[tex]\frac{1}{2} m(v^{2} - v_{e} ^{2} ) +GM_{E} m(\frac{1}{r} -\frac{1}{R_{E} } )[/tex]
The mass of the satellite = m
Altitude = h = r
The radius of the Earth = [tex]R_{E}[/tex]
The Mass of the Earth = [tex]M_{E}[/tex]
Speed of the satellite = v
Speed of the Earth = [tex]v_{e}[/tex]
Gravitational Constant = G
The energy at the surface of the Earth will be the sum of the kinetic and potential Energy,
So,
[tex]E_{i} = K.E. +P.E.\\\\ E_{i} = \frac{1}{2} mv_{e} ^{2} -\frac{GM_{E} m}{r}[/tex]
Also, Energy in orbit,
[tex]E_{i} = K.E. +P.E.\\\\ E_{i} = \frac{1}{2} mv^{2} -\frac{GM_{E} m}{R_{E} }[/tex]
Now, The change in energy is the minimum energy required to place this satellite in orbit,
Then, the Change in Energy =
∆E [tex]= \frac{1}{2} m(v^{2} - v_{e} ^{2} ) +GM_{E} m(\frac{1}{r} -\frac{1}{R_{E} } )[/tex]
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