Taking the earth to be the uniform sphere of radius $6400 \; km$, calculate the total energy needed to raise a satellite of mass $1000 \; kg$ to a height of $600 \; km$ above the ground and to set it into a circular orbit at that altitude.

Given;

Radius of earth (R) = 6400 km = 6400000 m

Mass of satellite (m) = 1000 kg

Height of satellite (h) = 600 km = 600000 m

Total Energy needed (E) = ?

Energy needed = Increase in Potential Energy + Kinetic Energy at Orbit

= $-$$\frac{G\;M\;m}{r}$$\;-\;$$\frac{-\;G\;M\;m}{R}$$\;+\;$$\frac{1}{2}$$mv^2$

= $-$$\frac{G\;M\;m}{r}$$\;-\;$$\frac{-\;G\;M\;m}{R}$$\;+\;$$\frac{1}{2}$$m$$\frac{G\;M}{r}$                               [∵ r = R+h]

= $g\;R^2\;m[$$\frac{1}{R}$$\;-\;$$\frac{1}{2(R+h)}$$]$     =     $g\;m[R\;-\;$$\frac{R^2}{2(R+h)}$$]$

= $1000\;*\;10\;[6400000 \;-\;$$\frac{(6400000)^2}{2(6400000 \;+\; 600000)}]$

= $3.47\;*\;10^{10}\;J$

∴ The total Energy needed (E) = $3.47\;*\;10^{10}\;J$

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