Question

A $3180-kg$ spacecraft is in a circular orbit $2230km$ above the surface of Mars.
How much work must the spacecraft engines perform to move the spacecraft to a circular orbit that is $4170km$ above the surface?

Answer 1

$E=KE+PE=\frac{1}{2}{mv}^2-\frac{-GmM}{R}$

And $\frac{v^2}{r}=\frac{GM}{r^2}\Rightarrow v^2=\frac{GM}{r}$

$\therefore TE=\frac{1}{2}m\left(\frac{GM}{r}\right)-\frac{GmM}{r}=\frac{-1}{2}\frac{GmM}{r}$

$M\to$mass ofmars:-$6.42\times {10}^{23}kg$

$m=3180kg$

$r\to$Radius of mars:-$3.4\times {10}^{6}km$

First Orbit

$T{E}_1=\frac{-1}{2}\times \frac{6.67\times {10}^{-11}\times 3180\times 6.42\times {10}^{23}}{(3.4\times {10}^6+2.2\times {10}^6)}\approx -12\times {10}^{9}J$

Second orbit

$T{E}_2=-\left[\frac{6.67\times {10}^{-11}\times 3180\times 6.42\times {10}^{23}}{(3.4\times {10}^6+4.1\times {10}^6)}\right]\approx -9\times {10}^{9}J$

$\therefore$ Required work, $W=3\times {10}^{9}J$