Blog: A satellite of mass 2500 kg is orbiting the Earth i n an elliptical orbit. At the farthest point from the Earth, its altitude is 3600 km, while at...

Let's start with the equations you will need for this problem.

Gravitational potential energy, U:

$\displaystyle{U}=-\frac{{{G}\cdot{M}\cdot{m}}}{{r}}$

where: G = gravitational constant, m = satellite's mass, and M = Earth's mass

Aphelion is where the satellite is furthest away:

$\displaystyle{r}_{{1}}={R}_{{E}}+{3600}{k}{m}={9.967}\cdot{{10}}^{{6}}{m}$

Perihelion is where the satellite is the closest:

$\displaystyle{r}_{{2}}={R}_{{E}}+{1100}{k}{m}={7.467}\cdot{{10}}^{{6}}{m}$

Conservation of Energy:

$\displaystyle{K}_{{1}}+{U}_{{1}}={K}_{{2}}+{U}_{{2}}$

$\displaystyle\frac{{1}}{{2}}\cdot{m}\cdot{{v}_{{1}}^{{2}}}-{G}\cdot{m}\cdot\frac{{M}}{{r}_{{1}}}=\frac{{1}}{{2}}\cdot{m}\cdot{{v}_{{2}}^{{2}}}-{G}\cdot{m}\cdot\frac{{M}}{{r}_{{2}}}$

Conservation of Angular Momentum:

$\displaystyle{m}{v}_{{1}}{r}_{{1}}={m}{v}_{{2}}{r}_{{2}}$

Solve system...

Let's start with the equations you will need for this problem.

Gravitational potential energy, U:

$\displaystyle{U}=-\frac{{{G}\cdot{M}\cdot{m}}}{{r}}$

where: G = gravitational constant, m = satellite's mass, and M = Earth's mass

Aphelion is where the satellite is furthest away:

$\displaystyle{r}_{{1}}={R}_{{E}}+{3600}{k}{m}={9.967}\cdot{{10}}^{{6}}{m}$

Perihelion is where the satellite is the closest:

$\displaystyle{r}_{{2}}={R}_{{E}}+{1100}{k}{m}={7.467}\cdot{{10}}^{{6}}{m}$

Conservation of Energy:

$\displaystyle{K}_{{1}}+{U}_{{1}}={K}_{{2}}+{U}_{{2}}$

$\displaystyle\frac{{1}}{{2}}\cdot{m}\cdot{{v}_{{1}}^{{2}}}-{G}\cdot{m}\cdot\frac{{M}}{{r}_{{1}}}=\frac{{1}}{{2}}\cdot{m}\cdot{{v}_{{2}}^{{2}}}-{G}\cdot{m}\cdot\frac{{M}}{{r}_{{2}}}$

Conservation of Angular Momentum:

$\displaystyle{m}{v}_{{1}}{r}_{{1}}={m}{v}_{{2}}{r}_{{2}}$

Solve system of equations for v2.

$\displaystyle\frac{{1}}{{2}}\cdot{m}\cdot{{\left({v}_{{2}}\cdot\frac{{r}_{{2}}}{{r}_{{1}}}\right)}}^{{2}}-{G}\cdot{m}\cdot\frac{{M}}{{r}_{{1}}}=\frac{{1}}{{2}}\cdot{m}\cdot{{v}_{{2}}^{{2}}}-{G}\cdot{m}\cdot\frac{{M}}{{r}_{{2}}}$

$\displaystyle{{v}_{{2}}^{{2}}}={2}{G}{M}\cdot\frac{{\frac{{1}}{{r}_{{1}}}-\frac{{1}}{{r}_{{2}}}}}{{{{\left(\frac{{r}_{{2}}}{{r}_{{1}}}\right)}}^{{2}}-{1}}}$

Plug in numerical values to and solve to get the velocities.

$\displaystyle{v}_{{1}}={4387.3}\frac{{m}}{{s}}$ at aphelion, and $\displaystyle{5856.2}\frac{{m}}{{s}}$ at perihelion.

Next, plug in either position 1 or position 2 values to get E at BOTH aphelion and perihelion of $\displaystyle{E}={K}+{U}=\frac{{1}}{{2}}\cdot{m}\cdot{{v}_{{1}}^{{2}}}-{G}\cdot{m}\cdot\frac{{M}}{{r}_{{1}}}={7.603}\cdot{{10}}^{{10}}{J}$

Your angular momentum will also be the same for both aphelion and perihelion

$\displaystyle{J}={m}\cdot{r}_{{1}}\cdot{v}_{{1}}={1.09}{x}{{10}}^{{14}}\frac{{{k}{{g{{m}}}}^{{2}}}}{{s}}$

Therefore this problem was completly solved by the conservation of energy and angular momentum.

I included a link that attempts to solve this same problem in more detail. There numerical values are incorrect because they are missing a factor of two when they solve for the velocity.

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A satellite of mass 2500 kg is orbiting the Earth i n an elliptical orbit. At the farthest point from the Earth, its altitude is 3600 km, while at...

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A satellite of mass 2500 kg is orbiting the Earth i n an elliptical orbit. At the farthest point from the Earth, its altitude is 3600 km, while at...

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What are the problems with Uganda&#39;s government?

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