The equation that could be used to find the velocity of the satellite if it is placed in a low Earth orbit is [tex]v = \sqrt{\frac{Gm_p}{200 \ km} }[/tex].
The given parameters;
- mass of the planet, = [tex]m_p[/tex]
A low Earth orbit is an orbit around the Earth with altitude in the range of 250 km and 2000 km.
The gravitational potential or the work done in bringing the satellite to the low Earth orbit is given as;
[tex]U = \frac{GMm_p}{R} \\\\[/tex]
where;
- M is the mass of the Earth
- R is the altitude of the low Earth orbit
let the value of R = 400 km
Note: any value between 250 km and 2000 km can be used BUT 400 km is the only value within this range that can be found in the given options as you will see at the end of the solution.
[tex]U = \frac{GMm_p}{400 \ km}[/tex]
The kinetic energy of the satellite at the low Earth orbit is equal to the gravitational potential energy due to the altitude of the low Earth orbit.
[tex]K.E = U\\\\\frac{1}{2} Mv^2 = \frac{GMm_p}{400 \ km} \\\\v^2 = \frac{2Gm_p}{400 \ km}\\\\v^2 = \frac{Gm_p}{200 \ km}\\\\v = \sqrt{\frac{Gm_p}{200 \ km}}[/tex]
Thus, the equation that could be used to find the velocity of the satellite if it is placed in a low Earth orbit is [tex]v = \sqrt{\frac{Gm_p}{200 \ km} }[/tex].
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