Two earth satellites, a and b, each of mass m, are to be launched into circular orbits about earth's center. satellite a is to orbit at an altitude of 7580 km. satellite b is to orbit at an altitude of 22000 km. the radius of earth reis 6370 km. (a) what is the ratio of the potential energy of satellite b to that of satellite a, in orbit? (b) what is the ratio of the kinetic energy of satellite b to that of satellite a, in orbit? (c) which satellite (answer a or b) has the greater total energy if each has a mass of 18.2 kg? (d) by how much?

(a) 0.49

The potential energy of a satellite in orbit is given by

where G is the gravitational constant, M is the Earth's mass, m is the satellite's mass, R is the Earth's radius and h is the altitude of the satellite.

The potential energy of satellite A is

The potential energy of satellite B is

So the ratio of the potential energy of satellite B to that of satellite A is

The two satellites have same mass, , so the equation becomes

where

Substituting into the equation,

(b) 0.49

The kinetic energy of a satellite in orbit is given by

where U is the potential energy.

The kinetic energy of satellite A is

The kinetic energy of satellite B is

So the ratio of the kinetic energy of satellite B to that of satellite A is

and we already found the value of this ratio in part (a), so this ratio is also equal to 0.49.

(c) Satellite A

The total energy of a satellite in orbit is sum of potential energy and kinetic energy:

We can write therefore the total energy of satellite A as

and the total energy of satellite B as

The ratio of the total energy of satellite B to that of satellite A is therefore

which implies that satellite A has greater total energy then satellite B.

(d) by

In order to find the difference between the total energies of the two satellite, we must calculate the exact value for each satellite. Using m = 18.2 kg and (Earth's mass), we find:

So the difference between the two energies is

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