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 6380 km. satellite b is to orbit at an altitude of 22700 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 35.0 kg? (d) by how much?

(a) 0.439

The potential energy of a satellite in orbit is given by

where

G is the gravitational constant

m is the mass of the satellite

M is the mass of the Earth

R is the Earth's radius

h is the altitude of the satellite

If we call

the potential energy of satellite A, with

being its altitude, and

the potential energy of satellite B, with

being the altitude of satellite B

and

being the Earth's radius

The ratio between the potential energy of satellite B to that of satellite A will be

(b) 0.439

The kinetic energy of a satellite in orbit has a similar expression to the potential energy

As before, if we call

the kinetic energy of satellite A, with

being its altitude, and

the kinetic energy of satellite B, with

being the altitude of satellite B,

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

(c) Satellite B

The total energy of each satellite is given by the sum of the potential energy and the kinetic energy:

For satellite A we have:

While for satellite B we have

We see that the total energy is inversely proportional to the altitude of the satellite: therefore, the higher the satellite, the smaller the energy. So, satellite A will have the greater total energy (in magnitude), since ; however, the value of the total energy is negative, so actually satellite B will have a greater energy than satellite A.

(d)

The total energy of satellite A is

with

while the total energy of satellite B is

with

So the difference between the two energies is

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