Question

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?

Answer 1

(a) 0.439

The potential energy of a satellite in orbit is given by

`U=-(GmM)/(R+h)`

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

`U_A=-(GmM)/(R+h_A)`

the potential energy of satellite A, with

`h_A = 6380 km = 6.38\cdot 10^6 m`

being its altitude, and

`U_B=-(GmM)/(R+h_B)`

the potential energy of satellite B, with

`h_B = 22700 km = 22.7\cdot 10^6 m`

being the altitude of satellite B

and

`R=6370 km = 6.37 \cdot 10^6 m` being the Earth's radius

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

`(U_B)/(U_A)=(R+h_A)/(R+h_B)=(6.37\cdot 10^6 m+6.38\cdot 10^6 m)/(6.37\cdot  10^6 m+22.7\cdot 10^6 m)=0.439`

(b) 0.439

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

`K=(1)/(2) (GmM)/(R+h)`

As before, if we call

`K_A=(1)/(2) (GmM)/(R+h_A)`

the kinetic energy of satellite A, with

`h_A = 6380 km = 6.38\cdot 10^6 m`

being its altitude, and

`K_B=(1)/(2) (GmM)/(R+h_B)`

the kinetic energy of satellite B, with

`h_B = 22700 km = 22.7\cdot 10^6 m`

being the altitude of satellite B,

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

`(K_B)/(K_A)=(R+h_A)/(R+h_B)=(6.37\cdot 10^6 m+6.38\cdot 10^6 m)/(6.37\cdot  10^6 m+22.7\cdot 10^6 m)=0.439`

(c) Satellite B

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

`E= U+K = -(GMm)/(R+h)+(1)/(2) (GMm)/(R+h)=-(1)/(2)(GMm)/(R+h)`

For satellite A we have:

`E_A = -(1)/(2)(GMm)/(R+h_A)`

While for satellite B we have

`E_B = -(1)/(2)(GMm)/(R+h_B)`

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
`h_A < h_B`; however, the value of the total energy is negative, so actually satellite B will have a greater energy than satellite A.

(d)
`3.07\cdot 10^8 J`

The total energy of satellite A is

`E_A = -(1)/(2)(GMm)/(R+h_A)`

with

`h_A = 6380 km = 6.38\cdot 10^6 m`

while the total energy of satellite B is

`E_B = -(1)/(2)(GMm)/(R+h_B)`

with

`h_B = 22700 km = 22.7\cdot 10^6 m`

So the difference between the two energies is

`E_B - E_A = -(1)/(2)((6.67\cdot 10^(-11)(35 kg)(5.98\cdot 10^(24) kg))/(6.37\cdot 10^6 m +22.7\cdot 10^6 m)-(-(1)/(2)((6.67\cdot 10^(-11)(35 kg)(5.98\cdot 10^(24) kg))/(6.37\cdot 10^6 m +6.38\cdot 10^6 m))=3.07\cdot 10^8 J`