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$\mathbf{6370}\mathbf{km}$. Satellite B is to orbit at an altitude of $\mathbf{19110}\mathbf{\text{km}}$ . The radius of Earth ${\mathbf{\text{R}}}_{\mathbf{E}}$is $\mathbf{6370}\mathbf{\text{km}}$ .

(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 has the greater total energy if each has a mass of$\mathbf{14}\mathbf{.}\mathbf{6}\mathbf{}\mathbf{kg}$ ?

(d) By how much?

Short answer

1. $\left(\frac{{\text{U}}_{\text{b}}}{{\text{U}}_{\text{a}}}\right)=\frac{1}{2}$
2. $\left(\frac{{\text{K}}_{\text{b}}}{{\text{K}}_{\text{a}}}\right)=\frac{1}{2}$
3. ${\text{E}}_{\text{b}}>{\text{E}}_{\text{a}}$
4. ${\text{E}}_{\text{a}}$ Is greater than ${\mathrm{E}}_{\mathrm{b}}$ by $1.1\times {10}^8\mathrm{J}$

Step-by-step solution

Listing the given quantities

Mass of satellite A (M_a) and Mass of satellite B${\mathrm{M}}_{\mathrm{a}}={\mathrm{M}}_{\mathrm{b}}=\text{m}=14.6\mathrm{kg}$.

Height of satellite A${\text{h}}_{\text{a}}=6370\mathrm{km}\left(\frac{{10}^3\mathrm{m}}{1\mathrm{km}}\right)=6370\times {10}^3\mathrm{m}$

Height of satellite B${\text{h}}_{\text{b}}=19110\mathrm{km}\left(\frac{{10}^3\mathrm{m}}{1\mathrm{km}}\right)=19110\times {10}^3\mathrm{m}$

The radius of the Earth ${\text{R}}_{\text{E}}=6370\mathrm{km}\left(\frac{{10}^3\mathrm{m}}{1\mathrm{km}}\right)=6370\times {10}^3\mathrm{m}$

Understanding the potential and kinetic energy

By using potential energy and kinetic energy formulae, we can find out

$\left(\frac{{\text{U}}_{\text{b}}}{{\text{U}}_{\text{a}}}\right)$And$\left(\frac{\text{Kb}}{{\text{K}}_{\text{a}}}\right)$

By adding potential energy and kinetic energy, we can find the total energy.

Formula:

$\mathrm{U}=\frac{-\mathrm{GMm}}{(\mathrm{R}+\mathrm{h})}$

$\mathrm{K}=\frac{\mathrm{GMm}}{2(\mathrm{R}+\mathrm{h})}$

$\mathrm{E}=\mathrm{U}+\mathrm{K}$

$\mathrm{E}=-\frac{\mathrm{GMm}}{2(\mathrm{R}+\mathrm{h})}$

(a) Calculation of the ratio of the potential energy of satellite B to that of satellite A

$\left(\frac{{\text{U}}_{\text{b}}}{{\text{U}}_{\text{a}}}\right)$

${\text{U}}_{\text{b}}\text{=}\frac{\text{-GMm}}{{\text{R}}_{\text{E}}{\text{+h}}_{\text{b}}}$ (1)

${\text{U}}_{\text{a}}\text{=}\frac{\text{-GMm}}{{\text{R}}_{\text{E}}{\text{+h}}_{\text{a}}}$ (2)

Dividing equations 1 and 2

$\frac{{\text{U}}_{\text{b}}}{{\text{U}}_{\text{a}}}\text{=}\frac{\text{-GMm}}{{\text{R}}_{\text{E}}\text{+hb}}\text{×}\frac{{\text{R}}_{\text{E}}\text{+ha}}{\text{-GMm}}$

$\begin{array}{c}\frac{{\text{U}}_{\text{b}}}{{\text{U}}_{\text{a}}}\text{=}\frac{{\text{R}}_{\text{E}}\text{+ha}}{{\text{R}}_{\text{E}}\text{+hb}}\\ \text{=}\frac{\text{6370\hspace{0.17em}km+6370\hspace{0.17em}km}}{\text{6370\hspace{0.17em}km+19110\hspace{0.17em}km}}\end{array}$

$\frac{{\text{U}}_{\text{b}}}{{\text{U}}_{\text{a}}}=\frac{1}{2}$

${\text{U}}_{\text{a}}=2{\text{U}}_{\text{b}}$ (3)

(b) Calculation of the ratio of the kinetic energy of satellite B to that of satellite A

Calculation for$\left(\frac{{\text{K}}_{\text{b}}}{{\text{K}}_{\text{a}}}\right)$

${\text{K}}_{\text{b}}\text{=}\frac{\text{GMm}}{\text{2}\left({\text{R}}_{\text{E}}{\text{+h}}_{\text{b}}\right)}$ (4)

${\text{K}}_{\text{a}}\text{=}\frac{\text{GMm}}{\text{2}\left({\text{R}}_{\text{E}}{\text{+h}}_{\text{a}}\right)}$ (5)

Dividing equation (4) by (5)

$\frac{{\text{K}}_{\text{b}}}{{\text{K}}_{\text{a}}}\text{=}\frac{{\text{R}}_{\text{E}}{\text{+h}}_{\text{a}}}{{\text{R}}_{\text{E}}{\text{+h}}_{\text{b}}}$

$\frac{{\text{K}}_{\text{b}}}{{\text{K}}_{\text{a}}}\text{=}\frac{\text{1}}{\text{2}}$

${\text{K}}_{\text{a}}{\text{=2K}}_{\text{b}}$

(c) Which satellite has the greater total energy, satellite A (Ea) or satellite B (Eb)? 

Comparing the total energy of satellite A and satellite B

The satellite with a smaller value of the radius of orbit would have larger energy. Also, the value of E is negative. So, the satellite with the smallest value of magnitude would have the largest energy. Therefore, satellite B has the largest energy.

(d) how much is the total energy of one satellite greater than the other? 

Calculation for${\mathrm{E}}_{\mathrm{a}}$and${\mathrm{E}}_{\mathrm{b}}$

$\begin{array}{c}{\text{E}}_{\text{a}}=-\frac{\text{GMm}}{\text{2}\left({\text{R}}_{\text{E}}{\text{+h}}_{\text{a}}\right)}\\ =-\frac{6.67\times {10}^{-11}\mathrm{N}\cdot {\mathrm{m}}^2/{\mathrm{kg}}^2\times 5.98\times {10}^{24}\mathrm{kg}\times 14.6\mathrm{kg}}{2(6370+6370){10}^3\mathrm{m}}\\ =-2.2\times {10}^8\mathrm{J}\end{array}$

$\begin{array}{c}{\text{E}}_{\text{b}}=\frac{{\text{E}}_{\text{a}}}{\text{2}}\\ =\frac{-2.2\times {10}^8\mathrm{J}}{2}\\ =-1.1\times {10}^8\mathrm{J}\end{array}$

$\begin{array}{c}\left|{\text{E}}_{\text{a}}\right|\text{-}\left|{\text{E}}_{\text{b}}\right|=2.2\times {10}^8\mathrm{J}-1.1\times {10}^8\mathrm{J}\\ =1.1\times {10}^8\mathrm{J}\end{array}$

${\mathrm{E}}_{\mathrm{a}}$ Is greater than ${\mathrm{E}}_{\mathrm{b}}$ by $1.1\times {10}^8\mathrm{J}$