Question

A satellite is placed in a noncircular orbit about the Earth. The farthest point of its orbit (apogee) is 4 Earth radii from the center of the Earth, while its nearest point (perigee) is 2 Earth radii from the Earth's center. If we define the gravitational potential energy \(U\) to be zero for an infinite separation of Earth and satellite, find the ratio $U_{\text {perigce }} / U_{\text {apogec }}$

Short answer

Answer: The ratio of the gravitational potential energy at the perigee point to the apogee point is 2:1.

Step-by-step solution

Define the gravitational potential energy formula

The gravitational potential energy is given by the formula: \(U = -\frac{G * m_{1} * m_{2}}{r}\) where: - \(U\) is the gravitational potential energy - \(G\) is the gravitational constant, approximately \(6.674 \times 10^{-11} \ Nm^2/kg^2\) - \(m_1\) is the mass of the satellite - \(m_2\) is the mass of the Earth, approximately \(5.972 \times 10^{24} \ kg\) - \(r\) is the distance between the centers of the two masses

Use the potential energy formula to find the energy at the perigee point

At the perigee point, the satellite is \(2\) Earth radii away from the center of the Earth. Since the radius of the Earth is approximately \(6.371 \times 10^6 \ m\), the perigee distance is \(2 * 6.371 \times 10^6 \ m\). Now we can find the potential energy at the perigee point: \(U_{perigee} = -\frac{G * m_{1} * m_{2}}{2*6.371 \times 10^6} \)

Use the potential energy formula to find the energy at the apogee point

At the apogee point, the satellite is \(4\) Earth radii away from the center of the Earth. So, the apogee distance is \(4 * 6.371 \times 10^6 \ m\). Now we can find the potential energy at the apogee point: \(U_{apogee} = -\frac{G * m_{1} * m_{2}}{4*6.371 \times 10^6} \)

Compute the ratio of the potential energy at the perigee to the apogee points

We need to find the ratio \(U_{perigee} / U_{apogee} \). We can divide \(U_{perigee}\) by \(U_{apogee}\) as follows: \(\frac{U_{perigee}}{U_{apogee}} = \frac{-\frac{G * m_{1} * m_{2}}{2*6.371 \times 10^6}}{-\frac{G * m_{1} * m_{2}}{4*6.371 \times 10^6}}\) The masses and the gravitational constant will cancel out: \(\frac{U_{perigee}}{U_{apogee}} = \frac{-\frac{1}{2*6.371 \times 10^6}}{-\frac{1}{4*6.371 \times 10^6}} = \frac{4*6.371 \times 10^6}{2*6.371 \times 10^6}\) Finally, the ratio is: \(\frac{U_{perigee}}{U_{apogee}} = 4 / 2 = 2\) So, the ratio of the gravitational potential energy at the perigee point to the apogee point is 2:1.