Question

A 1 003-kg satellite orbits the earth at a constant altitude of 105-km. (a) how much energy must be added to the system to move the satellite into a circular orbit with altitude 202 km?

Answer 1

(a) The energy required to move the satellite into a circular orbit with an altitude of 202 km is
`\(1.81 * 10^(11) \, \text{J}\).`

Step-by-step explanation:

To calculate the energy required, we use the gravitational potential energy formula:

`\[ U = -(G \cdot M \cdot m)/(r) \]`

where \( G \) is the gravitational constant
`(\(6.674 * 10^(-11) \, \text{Nm}^2/\text{kg}^2\)),`
`\( M \)` is the mass of the Earth
`(\(5.972 * 10^(24) \, \text{kg}\)), \( m \)` is the mass of the satellite
`(\(1.003 \, \text{kg}\)),` and
`\( r \)` is the distance from the center of the Earth to the satellite.

The initial altitude is
`\(105 \, \text{km} + \text{Earth's radius}\),` and the final altitude is
`\(202 \, \text{km} + \text{Earth's radius}\).` Converting these altitudes to meters gives the initial and final distances from the center of the Earth.

The energy required is the difference in potential energy:

`\[ \Delta U = U_{\text{final}} - U_{\text{initial}} \]`

Substituting the values into the formula and simplifying, we get:

`\[ \Delta U = -\frac{G \cdot M \cdot m}{r_{\text{final}}} + \frac{G \cdot M \cdot m}{r_{\text{initial}}} \]`

After plugging in the known values and performing the calculations, the result is
`\(1.81 * 10^(11) \, \text{J}\).` This positive value indicates that energy must be added to the system to move the satellite into the higher circular orbit.