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A 970-kg satellite orbits the Earth at a constant altitude of 94 km.

How much energy must be added to the system to move the satellite into a circular orbit with an altitude of 205 km?

Answer :

The energy needed to move the satellite to a higher orbit is approximately [tex]\(2.94 \times 10^{10} \, \text{J}\)[/tex].

To determine the energy needed to move the satellite to a higher orbit, we can use the gravitational potential energy formula for an object in orbit:

[tex]\[ U = -\frac{GMm}{r} \][/tex]

where:

- U is the gravitational potential energy,

- G is the gravitational constant [tex](\(6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2\))[/tex],

- M is the mass of the Earth [tex](\(5.972 \times 10^{24} \, \text{kg}\))[/tex],

- m is the mass of the satellite [tex](\(970 \, \text{kg}\))[/tex],

- r is the distance from the center of the Earth to the satellite.

The initial orbital radius [tex](\(r_1\))[/tex] is the sum of the Earth's radius and the initial altitude:

[tex]\[ r_1 = R_{\text{Earth}} + h_1 \][/tex]

The final orbital radius [tex](\(r_2\))[/tex] is the sum of the Earth's radius and the final altitude:

[tex]\[ r_2 = R_{\text{Earth}} + h_2 \][/tex]

The change in potential energy [tex](\( \Delta U \))[/tex] is given by:

[tex]\[ \Delta U = U_2 - U_1 \][/tex]

Substitute the expressions for U and the values into the formula:

[tex]\[ \Delta U = -\frac{GMm}{r_2} + \frac{GMm}{r_1} \][/tex]

Now, calculate the values and solve for [tex]\( \Delta U \)[/tex]:

[tex]\Delta U[/tex] = [tex]-\frac{(6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2)(5.972 \times 10^{24} \, \text{kg})(970 \, \text{kg})}{r_2} + \frac{(6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2)(5.972 \times 10^{24} \, \text{kg})(970 \, \text{kg})}{r_1}[/tex]

Substitute the values of [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex], which are [tex]\(R_{\text{Earth}} + h_1\)[/tex] and [tex]\(R_{\text{Earth}} + h_2\)[/tex], respectively.

Now, plug in the known values and calculate the change in potential energy [tex](\( \Delta U \))[/tex].

[tex]\[ \Delta U \approx 2.94 \times 10^{10} \, \text{J} \][/tex]

Therefore, the energy that must be added to the system to move the satellite into a circular orbit with an altitude of 205 km is approximately [tex]\(2.94 \times 10^{10} \, \text{J}\)[/tex].

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