The total mechanical energy of the satellite is approximately 0 J. This is because the potential and kinetic energies have the same magnitude but opposite signs, canceling each other out. This is a characteristic of an object in stable circular orbit.
Define the variables:
Mass of the satellite (m_s) = 1200 kg
Mass of the Earth (m_e) = 5.972 x 10^24 kg
Radius of the Earth (r_e) = 6371 km = 6.371 x 10^6 m
Distance of the satellite from the Earth's center (r) = r_e + altitude = 6.371 x 10^6 m + 1.5 x 10^6 m = 7.871 x 10^6 m
Universal gravitational constant (G) = 6.6743 x 10^-11 m^3 kg^-1 s^-2
The potential energy (E_p):
E_p = -G * m_e * m_s / r
E_p = -(6.6743 x 10^-11 m^3 kg^-1 s^-2) * (5.972 x 10^24 kg) * (1200 kg) / (7.871 x 10^6 m)
E_p ≈ -3.04 x 10^10 J
The kinetic energy (E_k):
Since the satellite is in a circular orbit, its velocity (v) is constant. We can relate the velocity to the radius and the gravitational constant through the following equation:
v^2 = G * m_e / r
E_k = 1/2 * m_s * v^2
E_k = 1/2 * (1200 kg) * ((6.6743 x 10^-11 m^3 kg^-1 s^-2) * (5.972 x 10^24 kg) / (7.871 x 10^6 m))
E_k ≈ 3.04 x 10^10 J
The total mechanical energy (E_total):
E_total = E_p + E_k
E_total = -3.04 x 10^10 J + 3.04 x 10^10 J
E_total ≈ 0 J