Question

A satellite of mass mS orbits the Earth (with mass m_E and radius R_E) with a velocity v and an altitude h. The gravitational force F_G and the centripetal force F_C are given by:

F_G = G(m_S · m_E)/(R_E + h)^2 ,

F_C =(m_S · v^2)/(R_E + h) ,

G = const.

(a) Find an equation for the kinetic energy E_kin(h) of a satellite with an altitude h.

(b) Based on the kinetic energy, how much liquid hydrogen (energy density: 10^6J/Litre) is at least needed to bring a small 1 kg satellite in an orbit of 400 km. (Use literature to find m_E, R_E, G.)

Answer 1

(a) For satellite, at an altitude of h, the centripetal force is supplied by the gravitational force. Therefore, you can write,

Therefore, the kinetic energy (E_K) of the satellite will be

(b) The mass of Earth is m_E = 5.972 x 10²⁴ Kg, m_S = mass of satellite = 1 Kg, h = height of the satellite = 400 Km, R_E = radius of Earth = 6378 Km, Hence (R_E + h) = (6378 + 400) Km =  6778000 m,

G = gravitational constant = 6.674 x 10^-11 m³ Kg^-1 s^-2. Therefore, the kinetic energy will be

For a satellite, the potential energy will be twice the kinetic energy but negative in sign. Hence, total energy of the satellite will be same as the kinetic energy in magnitude. Therefore, the liquid hydrogen (energy density: 10⁶J/Litre) is at least needed to bring a small 1 kg satellite in an orbit of 400 km