The Space X engineering group has been asked to design a mission to send a satellite in orbit around the Earth. The satellite has a mass m_s = 500 kg, and the rocket has a mass m_R = 2000 kg. The satellite is required to revolve with an elliptic orbit around the Earth with velocity v_B = 25 km/s when it is located in point B, at a distance to the earth r_B = 10 Mm. Assuming the mass of the Earth is equal to m_E = 5.976 middot 10^24 kg the engineers must determine the propulsive force required to put the satellite in orbit. For the completion of the mission, we define three phases: launch (when the system leaves the ground), ejection (when the satellite separates from the rocket), orbit (when the satellite starts its motion around the earth). To solve this problem, we will analyze the three phases in reverse order (i.e. orbit first, then ejection, then launch) a. Phase 3: Determine the distance from the Earth r_A and the initial velocity v_A of the satellite so that the requirements for v_B and r_B are satisfied. Assume that phi_4, i.e. the angle between the tangent to the orbit and the vector connecting the Earth’s center to point A, equals to 80 deg. Use both the conservation of angular momentum and the conservation of energy to solve for the two unknowns v_A and r_A. b. Phase 2: Assume that during the ejection phase, the rocket gains a velocity v_R0 = 40 m/s directed toward the center of the earth. Determine the velocity of the system rocket + satellite v_RS such that the final velocity of the satellite is v_A (computed in a.) c. Phase 1: Assuming that gravity is constant during the launch phase (g = 9.81 m/s^2), determine the flight time and propulsive force needed to launch system rocket + satellite such that the system reaches the required distance r_A with the previously determined v_RS velocity (Computed in b.). (Assume that the mass of the system does not change, the propulsive force is constant and the initial velocity is zero).
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