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Useful Earth data: radius = 6378.14 km, km3/s2,

Problem 1: A satellite is launched into Earth orbit by a Delta II launch vehicle (LV). The Delta LV’s engines do not perform as expected, and at upper-stage burnout the satellite has the following states: altitude is 250 km, inertial velocity is 7.58 km/s, flight-path angle is 0.6 deg. The satellite mass after burn-out and separation is 2,300 kg.

a) (20 pts) Determine the velocity impulse (V) required to obtain a circular orbit when the satellite reaches its apogee.
b) (10 pts) If the satellite uses its on-board propulsion system for the circularization burn, determine the propellant mass required for the circularization burn (assume the on-board hydrazine engine has an Isp = 280 sec).
Problem 2: (45 pts) A new heat shield is being tested experimentally by launching a payload that consists of a re-entry capsule with mass m = 200 kg, drag coefficient CD = 1.7, and radius r = 0.65 m. Design an acceptable engine cut-off state (r, V, and ) that satisfies the following mission constraints: 1) the re-entry capsule’s maximum deceleration is 12 g, and 2) the flight time from engine cut-off to entry interface (EI) must be greater than 15 min to allow payload separation procedures. In addition, determine the inertial velocity and flight-path angle at EI.

( Problem 3 – next page )

Problem 3: (25 pts) A spacecraft design similar to GEOSAT proposes to use passive gravity-gradient stabilization for pitch attitude control (motion about the y-axis). The satellite consists of a tip mass, separated from the main satellite body mass by a rigid boom with length 10 m. The z-body axis is along the boom, and pitching motion is along the y-body axis (which is opposite the angular momentum vector of the orbit; the +y-axis is into the page). The spacecraft is in circular LEO (altitude = 800 km). The satellite is axis-symmetric, and the moments of inertia are kg-m2, and kg-m2.

Compute (by hand) the pitching motion of the spacecraft in LEO for an initial pitch angle of 5 deg. Assume that the only torque acting on the satellite is the gravity-gradient torque and that the initial angular rates are zero and that no roll/yaw motion exists. Sketch the pitching motion vs. time for one orbital revolution.

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