University of Colorado Colorado Springs
MAE 4410: Fundamentals of Astrodynamics
Homework #9
Due: 28 April 2008
Understanding ballistic missile trajectories
1. GEOMETERY OF A BALLISTIC TRAJECTORY: You should be acquainted with the six initial conditions of a ballistic trajectory (RBurnout, VBurnout, ΦBurnout, βBurnout, LBurnout, [pic]Burnout). These conditions exist at the point where the missile rocket motor shut off—after a period of thrusting and maneuvering through the atmosphere as shown in Figure 1.
[pic]
Figure 1: Ballistic Trajectory Geometry
2. CALCULATE THE RANGE ANGLE (Λ) FOR THE FOLLOWING SCENARIO: Imagine you are the Chief of Safety for a test launch of an Intercontinental Ballistic Missile (ICBM). The launch site will be Vandenberg Air Force Base (VAFB) just north of Los Angeles (Lo=35°N, lo=121°W). The target is an island in the South Pacific Ocean (Figure 2)—Yap Island (Lt=10°N, lt=140°E).
[pic]
Figure 2: Yap Island Targeted
a) The contractor planning the test assures you that the test vehicle, which can achieve a burnout velocity (VBurnout) of 7200 m/s, will hit the target based on their calculated initial conditions without error. However, as the Chief of Safety, you must verify their numbers. Determine what the Range Angle (Λ), Range in km, and Burnout Azimuth (βBurnout). Recall from Appendix E the equation is:
[pic]
[pic]_____________ or ____________
b) Remember these two answers correspond to the short and the long way around the Earth. Using the shortest path, find the Range and possible launch azimuths.
[pic]
[pic]___________________ [pic]
[pic]
[pic]_____________ or ____________
c) One of these azimuths is correct for the short path, the other for the long path around the Earth.
3. DRAW THE TRAJECTORY’S GROUND TRACK: Using the Polar Map given in Figure 3, draw the ground track for the short path from VAFB to Yap Island. Assume the boost phase places the missile directly above the launch site and the re-entry occurs directly above the target. Annotate the both the Range Angle ([pic], centered at the pole) and launch azimuth ([pic], centered on your launch site) on your drawing.
Based on your figure, do you expect the launch azimuth to be;
a) 0 ................
................
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