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exam1-solution - ENAE404 Spring 2007 Answers to Exam#1 75...

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ENAE404 Spring 2007 Answers to Exam #1 March 13, 2007 75 minutes L. Healy 1. [20 points] For each of the following earth-orbiting satellites, identify whether the orbit is circular, elliptical, parabolic, or hyperbolic. Give a brief justification. (a) Position vector is 7000 ˆ I +8000 ˆ J +750 ˆ K km at 03:21:19 UTC and again at 08:23:19 UTC the same day. Elliptical ; because the orbit returns on itself, it is closed (circu- lar or elliptical). The possibility of circular orbit can be eliminated as follows. The time interval 5 hours 2 minutes must be a multi- ple of the period. If it is equal to the period, then the semimajor axis is a = 1 . 491 × 10 4 km, if it is twice the period, a = 9393 km. Any higher multiple and the semimajor axis would be less. But the actual geocentric distance is r = 1 . 066 × 10 4 km. Since this is not the semimajor axis r = a , the orbit cannot be circular and so must be elliptical. Note: several students determined that the semimajor axis of a circular orbit with the period equal to the time interval did not match that of a circular orbit. They then concluded that it was an elliptical orbit. However, they did not consider that multiple orbits (two, three, etc.) could have occured in the time interval stated. (b) Energy ε = 12 . 4 km 2 / s 2 and angular momentum magnitude h = 62000 km 2 / s . Hyperbolic , because the energy is positive.
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