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hw1 - A space probe is in a circular parking orbit at an...

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MAE 180B (Spring 2008) - HW #1 Due: Wednesday, April 23, 2008 Note : In the following problems, vectors are shown in bold (for example, vector V , or unit vectors i , j , k , etc.) Problem 1 The position and velocity of a satellite at a given instant are described by: r = 12,000 i + 12,000 j + 12,000 k (in km units), V = - 3.4 i + 1.7 j + 3.4 k (in km/s units). Here, the unit vectors i , j , k correspond to the geocentric coordinate system X-Y-Z, which has its origin at the Earth’s center (see Fig. 6-3). Find the specific angular momentum vector (in units of km 2 /s), the magnitude of this vector, and the total specific mechanical energy of the satellite (in units of km 2 /s 2 ). Problem 2 For an Earth satellite, the perigee distance is 5,540 miles and the apogee distance is 7,820 miles. (a) Calculate the semi-major axis and the eccentricity of the orbit; (b) Calculate the specific mechanical energy (in miles 2 /s 2 ); (c) Calculate the length of the position vector (in miles) at a true anomaly of 135 ° . Problem 3
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Unformatted text preview: A space probe is in a circular parking orbit at an altitude of 225 km above the Earth. The probe is ready to be launched on a parabolic escape trajectory from the parking orbit. Calculate the minimum escape speed required to leave the parking orbit. Assume that the Earth is spherical with the Mean Equatorial Radius of 6,380 km. Problem 4 Consider the Quadrantal Spherical Triangle of Fig. 5-7(c), and assume that the reference meridian is the great circle just to the left of point B in the diagram. (a) Show the locations of points A, B, and C in degrees in the form (Az , El) ; (b) Calculate the area of the triangle in steradians (you need to use Appendix D); (c) Calculate the unit vector coordinates of points A, B, and C. Problem 5 A radar station locates an object at r = - 6,400 i- 6,400 j- 6,400 k (in km units). The velocity of the object is registered as V = 2.5 i- 2.5 j + 2.5 k (in km/s units). Calculate all of the orbital elements listed in Table 6.2....
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