Parabolic Trajectories

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CHAPTER Parabolic trajectories (e =1) CHAPTER CONTENT Page 148 / 338 9- PARABOLIC TRAJECTORIES (e =1) Page 149 / 338 9- PARABOLICTRAJECTORIES (e =1) If the eccentricity equals 1. then the orbit equation becomes: (1) If For a parabolic trajectory the conservation of energy is It means that the speed anywhere on a parabolic path is: (2) Page 150 / 338 9- PARABOLICTRAJECTORIES (e =1) If the body is launched on a parabolic trajectory; it will coast to infinity, arriving there with zero velocity relative to . It will not return Parabolic paths are therefore called escape trajectories. At a given distance r from , the escape velocity is: (4) Page 151 / 338 9- PARABOLICTRAJECTORIES (e =1) Let be the speed of a satellite in a circular orbit of radius then : (4) ( NOTE 16, P66, {1}) For the parabola, the flight path angle takes the form: Using the trigonometric identities We can write (5) Page 152 / 338 9- PARABOLICTRAJECTORIES (e =1) That is, on parabolic trajectories the flight path angle is one-half the true anomaly Page 153 / 338 9- PARABOLICTRAJECTORIES (e =1) Recall that the parameter of an orbit: . Substitute this expression into equation(1) and then 2a plot r in a cartesian 1 cos coordinate system centered at the focus, we will get: From the figure it is clear that: (6) (7) Page 154 / 338 9- PARABOLICTRAJECTORIES (e =1) Therefore Working to simplify the right-hand side, we get: It follows that: (10) This is the equation of a parabola in a cartesian coordinate system whose origin serves as the focus. Page 155 / 338 9- PARABOLICTRAJECTORIES (e =1) EXAMPLE ?.1 The perigee of a satellite in parabolic geocentric trajectory is 7000km. Find the distance d between point and n the orbit which are 8000km and 16000km, respectively, from the center of the earth. first, let us calculate the angular momentum of the satellite by evaluating the orbit equation at perigee, Page 156 / 338 9- PARABOLICTRAJECTORIES (e =1) EXAMPLE ?.1 From which To find the length of the chord we must use the law of cosines from trigonometry, The true anomalies of points and are found using the orbit equation: Therefore,: Page 157 / 338.

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