Path of a UFO Background: Two tracking stations make a sighting of a UFO and 7 s
ID: 2981476 • Letter: P
Question
Path of a UFO Background: Two tracking stations make a sighting of a UFO and 7 seconds later make a second sighting of the same UFO. You will find an equation for the path of the UFO in this problem. station 1 sights the UFO 30 degrees south of west with an angle of 0.83 degrees above the horizontal in first sighting. In second, sees it at due west, 1.32 degrees above horizontal. (3500,4500,800)= position station 2 sights the UFO due south, 4.70 degrees above the horizontal in first sighting. In the second sighting, sees it at 80 degrees south of west, 13 degrees above the horizontal. a. Let v1 and v2 be vectors that point from the tracking stations 1 and 2 towards the UFO, respectively for the 1st sighting. These vectors are not unique, so choose them to have the form < a, -1, b >. Use at least 3 significant digits in your computations to minimize errors. Here is some help finding v1 : From Station 1, draw a right triangle using the bearing 30 degrees south of west and using the vertical (y-component) of -1. Solve for the adjacent side of this triangle for the x-component of the vector. ???? To find the z-component of v1, find the hypotenuse of the triangle above and then use that hypotenuse as the base of a triangle whose hypotenuse is inclined in the z-direction 0.83 degrees. Find the z-component of that triangle. To check your work v1 = < -1. 732, -1, 0.029 > b. Find the two equations (parametric form) of the lines from the stations 1 and 2 to the UFO for the first sighting. c. Find the point of intersection, P1, between the lines in part b. This intersection is the position of the UFO at the time of the first sighting. Notice that the two lines might not quite intersect, but if the bearings are accurate you should be able to find a point that very nearly lies on both lines. d. Let v3 and v4 be vectors that point from the tracking stations 1 and 2 towards the UFO, respectively for the 2nd sighting. These vectors are not unique. Since v3 is due west, let it have the form < -1, b, c >. Similarly v4 is south of west and can have the form < d, -1, e >. e. Find the two equations (parametric form) of the lines from the stations 1 and 2 to the UFO for the 2nd sighting. f. Find the point of intersection, P2, between the lines in part e. This intersection is the position of the UFO at the time of the 2nd sighting. g. If we assume that the UFO travels in a straight line between P1 and P2 at a constant speed, we can find the vector equation r(t) of this line using the two points. (t = 0 at P1) h. If 7 seconds have passed between the 1st and 2nd sighting, how fast is the UFO traveling? i. Find the speed of the equation, r(t). What adjustment do you need to perform (mathematically) to adjust the speed to match what you found in part h? Make this adjustment and write a new equation for r(t).Explanation / Answer
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