Two tracking stations make a sighting of a UFO. Seven seconds later both trackin
ID: 2894988 • Letter: T
Question
Two tracking stations make a sighting of a UFO. Seven seconds later both tracking stations make another sighting of the UFO. Based on these two sightings, the position and velocity of the UFO are determined. One second after the second sighting, a probe is launched at precisely the correct speed and heading to intercept the UFO. How is it done? Measuring all distances in meters, Figure 1 shows the coordinates and the horizontal locations of the tracking stations and the launch site. The elevations are given by the third coordinates. Table 1 shows the bearings of the UFO taken by the tracking stations in each of two sightings. For example, tracking station 1 first sights the UFO at 30 degree south of due west at an angle of 0.83 degree above the horizontal. (Note that the stations are on hills.) One second after the second sighting the probe is launched (during which time the UFO continues to move). We assume that the probe reaches its constant speed instantaneously. Assume also that the probe intercepts the UFO T seconds after the probe is launched. Let an equation of the line along which the probe travels have the form R(t) = R_0 +(a, b, c)t, where t = 0 is the time the probe is launched and R_0 = (400, 500, 300) is the position of the launch site. Explain why the condition for the intersection of the probe and the UFO is r(8 + T) = R(T).Explanation / Answer
From the knowledge of the position and velocity values at two instants, based on two points, we can obtain the posiiton and velocity vectors of the trajectory of the UFO.
We do this using
r = r1 + t<r2 - r1>
and
v = v1 + s<v2 - v1>
So, now, using the r and the v vectors,
we know where the UFO's gonna be 1 second after the second sighting.
We can simply plug in t = 8(8 because t = 0 was the first sighting
and t = 7 was second sighting and then 1 second later would be t = 8)
and find the values of r and v.
The probe that we have is designed by us and therefore, we know the info of the position and velocity vectors and therefore, we can find the coordinates where our probe and the ufo are likely to intersect and then shoot off the probe after programming in the details.
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10)
We know that the probe is gonna meet the UFO T seconds after the ufo was first sighted.
The posiiton of the probe when t = T is :
R(T) = R0 + <a,b,c>T where T here is a constant value
Now, since the ufo has already traveled a good 8 seconds,
we have r(t) when t = T as r(8 + T)
And for them to intersect,
r(8 + T) = R(T)
Hence proved!
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