Example Aircraft collision avoidance Consider two aircrafts, A and B, flying horizontally at the same level with constant velocities u a and uB Aircraft B is capable of determining by measurement the relative position and velocity of A relative to B 1. e, TA/B and U A/B. We want to know whether it is possible to determine from this measurement whether the two aircraft are on a collision course TA Take the origin O at the hypothetical point of collision where the two trajectories intersect. If collision is to occur. the time taken for both aircraft to reach O will be the same. Therefore Now consider the velocity triangle formed by UA=UB+UA/B B A/B Clearly, UA is parallel to rA, and uB is parallel to rB. In addition, if collision is to occur, UA/B will be parallel to TA/B, since the position and velocity triangles will, in this case, be similar. We see, therefore that the condition for the two planes to collide is that the relative velocity between them is parallel to the relative position vector. If aircraft B uses a polar coordinate system(common for radar measurements) then the relative position of A with respect to B will be of the form TA/B=rer. Therefore, the velocity so has to be of that form, which implies that ve= re=0, or 6= constant. That is, the angle under which B sees A (sometimes called the bearing angle) should not changeExample Aircraft collision avoidance Consider two aircrafts, A and B, flying horizontally at the same level with constant velocities vA and vB. Aircraft B is capable of determining by measurement the relative position and velocity of A relative to B, i.e., rA/B and vA/B. We want to know whether it is possible to determine from this measurement whether the two aircraft are on a collision course. Take the origin O at the hypothetical point of collision where the two trajectories intersect. If collision is to occur, the time taken for both aircraft to reach O will be the same. Therefore, rA vA = rB vB . Now consider the velocity triangle formed by vA = vB + vA/B. Clearly, vA is parallel to rA, and vB is parallel to rB. In addition, if collision is to occur, vA/B will be parallel to rA/B, since the position and velocity triangles will, in this case, be similar. We see, therefore, that the condition for the two planes to collide is that the relative velocity between them is parallel to the relative position vector. If aircraft B uses a polar coordinate system (common for radar measurements), then the relative position of A with respect to B will be of the form rA/B = rer. Therefore, the velocity also has to be of that form, which implies that vθ = r ˙θ = 0, or θ = constant. That is, the angle under which B sees A (sometimes called the bearing angle) should not change. 5