You are working as an assistant to an air-traffic controller...

You are working as an assistant to an air-traffic controller at the local airport, from which small airplanes take off and land. Your job is to make sure that airplanes are not closer to each other than a minimum safe separation distance of $2.00 \mathrm{~km}$. You observe two small aircraft on your radar screen, out over the ocean surface. The first is at altitude $800 \mathrm{~m}$ above the surface, horizontal distance $19.2 \mathrm{~km}$, and $25.0^{\circ}$ south of west. The second aircraft is at altitude $1100 \mathrm{~m},$ horizontal distance $17.6 \mathrm{~km},$ and $20.0^{\circ}$ south of west. Your supervisor is concerned that the two aircraft are too close together and asks for a separation distance for the two airplanes. (Place the $x$ axis west, the $y$ axis south, and the $z$ axis vertical. $)$

Key Concepts

Cartesian Coordinate System

This concept involves defining a three-dimensional coordinate system with mutually perpendicular axes (in this case, x, y, and z). In problems like this, the axes are assigned specific directions (e.g., west, south, and vertical) to establish a consistent frame of reference for positioning objects in space.

Vector Component Decomposition

This concept entails breaking down a vector quantity into its components along the chosen coordinate axes. When a measurement is given using a magnitude and a directional angle, trigonometric functions are used to resolve it into x (west) and y (south) components, while any vertical component is directly accounted for as the z value.

Trigonometric Conversion

This key concept is the use of trigonometric functions, specifically sine and cosine, to convert a directional magnitude (such as a horizontal distance at a certain angle) into its orthogonal components in a predefined coordinate system. It is fundamental for analyzing directions and distances in both navigation and physics problems.

3D Distance Calculation

This concept involves applying the three-dimensional version of the Pythagorean theorem to calculate the distance between two points in space. The calculation considers the differences in all three coordinate directions (x, y, and z) to determine the straight-line or Euclidean distance, ensuring that all aspects of the separation are properly accounted for.

Transcript

00:01 Okay, question 3 .42.

00:04 You're working as an air traffic controller assistant.

00:08 You want to figure out the distance between two aircraft.

00:11 It needs to be greater than or equal to two kilometers.

00:16 So let's draw the positions of each aircraft and then see if we can figure out a way to determine how far they are from each other.

00:25 So we're told x is west, y is south, x, y, put the coordinate in parentheses, i guess.

00:38 And then z is vertical the z coordinate.

00:47 Okay, so one aircraft is observed to be an altitude of 800 meters and at a horizontal distance, 19 .2 kilometers, 25 degrees south of west.

01:04 So let's do the 25 degrees south of west first.

01:12 Call that, well, let's actually make the, we'll exaggerate the angle just a little bit here.

01:24 We'll call that angle theta 1.

01:28 And let's just say the airplane's right there.

01:31 So that distance from that point to the origin is a distance, we'll call d1.

01:42 And then it's at an angle theta 1, and then an altitude of 8 meters.

01:49 Okay, the second is at an altitude of 1 ,000 ,000.

01:52 100 meters, so higher, a horizontal distance of 17 .6 kilometers, we'll call that d2, and 20 degrees south of west.

02:02 So this one is, you know, maybe over here.

02:13 I'll call this angle theta 2.

02:16 So a little bit closer to the origin and closer to the west axis.

02:22 So let's, and then a distance d2.

02:28 Let's kind of summarize what we've done here.

02:36 So aircraft 1, aircraft 2.

02:42 So aircraft 1 has a position vector, we'll call it r1.

02:47 Its position vector in the x direction is going to be d1, cosine theta 1, right? that's in the west direction, so you take the hypotenuse d1, multiply by cosine theta 1, that will give you the x component...

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