Question

An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 800 m, horizontal distance 19.2 km, and 25.0° south of west. The second aircraft is at altitude 1 100 m, horizontal distance 17.6 km, and 20.0° south of west. What is the distance between the two aircraft? (Place the x axis west, the y axis south, and the z axis vertical.)

Answer 1

300 mm 2265 m 800 m 800 m 2265 m Horizontal Plane parallel to Ground Vertical Plane

Left side figure shows the position of airplanes A and B detected by air-traffic controller observing from O.

We write position vector of A as, $\underset{OA}{\rightarrow}$ = 19.2 cos25 $\hat{i}$ + 19.2 sin25 $\hat{j}$ km

We write position vector of B as, $\underset{OB}{\rightarrow}$ = 17.6 cos20 $\hat{i}$ + 17.6 sin20 $\hat{j}$ km

Vector $\underset{AB}{\rightarrow}$ = $\underset{OB}{\rightarrow}$ - $\underset{OA}{\rightarrow}$ = (17.6 cos20 - 19.2 cos25) $\hat{i}$ + (17.6 sin20 - 19.2 sin25 ) $\hat{j}$ km

$\underset{AB}{\rightarrow}$ = -0.863 $\hat{i}$ -2.095 $\hat{j}$ km

Distance between A and B = | $\underset{AB}{\rightarrow}$ | = ( 0.863 $\times$ 0.863 + 2.095 $\times$ 2.095 )^1/2 = 2.265 km

Right side figure shows the vertical plane , where the air planes height from ground are seen.

Distance between airplanes A and B in space = ( 2265 $\times$ 2265 + 300 $\times$ 300 )^1/2 = 2285 m