Question

An airplane undergoes the following displacements: First, it flies 72 km in a direction 30° east of north. Next, it flies 48 km due south. Finally, it flies 100 km 30° north of west. Using analytical methods, determine how far the airplane ends up from its starting point. (Show all work)

Answer 1

case 1 : airplane flies 72km in a direction 30° east of North. let initial position of airplane is O and it reaches point A.

then, OA = 72km makes an angle 30° with positive y-axis.

in vector form, OA = 72(sin30° i + cos30° j) = 36(i + √3 j) km

case 2 : next it flies 48km due to south.

i.e., AB = 48 along negative y-axis.

in vector form, AB = -48 j km

case 3 : finally, it flies 100 km 30° north of West.

i.e., BC = 100km makes an angle 30° with negative x-axis.

in vector form, BC = 100(-cos30°i + sin30°j) = 50(-√3i + j) km

so, seperation between starting position to final position = OA + AB + BC

= 36(i + √3j) + (-48j) + 50(-√3i + j)

= (36 - 50√3)i + (36√3 + 2)j

magnitude of seperation = √{(36 - 50√3)²+(36√3 + 2)²}

≈ 81.86 km