Components: An airplane undergoes the following displacements: First, it flies 66 km in a direction 30° east of north. Next, it flies 49 km due south. Finally, it flies 100 km 30° north of west. Using vector components, determine how far the airplane ends up from its starting point.
A) 79 km
B) 81 km
C) 82 km
D) 78 km
E) 76 km

Question

Quizplus · Accepted Answer

To solve this, we break down each displacement into its north-south and east-west components and then sum these components separately. 1. First displacement: 66 km at 30° east of north.   - North component = 66 * cos(30°) = 57.2 km   - East component = 66 * sin(30°) = 33 km2. Second displacement: 49 km due south.   - South component = -49 km (negative because it's south)   - East component = 0 km3. Third displacement: 100 km at 30° north of west.   - North component = 100 * cos(30°) = 86.6 km   - West component = -100 * sin(30°) = -50 km (negative because it's west)Summing up the components:- North-South = 57.2 km (north) + (-49 km) (south) + 86.6 km (north) = 94.8 km (north)- East-West = 33 km (east) + 0 km + (-50 km) (west) = -17 km (west)The total displacement vector's magnitude can be found using the Pythagorean theorem:$$ \sqrt{(94.8)^2 + (-17)^2} = \sqrt{8990.24 + 289} = \sqrt{9279.24} \approx 79 \, \text{km} $$

Components: an Airplane Undergoes the Following Displacements: First, It Flies