Question

A wildlife researcher is tracking a flock of geese. The geese fly 4.0 km due west, then turn toward the north by 30 degrees and fly another 4.5 km. What is the total displacement of the geese?

1) 4.5 km
2) 4.9 km
3) 5.0 km
4) 5.2 km

Answer 1

The geese's total displacement shows their direct distance from their starting point after flying west, then turning north by 30 degrees and flying further. Vector addition and trigonometry determine their displacement.

Step-by-step explanation:

The total displacement of the geese following their two-part journey can be calculated using vector addition. Initially, the geese fly 4.0 km due west, which is represented as a vector pointing directly to the west with a magnitude of 4.0 km. Then, they turn towards the north by 30 degrees and fly another 4.5 km. This second vector can be decomposed into northward and westward components using trigonometry:

• 
  
• Northward component = 4.5 km × cos(30°)
• 
  
• Westward component = 4.5 km × sin(30°)

Adding these components to the initial 4.0 km west vector and using the Pythagorean theorem, the magnitude of the total displacement vector can be calculated. The resulting magnitude, which is the direct distance from the starting point to the final position of the geese, is the hypotenuse of the triangle formed by the northward and westward components.