91Ó°ÊÓ

Displacement refers to the overall change in position of an object. It is a vector quantity, meaning it has both magnitude and direction. In the scenario of the bird watcher, each path through the woods contributes to the total displacement. While distance measures the total ground covered, displacement focuses on the shortest path between the starting and ending points.
The bird watcher's journey is broken into three vectors: walking east, south, and in a direction north of west. By adding these vectors together using vector addition, we can find the resultant vector that represents the bird watcher's total displacement.
Mathematically, the displacement is calculated by summing up the east-west and north-south components separately. The components are added as follows:

• East-west direction: 0.50 km (east) - 1.76 km (west)
• North-south direction: -0.75 km (south) + 1.23 km (north)

Converting these into a single vector yields the displacement vector (\(-1.26 \text{ km}, 0.48 \text{ km}\)). This vector gives both the magnitude and direction of the bird watcher's displacement from the starting to the final position.

Average velocity is a measure of how quickly an object changes its position, taking into account the direction of movement. It is distinct from speed, which only considers magnitude. To find the average velocity of the bird watcher, we use the total displacement vector (\(-1.26 \text{ km}, 0.48 \text{ km}\)) calculated earlier and the total time of the journey, which is 2.50 hours.
Thus, the average velocity vector (\( \text{\vec{V}}_{avg} \)) is obtained by dividing each component of the displacement by the total time. For magnitude:

• \( \text{Magnitude of } \vec{V}_{avg} = \frac{1.35 \text{ km}}{2.50 \text{ h}} \approx 0.54 \text{ km/h} \)

The direction remains the same as the direction determined for the displacement vector, which is 20.84° north of west. Understanding average velocity helps one identify not just how fast, but also in which direction the bird watcher is moving on average during the trip.

The Pythagorean theorem is a mathematical principle employed to determine the length of the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this exercise, the theorem helps us calculate the magnitude of the displacement vector (\(|\vec{D}|\)). This is done by treating the east-west and north-south components as the two shorter sides of a right triangle:

• \( |\vec{D}| = \sqrt{(-1.26)^2 + (0.48)^2} \)
• \( |\vec{D}| \approx \sqrt{1.5876 + 0.2304} \approx \sqrt{1.818} \approx 1.35 \text{ km} \)

By using the Pythagorean theorem, we derive the shortest possible straight-line distance between the bird watcher's starting and ending points, capturing the essence of displacement.

Vector components break down a vector into smaller, more manageable parts. These parts usually align with specific coordinate axes, such as east-west and north-south in this scenario. It simplifies complex vector problems by transforming a vector into scalar components that can be easily addedVector components are particularly helpful when the direction of a vector is not aligned with standard axes, as shown by the 2.15 km walk at 35° north of west. Here, the vector is broken down as:

• Westward component: \( \cos(35°) \times 2.15 \text{ km} \approx 1.76 \text{ km} \)
• Northward component: \( \sin(35°) \times 2.15 \text{ km} \approx 1.23 \text{ km} \)

The decomposition facilitates vector addition by outlining clear steps for combining various directions of movement into one coherent displacement vector. Understanding and using vector components is essential in physics for analyzing and solving vector-related problems efficiently.