A person walks in the following pattern: 2.8 km north, then 2.7 km west, and finally 6.9 km south. Construct the vector diagram that represents this motion and from it judge how far and in what direction a bird would fly in a straight line from the same starting point to the same final point. (Choose east as the +x direction.)
To construct the vector diagram, we can represent the person's motion as a series of vectors.
Let's start by drawing a coordinate system with east as the +x direction and north as the +y direction. We can choose any scale for the diagram as long as we maintain the relative lengths of the vectors.
1. Start at the origin (0,0) and draw a vector of 2.8 km in the positive y-direction (north). Label this vector as A.
2. From the end point of vector A, draw a vector of 2.7 km in the negative x-direction (west). Label this vector as B.
3. From the end point of vector B, draw a vector of 6.9 km in the negative y-direction (south). Label this vector as C.
4. Connect the starting point (origin) to the end point of vector C. This final vector represents the displacement of the person. Label this vector as R.
The vector diagram should now show vectors A, B, C, and R.
To find out how far and in what direction a bird would fly in a straight line from the same starting point to the same final point, we can calculate the magnitude and direction of vector R using the Pythagorean theorem and trigonometry.
1. Calculate the magnitude of vector R:
magnitude(R) = sqrt( (2.7 km)^2 + (2.8 km - 6.9 km)^2 )
2. Calculate the direction of vector R:
direction(R) = atan( (6.9 km - 2.8 km) / 2.7 km )
The magnitude of vector R represents the distance the bird would fly in a straight line, and the direction of vector R represents the direction the bird would fly from the same starting point to the same final point.
To construct the vector diagram representing the person's motion, we need to consider the displacement in terms of both the magnitude and direction for each step:
1. Northward: The person walked 2.8 km north. We represent this as a vector pointing upwards with a magnitude of 2.8 km.
2. Westward: The person then walked 2.7 km west. We represent this as a vector pointing to the left with a magnitude of 2.7 km.
3. Southward: Finally, the person walked 6.9 km south. We represent this as a vector pointing downwards with a magnitude of 6.9 km.
To construct the vector diagram, we draw these vectors head-to-tail:
```
N
↑
│
2.8 km│ 6.9 km
│
← 2.7 km
│
│
```
With the vectors drawn, we can find the resultant or net displacement by drawing a straight line from the starting point to the ending point of the person's motion:
```
N
↑
│
│
2.8 km│ 6.9 km
│
← 2.7 km
│
│
│━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
```
Now, to determine how far and in what direction a bird would fly in a straight line from the same starting point to the same final point, we need to find the magnitude and angle of this resultant vector:
1. Magnitude: We can measure the length of the resultant vector using a ruler or a scale. Let's say it measures approximately 7.6 cm.
2. Direction: We measure the angle between the resultant vector and the positive x-axis using a protractor. Using east as the +x direction, we find that the angle measures approximately 58 degrees.
Therefore, the bird would fly approximately 7.6 km in a straight line at an angle of approximately 58 degrees east of the +x direction.