On a 3 day hike a hiker walks 40 km due south the first day, 30 km at an angle of 40 degrees south of east the second day, and 25 km at an angle of 45 degrees north of west the third day. What is the person's displacement from her original position?
To find the person's displacement from their original position, we need to calculate the vector sum of the distances they traveled on each day.
1. Convert the distances and angles into their respective vector components:
- Day 1: 40 km due south = (-40, 0) km
- Day 2: 30 km at 40 degrees south of east
- The horizontal component = 30 km * cos(40°)
- The vertical component = 30 km * sin(40°)
- So the vector components for Day 2 are approximately (22.99, -19.28) km
- Day 3: 25 km at 45 degrees north of west
- The horizontal component = 25 km * cos(45°)
- The vertical component = 25 km * sin(45°)
- So the vector components for Day 3 are approximately (-17.68, 17.68) km
2. Add up the vector components:
- Horizontal component: -40 + 22.99 - 17.68 ≈ -34.69 km
- Vertical component: 0 - 19.28 + 17.68 ≈ -1.60 km
3. The resulting vector has a horizontal component of approximately -34.69 km and a vertical component of approximately -1.60 km. Thus, the person's displacement from their original position is (-34.69, -1.60) km.
So, the person's displacement from their original position is approximately 34.69 km to the west and 1.60 km to the south.