You start from an acorn and walk 300 yards do north. Then, you turn N 63° E and walk 260 yards. Then, you turn S 55° E and walk 240 yards. Then you walk back to the little acorn.
and lo! it is a mighty oak.
Take some graph paper and plot the course. Recall that for each section
x = d*cos(theta)
y = d*sin(theta)
To determine your final location after following the described path, we need to break down the movements and calculate the displacement relative to the starting point.
1. Start by visualizing the directions using a compass rose. The initial direction is due north (0°), then N 63° E, and finally S 55° E.
2. Let's calculate the north-south (N-S) displacement:
- You walked 300 yards due north, so your N-S displacement is +300 yards.
- Then, you turned N 63° E, which means you deviated 63° to the east from the northern direction. This movement does not change the N-S displacement.
- Next, you turned S 55° E, meaning you deviated 55° to the east from the southern direction. This movement also does not affect the N-S displacement.
- Finally, you walked back to the starting point, which means you canceled any N-S displacement. Thus, the final N-S displacement is 0 yards.
3. Now, let's calculate the east-west (E-W) displacement:
- You did not make any east-west movement during the first leg (walking due north), so the E-W displacement remains 0 yards.
- In the second leg, turning N 63° E and walking 260 yards, you moved both to the north and to the east. To find the E-W displacement, we need to calculate the eastward and northward components of this movement:
- The eastward component can be found by multiplying the total distance (260 yards) by the cosine of the angle (63°).
- The northward component can be found by multiplying the total distance (260 yards) by the sine of the angle (63°).
- The E-W displacement is the difference between the eastward and westward components of the movement.
- In the third leg, turning S 55° E and walking 240 yards, you moved both to the south and to the east. Again, calculate the eastward and southward components of this movement:
- The eastward component can be found by multiplying the total distance (240 yards) by the cosine of the angle (55°).
- The southward component can be found by multiplying the total distance (240 yards) by the sine of the angle (55°).
- The E-W displacement is the sum of the eastward and westward components of the movement.
4. Add up all the E-W components from each leg and subtract all the N-S components to find the overall displacement.
By following these steps, you can calculate the final location after completing the described path.