In an attempt to escape his island, Gilligan builds a raft and sets to sea. The wind shifts a great deal during the day, and he is blown along the following straight lines:
2.5 km 45° north of west; then
4.70 km 60° south of east; then
5.1 km straight east; then
7.2 km 55° south of west; and finally
2.8 km 10° north of east.
What is his final position relative to the island?
2.5 @ W45N = -1.77i + 1.77j
4.7 @ E60S = 2.35i - 4.07j
do the others likewise.
Then add them all up and convert back to a distance and bearing
All angles are measured CCW from +x-axis.
Disp. = 2.5[135o]+4.7[300]+5.1+
7.2[235]+2.8[10o].
Disp. = -1.77+1.77i + 2.35-4.07i + 5.1 + 4.13-5.90i + 2.76+0.486i
Disp. = 12.6 - 7.71i = 14.8km[ -31.5o] = 14.8km[31.5o] S. of E.
To determine Gilligan's final position relative to the island, we can break down his movements step-by-step and calculate the resultant displacement.
Step 1:
Gilligan travels 2.5 km 45° north of west. This can be represented as a vector, V1, with a magnitude (length) of 2.5 km and a direction of 45° north of west.
Step 2:
From his first position, Gilligan is blown 4.70 km 60° south of east. This can be represented as a vector, V2, with a magnitude of 4.70 km and a direction of 60° south of east.
Step 3:
Gilligan then travels 5.1 km straight east. This can be represented as a vector, V3, with a magnitude of 5.1 km and a direction of due east.
Step 4:
From his third position, Gilligan is blown 7.2 km 55° south of west. This can be represented as a vector, V4, with a magnitude of 7.2 km and a direction of 55° south of west.
Step 5:
Finally, Gilligan travels 2.8 km 10° north of east. This can be represented as a vector, V5, with a magnitude of 2.8 km and a direction of 10° north of east.
To find Gilligan's final position, we need to add up all the vectors:
V_final = V1 + V2 + V3 + V4 + V5
First, we can resolve all the vectors into their horizontal (x) and vertical (y) components:
V1 = (-2.5 km * cos(45°), 2.5 km * sin(45°))
V1 = (-1.77 km, 1.77 km)
V2 = (4.70 km * cos(30°), -4.70 km * sin(30°))
V2 ≈ (4.08 km, -2.35 km)
V3 = (5.1 km, 0 km)
V4 = (-7.2 km * cos(35°), -7.2 km * sin(35°))
V4 ≈ (-5.88 km, -4.12 km)
V5 = (2.8 km * cos(10°), 2.8 km * sin(10°))
V5 ≈ (2.74 km, 0.48 km)
Now, we can add up the respective components:
V_final = V1 + V2 + V3 + V4 + V5
V_final = (-1.77 km + 4.08 km + 5.1 km - 5.88 km + 2.74 km, 1.77 km - 2.35 km + 0 km - 4.12 km + 0.48 km)
V_final = (4.27 km, -4.02 km)
Therefore, Gilligan's final position relative to the island is approximately 4.27 km east and 4.02 km south.
To find Gilligan's final position relative to the island, we need to analyze his movements step by step and keep track of his displacement vector.
1. Gilligan is blown 2.5 km 45° north of west.
We can break down this movement into horizontal (west) and vertical (north) components using trigonometry.
Horizontal component = 2.5 km * cos(45°) = 2.5 km * 0.7071 = 1.7678 km west
Vertical component = 2.5 km * sin(45°) = 2.5 km * 0.7071 = 1.7678 km north
So, Gilligan's position after this movement is 1.7678 km west and 1.7678 km north.
2. Gilligan is blown 4.70 km 60° south of east.
Again, we break down this movement into horizontal (east) and vertical (south) components.
Horizontal component = 4.70 km * cos(60°) = 4.70 km * 0.5 = 2.35 km east
Vertical component = 4.70 km * sin(60°) = 4.70 km * 0.866 = 4.0686 km south
Adding these components to Gilligan's previous position, his new position is 1.7678 km west + 2.35 km east and 1.7678 km north - 4.0686 km south.
Simplifying, his new position is 0.5822 km east and -2.3008 km south.
3. Gilligan moves 5.1 km straight east.
This movement only adds to the eastward component of his position.
Adding 5.1 km to Gilligan's eastward position, his new position becomes 0.5822 km east + 5.1 km east and -2.3008 km south.
Simplifying, his new position is 5.6822 km east and -2.3008 km south.
4. Gilligan is blown 7.2 km 55° south of west.
As before, we break down the movement into horizontal (west) and vertical (south) components.
Horizontal component = 7.2 km * cos(55°) = 7.2 km * 0.5736 = 4.1352 km west
Vertical component = 7.2 km * sin(55°) = 7.2 km * 0.8192 = 5.9088 km south
Adding these components to Gilligan's current position, his new position becomes 5.6822 km east + 4.1352 km west and -2.3008 km south - 5.9088 km south.
Simplifying, his new position is 1.547 km west and -8.2096 km south.
5. Gilligan moves 2.8 km 10° north of east.
Breaking down this movement:
Horizontal component = 2.8 km * cos(10°) = 2.8 km * 0.9848 = 2.7554 km east
Vertical component = 2.8 km * sin(10°) = 2.8 km * 0.1736 = 0.485 km north
Adding these components to Gilligan's current position, his new position becomes 1.547 km west + 2.7554 km east and -8.2096 km south + 0.485 km north.
Simplifying, his new position is 1.2084 km east and -7.7246 km south.
Therefore, Gilligan's final position relative to the island is approximately 1.21 km east and 7.72 km south.