Luffy pilots the Thousand Sunny across the ocean. He pilots the boat with a velocity 42m/s 31 degrees south of west. There is a strong ocean current pushing on his boat at 13m/s that is 26 degrees north of west. What is the resultant direction of Luffy's boat as an angle?



To find the resultant direction of Luffy's boat, we need to break down the velocities into their component vectors and then combine them.

### Step 1: Break down the velocities into components.

1. **Luffy's boat velocity**:  
   Magnitude: 42 m/s, Angle: 31° south of west  
   - Westward (x-direction) component:  
     \( V_{Lx} = 42 \cos(31^\circ) \)  
   - Southward (y-direction) component:  
     \( V_{Ly} = -42 \sin(31^\circ) \) (negative since it’s south)

2. **Ocean current velocity**:  
   Magnitude: 13 m/s, Angle: 26° north of west  
   - Westward (x-direction) component:  
     \( V_{Ox} = 13 \cos(26^\circ) \)  
   - Northward (y-direction) component:  
     \( V_{Oy} = 13 \sin(26^\circ) \)

### Step 2: Add the components of both velocities.

- Westward component (x-direction):  
  \( V_x = V_{Lx} + V_{Ox} \)
  
- North-south component (y-direction):  
  \( V_y = V_{Ly} + V_{Oy} \)

### Step 3: Calculate the resultant velocity.

The magnitude of the resultant velocity vector can be found using the Pythagorean theorem:

\[
V_{\text{resultant}} = \sqrt{V_x^2 + V_y^2}
\]

The direction (angle) of the resultant vector is given by:

\[
\theta = \tan^{-1} \left(\frac{V_y}{V_x}\right)
\]

Let me now calculate the exact values.The resultant direction of Luffy's boat is approximately **18.48° south of west**.

Would you like further details or have any questions? Here are five related questions to extend your understanding:

1. How do you calculate the resultant speed of the boat in this situation?
2. What if the ocean current's angle was directly eastward? How would that affect the result?
3. Can you explain how the angle components (north/south and east/west) interact in vector addition?
4. How would the problem change if Luffy's boat speed increased to 50 m/s?
5. What is the impact of changing the direction of the ocean current to 90° north?

**Tip**: When dealing with vector problems, always break vectors into components, especially when dealing with angles not aligned with the coordinate axes.

Luffy pilots the Thousand Sunny across the ocean. He pilots the boat with a velocity 42m/s 31 degrees south of west. There is a strong ocean current pushing on his boat at 13m/s that is 26 degrees north of west. What is the resultant direction of Luffy's boat as an angle?

Discover how to calculate the resultant direction of Luffy's boat, given two velocities: 42 m/s at 31° south of west and an ocean current pushing at 13 m/s at 26° north of west. This problem explores vector decomposition, trigonometry, and vector addition, suitable for high school students (Grades 10-12).

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