A cruise ship sails due south at 5.50 m/s while a coast guard patrol boat heads 19.0° north of west at 6.00 m/s. What are the x-component and the y-component of the velocity of the cruise ship relative to the patrol boat? (Assume that the +x-axis is east and the +y-axis is north. Enter your answers in m/s.)

To solve this problem, we'll break the velocities of both the cruise ship and the coast guard patrol boat into their respective components. Then, we will find the relative velocity of the cruise ship with respect to the patrol boat.

### Step 1: Velocity components of the patrol boat
The patrol boat is moving at 6.00 m/s at an angle of \(19.0^\circ\) north of west.

- The x-component of the patrol boat's velocity (since west is negative x-direction):
  \[
  v_{p_x} = -6.00 \cdot \cos(19.0^\circ)
  \]

- The y-component of the patrol boat's velocity (since north is positive y-direction):
  \[
  v_{p_y} = 6.00 \cdot \sin(19.0^\circ)
  \]

### Step 2: Velocity components of the cruise ship
The cruise ship is moving due south at 5.50 m/s. Therefore:

- The x-component of the cruise ship's velocity is zero:
  \[
  v_{c_x} = 0
  \]

- The y-component of the cruise ship's velocity (since south is negative y-direction):
  \[
  v_{c_y} = -5.50
  \]

### Step 3: Relative velocity components
To find the relative velocity of the cruise ship with respect to the patrol boat, subtract the velocity components of the patrol boat from those of the cruise ship:

- The relative x-component:
  \[
  v_{rel_x} = v_{c_x} - v_{p_x} = 0 - (-6.00 \cdot \cos(19.0^\circ))
  \]

- The relative y-component:
  \[
  v_{rel_y} = v_{c_y} - v_{p_y} = -5.50 - (6.00 \cdot \sin(19.0^\circ))
  \]

Now let's compute these values.The components of the velocity of the cruise ship relative to the patrol boat are:

- **x-component**: \( 5.67 \, \text{m/s} \)
- **y-component**: \( -7.45 \, \text{m/s} \)

This means the cruise ship is moving 5.67 m/s to the east and 7.45 m/s south relative to the patrol boat.

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:
1. How do you calculate the magnitude of the relative velocity from the components?
2. What would change if the patrol boat were heading due west instead of north of west?
3. How do velocity vectors transform when the reference frame is moving?
4. How can you graphically represent the relative velocity in a coordinate plane?
5. How would this problem change if the cruise ship were moving north instead of south?

**Tip**: When calculating velocity components, always double-check the direction (sign) associated with each axis to avoid common mistakes.

Learn how to calculate the x-component and y-component of the velocity of a cruise ship relative to a coast guard patrol boat. This problem involves vector decomposition, trigonometry, and relative velocity, with step-by-step solutions provided. Suitable for high school students in grades 10-12.

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