This problem involves vector addition since the boat's velocity and acceleration are in different directions. Here’s how to solve it:

Given:

• Initial velocity: $v_i = 6.4 \, \text{m/s}$ [W 35° N]
• Acceleration: $a = 2.2 \, \text{m/s}^2$ [S]
• Time: $t = 4.0 \, \text{s}$

We need to calculate the final velocity of the boat by finding the initial and final velocity vectors and combining them.

Steps:

1. Resolve the initial velocity into components:

   The direction of the initial velocity is 35° North of West.

   • West component: $v_{i_x} = 6.4 \cos(35^\circ)$
   • North component: $v_{i_y} = 6.4 \sin(35^\circ)$

2. Acceleration effect:

   The acceleration is purely in the south direction, so it only affects the north-south component. The velocity change due to acceleration in the south direction is: $\Delta v_{y} = a \times t = 2.2 \, \text{m/s}^2 \times 4.0 \, \text{s} = 8.8 \, \text{m/s}$ This decreases the northward component.

3. Calculate the final velocity components:

   • West component remains unchanged: $v_{f_x} = v_{i_x}$
   • North-South component: $v_{f_y} = v_{i_y} - 8.8 \, \text{m/s}$

4. Find the final velocity: Use the Pythagorean theorem to find the magnitude of the final velocity: $v_f = \sqrt{v_{f_x}^2 + v_{f_y}^2}$ The direction of the final velocity is given by: $\theta = \tan^{-1}\left(\frac{v_{f_y}}{v_{f_x}}\right)$ where $\theta$ is measured west of south (depending on the signs of the components).

Let me calculate these values for you.The final velocity of the boat is approximately 7.33 m/s at an angle of 44.4° south of west.

Would you like a detailed breakdown of these calculations or have any further questions?

Here are 5 related questions for deeper exploration:

1. How would the result change if the boat had been accelerating northward instead of southward?
2. How does the time of acceleration impact the final velocity in this scenario?
3. What if the boat was initially moving directly west? How would the final velocity be affected?
4. How would you calculate the final velocity if the boat was accelerating in a different diagonal direction (e.g., southwest)?
5. What is the significance of breaking velocity into components in solving vector-related problems?

Tip: Always check the direction of each vector before summing components, as they might cancel or reinforce each other depending on their alignment.