To solve this problem, let's break it down step by step.

Given:

1. Initial velocity of the bird: $v_x = 4.00 \, \text{mph}$
2. Wind acceleration in the y-direction (north): $a_y = 0.400 \, \text{m/s}^2$
3. Time duration of the wind: $t = 4.00 \, \text{s}$
4. Conversion factor: $1 \, \text{mile} = 1609 \, \text{m}$

We are tasked with finding:

• The magnitude $r$ of the bird's displacement after 4 seconds.
• The direction $\theta$ (counterclockwise from the easterly direction).

Step 1: Convert initial velocity from mph to m/s

We first need to convert the bird's initial velocity from miles per hour to meters per second.

$v_x = 4.00 \, \text{mph} \times \frac{1609 \, \text{m}}{1 \, \text{mile}} \times \frac{1 \, \text{hour}}{3600 \, \text{s}} = 1.788 \, \text{m/s}$

Step 2: Calculate the bird's displacement in the x-direction

Since there is no acceleration in the x-direction (the bird was coasting), the displacement in the x-direction can be calculated as:

$x = v_x \cdot t = 1.788 \, \text{m/s} \times 4.00 \, \text{s} = 7.152 \, \text{m}$

Step 3: Calculate the bird's displacement in the y-direction

The bird accelerates northward (y-direction) due to the wind. The displacement in the y-direction is given by the equation:

$y = \frac{1}{2} a_y t^2$

Substituting the given values:

$y = \frac{1}{2} \times 0.400 \, \text{m/s}^2 \times (4.00 \, \text{s})^2 = 3.200 \, \text{m}$

Step 4: Calculate the magnitude of the displacement

The total displacement $r$ is the resultant vector from the x and y components:

$r = \sqrt{x^2 + y^2} = \sqrt{(7.152)^2 + (3.200)^2} = \sqrt{51.146 + 10.240} = \sqrt{61.386} \approx 7.836 \, \text{m}$

Step 5: Calculate the direction of the displacement

The direction $\theta$ is the angle counterclockwise from the easterly direction. It can be found using the tangent function:

$\theta = \tan^{-1} \left( \frac{y}{x} \right) = \tan^{-1} \left( \frac{3.200}{7.152} \right)$

$\theta \approx \tan^{-1}(0.447) \approx 24.05^\circ$

Final Answer:

• The magnitude of the bird's displacement $r$ is approximately 7.836 m.
• The direction $\theta$ is approximately 24.05° counterclockwise from the easterly direction.

Would you like more details or have any questions?

Here are some related questions to explore:

1. How would the result change if the wind had lasted for 8 seconds instead of 4?
2. What if the wind had a higher acceleration, say $0.800 \, \text{m/s}^2$?
3. How would the displacement change if the bird started traveling at a higher initial velocity?
4. What is the bird's velocity in the y-direction after 4 seconds?
5. How can we calculate the bird's total velocity at the end of the 4-second interval?

Tip: Always ensure that your velocity and acceleration units are consistent (e.g., meters per second) before calculating displacements and directions.