Final answer:
To solve this problem, we need to break down the initial velocity and acceleration vectors into their respective components. We can then use the equations of motion to find the duck's speed after a given time. This involves calculating the horizontal and vertical components of the final velocity using trigonometry and the Pythagorean theorem.
Step-by-step explanation:
To solve this problem, we need to break down the initial velocity and acceleration vectors into their respective components. The initial velocity has a magnitude of 0.7 m/s and is at an angle of 25° north of west. To calculate its horizontal component, we can use the sine function: vx = v * sin(θ) = 0.7 * sin(25°) = 0.297 m/s. Similarly, the vertical component can be calculated using the cosine function: vy = v * cos(θ) = 0.7 * cos(25°) = -0.642 m/s (negative because it's south of east).
Now, let's find the horizontal and vertical components of the acceleration. The acceleration has a magnitude of 0.5 m/s² and is at an angle of 41° south of east. The horizontal component of acceleration is given by ax = a * cos(θ) = 0.5 * cos(41°) = 0.382 m/s². The vertical component can be found using the sine function: ay = a * sin(θ) = 0.5 * sin(41°) = -0.322 m/s² (negative because it's south of east).
To find the duck's speed after a given time, we can use the equations of motion. Since the initial velocity is 0.7 m/s and the acceleration is constant, we can use the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Let's calculate the final velocity as follows:
vx = vix + axt = 0.297 + 0.382t
vy = viy + ayt = -0.642 - 0.322t
Now, we can find the magnitude of the velocity using the Pythagorean theorem:
v = sqrt(vx² + vy²)
Let's substitute the values and solve for v:
v = sqrt((0.297 + 0.382t)² + (-0.642 - 0.322t)²)