Final answer:
The magnitude and direction of the wind force F on the boat can be found by breaking down known forces into components, calculating the boat's acceleration, and applying Newton's second law in both x and y directions to solve for F's components. The magnitude of F is determined using the Pythagorean theorem, and the direction using arctangent.
Step-by-step explanation:
A student has asked how to find the magnitude and direction of the force F acting on a boat due to the wind over a time interval of 30 seconds, given that the boat's velocity changes from 2.0 m/s at 15° north of east to 4.0 m/s at 35° north of east, and also given the forces due to an auxiliary engine and water resistance. To solve this, we can apply Newton's second law of motion, F = ma, where F is the net force acting on an object, m is its mass, and a is its acceleration.
To begin with, we need to calculate the boat's acceleration by determining the change in velocity (both in magnitude and direction) and dividing it by the time interval. The initial and final velocities can be represented as vectors, and the change in velocity (Δv) can be found by vector subtraction of the initial velocity from the final velocity. Next, we break down all the given forces, including the unknown wind force F, into their x (eastward) and y (northward) components.
Since two forces (engine and water resistance) with known magnitudes and directions are acting on the boat, their components can be calculated using basic trigonometry. We then apply Newton's second law in both the x and y directions separately to set up a system of equations. By solving the system of equations, we can find the components of the wind force F.
Finally, the magnitude of F is found using the Pythagorean theorem applied to its components, and the direction is found by taking the arctangent of the ratio of the y component to the x component. It's important to adjust this angle based on the quadrant in which the vector lies.