How do you solve this physics problem? There are so many forces involved!?

Question

A 341-kg boat is sailing 13.0 ° north of east at a speed of 1.80 m/s. 22.0 s later, it is sailing 33.0 ° north of east at a speed of 4.50 m/s. During this time, three forces act on the boat: a 30.8-N force directed 13.0 ° north of east (due to an auxiliary engine), a 21.9-N force directed 13.0 ° south of west (resistance due to the water), and one due to the wind. Find the (a) the magnitude and (b) direction of the force due to the wind . Express the direction as an angle with respect to due east.

Answer 1

This problem will require you to solve using Newton's second law F=ma and working with vector math. I won't do the math for you, but I can help walk you through the process of thinking through the problem so you know how to tackle future problems like it. Don't get all intimidated. Just do it one step at a time and it'll unfold by itself.

Working backwards, you are asked to find the force to due wind. You are told that it and two other forces sum together to push the boat. You know two forces and are asked for the third. In order to find the third, you need to know the total sum of forces. So how to you find the total sum of forces acting on the boat? You use the information that the boat started out heading in one direction and after some time, it is heading in some other direction. The boat must have some acceleration in order to change direction. That acceleration must be due to the net forces acting on it. So all you have to do is find the acceleration of the boat, and you can get the net forces acting on it. Once you get that, you go back to find the force due to wind. To recap, first, you use the information given about the boat's velocities to find the net direction of acceleration (and thus the net force applied to it). Then, knowing the net force acting on the boat, you subtract from it the known forces of propulsion and drag to get the force from wind.

Let's start. You have the boat's original velocity V0 and the boat's velocity Vf after 22 seconds.

V0=1.80*cos(13) i + 1.80*sin(13) j
Vf=4.50*cos(33) i + 4.50*sin(33) j

where i and j are unit vectors in the X and Y directions, respectively.

The difference between the two velocities is the velocity caused by acceleration on the boat from a net force on it. To find this acceleration (and the net force), you first need to find the net velocity change V_delta.

V_delta=Vf-V0

Once you have V_delta, you can find the average acceleration by dividing it by time t. To find the net force acting on the boat F_total, you multiply the acceleration by the mass of the boat m. The expression for F_total is going to look like this:

F_total=m*V_delta / t

That's the vector sum of all the forces acting on the boat. To find the force from the wind F_w, you need to subtract from it the forces due to propulsion F_p and forces due to drag F_d. You have the equation F_total=F_w+F_p+F_d.

F_p=30.8*cos(13) i + 30.8*sin(13) j
F_d=21.9*cos(193) i + 21.9*sin(193) j

Subtract F_p and F_d from F_total and you will get the force due to wind F_w.

Answer 2

Use trig to break the forces into their X and Y components. If you do that for each ot the forces, it will be easy...