A 301-kg boat is sailing 16.5° north of east at a speed of 2.08 m/s. Thirty seconds later, it is sailing 37.2° north of east at a speed of 3.60 m/s. During this time, three forces act on the boat: a 32.7-N force directed 16.5° north of east (due to an auxiliary engine), a 21.7-N force directed 16.5° south of west (resistance due to the water), and W (due to the wind). Find the magnitude and direction of the force W. Express the direction as an angle with respect to due east. magnitude 1 N
direction 2 °
To find the magnitude and direction of the force W, we need to use the principle of vector addition. The net force acting on the boat is equal to the sum of all the individual forces acting on it.
First, let's break down all the given information:
- The boat's mass (m) is 301 kg.
- The boat's initial velocity (v1) is 2.08 m/s, making an angle of 16.5° north of east.
- Thirty seconds later, the boat's velocity (v2) is 3.60 m/s, making an angle of 37.2° north of east.
- Three forces are acting on the boat:
1. A force of 32.7 N directed 16.5° north of east (auxiliary engine force).
2. A force of 21.7 N directed 16.5° south of west (water resistance force).
3. The force W, directed due to the wind (magnitude and direction unknown).
Now, let's calculate the net force acting on the boat:
1. Decompose the given forces into their respective north and east components:
- Auxiliary engine force: F1 magnitude = 32.7 N, F1 direction = 16.5° north of east.
F1north = F1 * cos(16.5°) = 32.7 N * cos(16.5°)
F1east = F1 * sin(16.5°) = 32.7 N * sin(16.5°)
- Water resistance force: F2 magnitude = 21.7 N, F2 direction = 16.5° south of west.
F2north = F2 * sin(16.5°) = 21.7 N * sin(16.5°)
F2west = F2 * cos(16.5°) = 21.7 N * cos(16.5°) [since it's south of west, we take the cos component]
2. Calculate the change in momentum (Δp) of the boat using the difference in velocity:
Δp = m * (v2 - v1)
Δp represents the change in momentum, and m represents the mass of the boat.
3. Calculate the average net force (Favg) acting on the boat over the 30-second time interval:
Favg = Δp / Δt
Δt represents the time interval of 30 seconds.
4. Calculate the north and east components of the average net force:
Favg_north = F1north + F2north
Favg_east = -F1east + F2west
Since the boat is facing east, the east component is subtracted. The negative sign indicates that it is in the opposite direction.
5. Find the magnitude and direction of the force W:
The magnitude of force W can be calculated using the Pythagorean theorem:
|W| = sqrt(Favg_north^2 + Favg_east^2)
Direction of W = atan(Favg_east / Favg_north)
Now that we know the steps, you can plug in the given values into the equations to find the magnitude and direction of force W.