Answer: 8.11 degrees
Explanation:
To find the total work done on the boat by the wind over the given time interval, we can use the formula:
W = F * d
where W represents work, F represents force, and d represents displacement.
1. First, let's calculate the magnitude of the displacement, |d|, using the given displacement vector:
|d| = sqrt((3.13)^2 + (2.81)^2)
|d| = sqrt(9.7969 + 7.8961)
|d| = sqrt(17.693)
|d| ≈ 4.2 m (rounded to one decimal place)
2. Next, let's calculate the magnitude of the force, |F|, using the given force vector:
|F| = sqrt((257)^2 + (121)^2)
|F| = sqrt(66049 + 14641)
|F| = sqrt(80790)
|F| ≈ 284.4 N (rounded to one decimal place)
3. Now, we can calculate the total work, W, by multiplying the magnitude of the force by the magnitude of the displacement:
W = |F| * |d|
W ≈ 284.4 N * 4.2 m
W ≈ 1194.48 N·m (rounded to two decimal places)
Therefore, the total work done on the boat by the wind over this period of time is approximately 1194.48 N·m.
To find the angle between the direction of the wind force and the direction of the boat's motion during this time interval, we can use the dot product of the force vector and the displacement vector:
θ = cos^(-1)((F • d) / (|F| * |d|))
1. Calculate the dot product of F and d:
F • d = (257)(3.13) + (121)(2.81)
F • d ≈ 804.41 + 339.01
F • d ≈ 1143.42
2. Plug the values into the equation for θ:
θ = cos^(-1)(1143.42 / (284.4 * 4.2))
θ = cos^(-1)(0.9898)
θ ≈ 8.11 degrees (rounded to two decimal places)
Therefore, the angle between the direction of the wind force and the direction of the boat's motion during this time interval is approximately 8.11 degrees.
