Question

A small boat is crossing a lake on a windy day. During some interval of time, the boat undergoes the given displacement Δ → r = (3.23 m) ^ ı + (2.71 m) ^ È·. During the same interval of time, the wind exerts the given constant force → f = (281 N) ^ ı - (117 N) ^ È·. The total work done is 590.56 J. What is the angle between the direction of the wind force and the direction of the boat's motion during this time interval?

Answer 1

The angle between the direction of the wind force and the direction of the boat's motion during this time interval is approximately 48.2 degrees.

Step-by-step explanation:

To find the angle between the direction of the wind force and the direction of the boat's motion, we can use the dot product formula. The dot product of two vectors A and B is given by A · B = AB cos(θ), where AB is the magnitude of vector A multiplied by the magnitude of vector B, and θ is the angle between the two vectors. In this case, A = Δr = (3.23 m)î + (2.71 m)ĵ, and B = F = (281 N)î - (117 N)ĵ. The dot product of A and B is calculated as A · B = (3.23 m)(281 N) + (2.71 m)(-117 N) = 908.63 Nm - 316.07 Nm = 592.56 Nm. The magnitude of vector A is √((3.23 m)^2 + (2.71 m)^2) = 4.18 m, and the magnitude of vector B is √((281 N)^2 + (-117 N)^2) = 304.63 N. Plugging these values into the dot product formula, we have 592.56 Nm = (4.18 m)(304.63 N) cos(θ). Solving for θ, we find that θ ≈ 48.2 degrees.