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Understanding the concept of resultant force is crucial when analyzing the motion of objects influenced by multiple forces. A resultant force is effectively the single force that represents the vector sum of all the forces acting on an object, defining the net force that dictates the object's acceleration. In the provided exercise, the sailboat experiences two perpendicular forces: a wind force of 390 N north and water resistance of 180 N east.

To find the magnitude of the resultant force, we use Pythagoras' theorem, which is applicable here since the forces are at a right angle to each other. The equation for the resultant force (\(F_R\)) becomes: \[ F_{R} = \( \( \( \(F{\_wind}^{2} + F{\_water}^{2} \( \)\)\)\) \) = \( \( \( \( \sqrt{390^{2} + 180^{2}} \( \)\)\)\) \) = \( \( \( \( \sqrt{210,600} \( \)\)\)\) \)\]Making sense of this calculation is key to predicting the boat's movement.

At the heart of motion analysis is Newton's second law of motion, which provides a relationship between the forces acting on an object, its mass, and the resulting acceleration. It is succinctly expressed in the equation \( F = ma \) where \( F \) represents the net force acting on the object, \( m \) is the mass of the object, and \( a \) is the acceleration.

Applying this law to the exercise, once we have the magnitude of the resultant force, we determine the acceleration of the boat by rearranging the equation to solve for \( a \) — \( a = \frac{F}{m} \). Plugging in the values from the resultant force and the boat's mass, we can find out how fast the boat will accelerate in response to the combined forces of wind and water.

Vectors are mathematical representations of quantities that have both a magnitude and a direction. In physics, vector addition is commonly used to combine multiple forces acting on an object to find a single resultant force. Since forces are vector quantities, they follow the rules of vector addition.

For instance, if two forces are perpendicular, as in the case of the sailboat, their vector addition is akin to the sides and hypotenuse of a right triangle. Therefore, to obtain the direction of the resultant force, we use trigonometry—specifically the tangent function—which relates the sides of a right triangle to its angles. The angle \( \theta \) east of north that gives us the direction of the total acceleration can be found using the equation \( \theta = \tan^{-1}(\frac{F_{water}}{F_{wind}}) \). The calculated angle helps us understand not only how fast the boat is accelerating but in what direction it will travel.