91Ó°ÊÓ

In physics, a velocity vector describes both the speed and direction of a moving object. It has two components: horizontal and vertical, generally represented in a 2D coordinate system. For a moving boat, like in the problem, you would split its movement into two directions using trigonometry. The initial velocity is given as 2.00 m/s at 15.0° north of east. The components would therefore be determined using:

• Horizontal component: \( v_{i,x} = 2.00 \cos(15^{\circ}) \)
• Vertical component: \( v_{i,y} = 2.00 \sin(15^{\circ}) \)

The same breakdown applies for the final velocity of the boat. This approach allows us to work with both the speed and direction accurately. Remember, the change in these velocity components over time helps us analyze how forces affect the boat's motion.

The net force is the overall force acting on an object when all individual forces are combined. According to Newton's Second Law, this net force determines the acceleration of an object. In our example, several forces act on the boat:

• The engine force pulling the boat forward.
• Water resistance pushing against the movement.
• Wind, adding an additional unseen force.

By calculating the net force, we establish how these forces interact to change the boat's speed and direction. For instance, the net force in the x-direction \( F_{\text{net}, x} \) and y-direction \( F_{\text{net}, y} \) can be derived from the components of each involved force. This analysis is crucial in predicting how the boat will behave as it sails.

Acceleration describes the rate of change of velocity of an object over time. It is key to linking force and motion through Newton's Second Law: \( \overrightarrow{F} = m\overrightarrow{a} \). Here, \( m \) is mass, \( \overrightarrow{F} \) is force, and \( \overrightarrow{a} \) is acceleration. For the boat, calculate the average acceleration during the time between its initial and final states.
The changes in velocity components are given by:

• \( a_x = \frac{\Delta v_x}{30} \)
• \( a_y = \frac{\Delta v_y}{30} \)

These tell us how fast the boat's velocity is changing in each direction. With these accelerations, apply Newton’s law to find the net forces affecting the boat's movement. Understanding acceleration helps to see how force applications result in speed and directional changes.

Wind force is the typically unseen yet influential force that acts on a moving object like a boat. In scenarios where a boat's velocity changes without signs of visible force changes, wind is often a suspect.
To determine the impact of wind in our scenario, solve for the wind force \( \overrightarrow{F_W} \) by analyzing the net force components. Once other force influences are known, isolate the wind components:

• \( F_{W, x} \) and \( F_{W, y} \) come from the net force equations.

Compute the magnitude of this force with:\[ |\overrightarrow{F_W}| = \sqrt{(F_{W, x})^2 + (F_{W, y})^2} \]and find the direction using the angle:\[ \theta = \tan^{-1}\left(\frac{F_{W, y}}{F_{W, x}}\right) \]This method ensures understanding of how wind alters the boat's trajectory, integrating seamlessly with other forces in the problem.