91Ó°ÊÓ

When dealing with vector addition in physics, specifically for problems involving velocity, it's crucial to break down each vector into its respective components. This helps us to easily analyze and understand the vector's effect in each direction.

In this exercise, we consider two vectors: the plane’s velocity relative to the wind and the wind's velocity relative to the Earth. Each vector can be split into components along the x-axis (east-west direction) and the y-axis (north-south direction). This process is called vector decomposition.

• The plane's velocity, moving due south, is represented as a component only in the negative y-direction: \(-35 \hat{j}\) m/s.
• The wind's velocity, heading southwest, has components both in the negative x and y directions. Given that it's heading exactly between south and west, these components are equal: \(-10/\sqrt{2} \hat{i} - 10/\sqrt{2} \hat{j}\) m/s.

By breaking down each velocity into its x and y components, we simplify the process of vector addition, which is crucial for finding the plane's resultant velocity with respect to the Earth.

Understanding both the magnitude and direction of a resultant vector is key to describing the overall motion of an object. Once we have the velocity components, we can find the resultant vector, which gives us this information.

The magnitude of the plane's velocity relative to the Earth is found using the Pythagorean Theorem. By squaring each component of the vector, then summing them and taking the square root, we get the overall magnitude:\[|\overrightarrow{v}_{P/E}| = \sqrt{(\frac{-10}{\sqrt{2}})^2 + (\frac{10}{\sqrt{2}} + 35)^2}\].

This calculation provides the speed of the plane relative to Earth.The direction of the resultant vector is equally important. It describes the angle at which the plane travels compared to a reference direction, typically the y-axis in these problems. We use the tangent function to find this angle:\[\tan(\Theta) = \frac{|v_{P/E}^{x}|}{|v_{P/E}^{y}|}\],where \(v_{P/E}^{x}\) and \(v_{P/E}^{y}\) are the x and y components of the plane's velocity relative to Earth. Solving for \(\Theta\) gives the angle of motion, offering a complete picture of the velocity vector.

Vector decomposition is a fundamental technique in physics, particularly useful for understanding how different forces or velocities interact. By decomposing vectors into x and y components, we can better address problems involving complex movements like those in aviation or nautical contexts.

In this exercise, vector decomposition helps us distinguish and correctly interpret the plane's and wind's impact on overall direction and speed.

• The wind's vector, being 10 m/s directed southwest, is split into two equal components because southwest is a 45-degree direction. Each of these components is calculated by multiplying the speed by \(\cos(45^{\circ})\) or \(\sin(45^{\circ})\), which is \(\frac{1}{\sqrt{2}}\).
• The plane's vector in the south direction is straightforward, having only a y-component since it's moving directly south.

By setting the x and y axes as our reference directions, vector decomposition transforms complex vector operations into simple algebraic additions and subtractions, making it a reliable method in vector analysis.