The nose of an ultralight plane is pointed due south, and its airspeed indicator shows $35$\text{ m/s}$$ . The plane is in a $10$\text{ m/s}$$ wind blowing toward the southwest relative to the earth. In a vector-addition diagram, show the relationship of $$\overrightarrow{v}$_{P/E}$ (the velocity of the plane relative to the earth) to the two given vectors.

검증된 단계별 안내

Begin by identifying the vectors involved in the problem. The plane's airspeed vector, $ \mathbf{v}_{P/A} $, is 35 m/s due south, and the wind vector, $ \mathbf{v}_{A/E} $, is 10 m/s toward the southwest.

Understand that the velocity of the plane relative to the earth, $ \mathbf{v}_{P/E} $, is the vector sum of the plane's velocity relative to the air, $ \mathbf{v}_{P/A} $, and the air's velocity relative to the earth, $ \mathbf{v}_{A/E} $. This can be expressed as: $ \mathbf{v}_{P/E} = \mathbf{v}_{P/A} + \mathbf{v}_{A/E} $.

Draw the vector $ \mathbf{v}_{P/A} $ pointing directly south. This vector is vertical on the diagram, with a magnitude of 35 m/s.

Next, draw the vector $ \mathbf{v}_{A/E} $ pointing southwest. This vector forms a 45-degree angle with the south direction, as southwest is halfway between south and west. The magnitude of this vector is 10 m/s.

Finally, use the tip-to-tail method to add the vectors. Place the tail of $ \mathbf{v}_{A/E} $ at the tip of $ \mathbf{v}_{P/A} $. The resultant vector $ \mathbf{v}_{P/E} $ is drawn from the tail of $ \mathbf{v}_{P/A} $ to the tip of $ \mathbf{v}_{A/E} $. This vector represents the velocity of the plane relative to the earth.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Vector Addition

Vector addition is a method used to combine two or more vectors to determine a resultant vector. In this context, it involves adding the plane's velocity vector and the wind's velocity vector to find the plane's velocity relative to the earth. This is typically done using graphical methods like the head-to-tail method or by breaking vectors into components.

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular frame of reference. Here, it refers to the velocity of the plane relative to the earth, which is affected by both the plane's airspeed and the wind's speed and direction. Understanding relative velocity is crucial for determining how the plane's movement is influenced by external factors like wind.

Vector Components

Vector components are the projections of a vector along the axes of a coordinate system, typically the x and y axes. Breaking vectors into components simplifies calculations, especially when dealing with vectors at angles. In this problem, the wind's southwest direction requires resolving it into south and west components to accurately perform vector addition.