Draw each of the following vectors. Then find its x- and y-components. v=(440 m/s,30∘ below the positive x-axis)\mathbf{v} = (440 \, \text{m/s}, 30^\circ \, \text{below the positive } x\text{-axis})

Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

문제 6b

3장, 문제 6b

Draw each of the following vectors. Then find its x- and y-components. $v = (440 m/s, 30^{\circ} below the positive x -axis)$

검증된 단계별 안내

Step 1: Understand the problem. The vector v = 440 m/s is given with a direction of 30 degrees below the positive x-axis. This means the angle is measured clockwise from the positive x-axis. We need to find the x- and y-components of this vector.

Step 2: Recall the formulas for the components of a vector. The x-component of a vector is given by v_x = v * cos(θ), and the y-component is given by v_y = v * sin(θ). Here, θ is the angle the vector makes with the positive x-axis.

Step 3: Note the direction of the components. Since the angle is below the x-axis, the y-component will be negative. The x-component will remain positive because the vector points in the positive x-direction.

Step 4: Substitute the given values into the formulas. Use v = 440 m/s and θ = 30 degrees. For the x-component: v_x = 440 * cos(30). For the y-component: v_y = 440 * sin(30).

Step 5: Simplify the trigonometric expressions. Use the known values of cos(30) = √3/2 and sin(30) = 1/2 to express the components. This gives v_x = 440 * (√3/2) and v_y = 440 * (-1/2).

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주요 개념

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

Vectors

Vectors are quantities that have both magnitude and direction. In physics, they are often represented graphically as arrows, where the length of the arrow indicates the magnitude and the direction of the arrow indicates the direction of the vector. Understanding vectors is essential for analyzing motion and forces in two or three dimensions.

Components of a Vector

The components of a vector are the projections of that vector along the axes of a coordinate system, typically the x-axis and y-axis. For a vector given in polar form (magnitude and angle), the x-component can be found using the cosine of the angle, while the y-component is found using the sine of the angle. This breakdown allows for easier calculations in physics problems involving motion and forces.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. In the context of vectors, these functions are used to resolve a vector into its components. For example, if a vector makes an angle θ with the x-axis, the x-component is found using cos(θ) and the y-component using sin(θ), which is fundamental in analyzing vector quantities in physics.