A delivery truck travels 21 blocks north, 16 blocks east, and 26 blocks south. What is its final displacement from the origin? Assume the blocks are equal length.

검증된 단계별 안내

Define displacement as a vector quantity that has both magnitude and direction. To find the final displacement, we need to calculate the net change in position in both the north-south and east-west directions.

Calculate the net displacement in the north-south direction. The truck travels 21 blocks north and 26 blocks south. The net displacement in this direction is:

21 − 26

blocks.

Calculate the net displacement in the east-west direction. The truck travels 16 blocks east and does not travel west. Therefore, the net displacement in this direction is:

16

blocks.

Represent the net displacement as a vector in two dimensions. The north-south displacement is the y-component, and the east-west displacement is the x-component. The displacement vector is:

(x, y)=(16, 21−26)

Use the Pythagorean theorem to calculate the magnitude of the displacement vector. The formula is:

\sqrt{x^{2 + y^{2}}}

. Substitute the values of x and y to find the magnitude of the displacement. The direction can be found using the inverse tangent function:

\tan ⁻ 1 (y / x)

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주요 개념

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

Displacement

Displacement is a vector quantity that refers to the change in position of an object. It is defined as the shortest straight-line distance from the initial position to the final position, along with the direction. In this context, it helps determine how far and in which direction the delivery truck is from its starting point after its journey.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In this problem, the truck's movements in different directions can be represented as vectors, and by adding these vectors, we can find the overall displacement. This involves considering both the magnitude and direction of each movement.

Coordinate System

A coordinate system provides a framework for defining the position of points in space. In this scenario, we can use a Cartesian coordinate system where north and east are positive directions. By translating the truck's movements into coordinates, we can easily calculate the final position and thus the displacement from the origin.