6.2 - Displacement, velocity, and acceleration

In Chapters 3 and 4, we used position, velocity, and acceleration to describe motion in one dimension. In this section we broaden our scope to include the full, three-dimensional vector character of these quantities. Position, velocity, and acceleration are vectors; the tools we learned with one-dimensional motion can now be extended to three dimensions. The key to working in three dimensions is to use vector components to separate complex, three-dimensional problems into three separate but solvable one-dimensional problems—one along each coordinate axis. Read the text aloud
Displacement vector
Suppose you walk 10 m east, turn, go 5 m north, then turn and go 5 m west. Where do you end up? To answer the question, you must treat each leg of the walk as a separate displacement vector. The displacement vector describes a change in position, such as a move from one place to another. The position at the end of the walk is the result of adding the three displacements together. This resultant vector is also the final position vector. The illustration below shows the calculation done graphically and by components, resulting in d = (+5,+5) m. Note that the westward displacement vector has a negative x-component because the positive x-axis points east. Displacements are often given in compass coordinates in which the x-axis is east–west and the y-axis is north–south. Read the text aloud
Adding three displacement vectors together and the resulting displacement vector
Displacement vectorMost real displacements are not purely north, south, east, or west. Instead, they are at angles that can change over time. For purposes of calculation, however, it is useful to represent a displacement by components as if it were two separate displacements in x and y. For example, moving 7 m at 30° north of east puts you in the same place as moving 3.5 m north and then 6.06 m east. The displacement is written in component form as d = (+6.06,+3.50) m, even though the actual motion follows the diagonal path. Read the text aloud Show Coordinate measuring machine
The illustrations above depict displacement using a vector model, where each displacement is represented by a vector. In the vector model, more than one vector can be added graphically using the head-to-tail method or algebraically by adding the components. Read the text aloud
Components of displacement in the <i>x</i> and <i>y</i> directionsCalculating the components of a displacement vector is the same as for the force vector. If the magnitude of the displacement is d, then the components are dx = d cos θ and dy = d sin θ. Read the text aloud
Two displacement vectors are A = (+3,+9) m and B = (−6,+2) m.
  1. What is A + B?
  2. What is A − B?
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