Chapter 3: Q4PE (page 87)

Suppose you walk ${\bf{18}}.{\bf{0}}$m straight west and then ${\rm{25}}{\rm{.0}}$ m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B , then this problem asks you to find their sum R = A + B .)

Short Answer

Expert verified

The resultant vector is ${\rm{30}}{\rm{.8}}$ m.

Step by step solution

Determining the total distance

Displacement is defined as the shortest path between the initial and final positions. It is a vector quantity.

To calculate the distance from starting point to the final point, the resultant vector is

$\overrightarrow R = \overrightarrow A + \overrightarrow B $

Here$\overrightarrow A $is the total x-component of displacement and$\overrightarrow B $is the total y-component of displacement.

By vector law of addition, the resultant vector is

$|\vec R| = \sqrt {{A^2} + {B^2} + 2AB\cos \theta } $…………..(1)

Given data:

$\overrightarrow A = 18\;{\rm{m}}$
$\overrightarrow B = 25\;{\rm{m}}$
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Determining the resultant vector

Substituting the values in equation (1), we get$\begin{array}{c}|\vec R|\; = \sqrt {{{18}^2} + {{25}^2} + 2 \times 18 \times 25\cos {{90}^ \circ }} \\|\vec R|\; = \sqrt {{{18}^2} + {{25}^2}} \\|\vec R|\; = \sqrt {324 + 625} \\|\vec R|\; = \sqrt {949} \\|\vec R|\; = 30.8\;\;{\rm{m}}\end{array}$Therefore the resultant vector is $30.8$m.