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Q16PE

Expert-verifiedFound in: Page 122

Book edition
1st Edition

Author(s)
Paul Peter Urone

Pages
1272 pages

ISBN
9781938168000

**Solve the following problem using analytical techniques: Suppose you walk ${\mathbf{18}}{\mathbf{.}}{\mathbf{0}}{\mathit{m}}$straight west and then ${\mathbf{250}}{\mathbf{.}}{\mathbf{0}}{\mathit{m}}$ 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 and , as in Figure, then this problem asks you to find their sum ${\mathit{R}}{\mathbf{=}}{\mathit{A}}{\mathbf{+}}{\mathit{B}}$ .)**

**The two displacements and add to give a total displacement having magnitude and direction .**

**Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique.**

The distance from the starting point is $30.8m$, and the compass direction of a line connecting starting point to the final position is $54.2\xb0$north of west.

**Vectors, which have both magnitude and direction, are physical quantities. Two vectors cannot be added using simple algebraic rules; they can be added by using the triangle law of vector addition.**

Given data:

- A = $18.0m$
- B = .$25.0m$

The magnitude of the resultant vector is

$R=\sqrt{{A}^{2}+{B}^{2}}$

Here is $A$the magnitude of the vector $A$ (displacement towards west) and $B$ is the magnitude of the vector $B$(displacement towards north).

Substitute role="math" localid="1668686725834" $18.0m$ for $A$ and $25.0m$ for $B$.

$R=\sqrt{{(18.0m)}^{2}+{(25.0m)}^{2}}\phantom{\rule{0ex}{0ex}}=30.8m$

The direction of the compass is

$\theta ={\mathrm{tan}}^{-1}\left(\frac{B}{A}\right)$

Substitute$18.0m$ for $A$and $25.0m$for $B$ .

$\theta ={\mathrm{tan}}^{-1}\left(\frac{25.0m}{18.0m}\right)\phantom{\rule{0ex}{0ex}}=54.2\xb0$

Hence, the distance from the starting point is 30.8m and the compass direction of a line connecting starting point to the final position is $54.2\xb0$north of west.

The analytical methods are more concise, accurate, and precise than the graphical method, which is limited by the accuracy with which a drawing can be made.

Hence, the analytical methods are used because of more precision.

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