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Expert-verified Found in: Page 67 ### Physics Principles with Applications

Book edition 7th
Author(s) Douglas C. Giancoli
Pages 978 pages
ISBN 978-0321625922 # A car is driven 225 km west and then 98 km southwest (45°). What is the displacement of the car from the point of origin (magnitude and direction)? Draw a diagram.

The magnitude of the displacement of the car from the point of origin is 302.3 km, and the direction is $13.2°$south of the west.

See the step by step solution

## Step 1. Meaning of displacement

The net displacement of an object can be defined as the vector addition of the object's individual displacement in various directions. If the object's initial and final points are similar, the object's net displacement value is zero.

## Step 2. Given information

Given data:

The displacement of the car in the western direction is ${\stackrel{\to }{D}}_{west}=225\mathrm{km}$.

The displacement of the car in the southwestern direction $\left(\theta =45°\right)$is ${\stackrel{\to }{D}}_{south-west}=98\mathrm{km}$.

## Step 3. Vector diagram

The vector diagram for the problem can be drawn as follows: ## Step 4. Calculate the resultant displacement vector in the western direction

The resultant displacement vector in the western direction can be calculated as

$\begin{array}{c}{\stackrel{\to }{D}}_{R-west}={\stackrel{\to }{D}}_{west}+{\stackrel{\to }{D}}_{south-west}\mathrm{cos}\theta \\ {\stackrel{\to }{D}}_{R-west}=\left(225\mathrm{km}\right)+\left(98\mathrm{km}\right)\mathrm{cos}\left(45°\right)\\ {\stackrel{\to }{D}}_{R-west}=294.3\mathrm{km}\end{array}$

## Step 5. Calculate the resultant displacement vector in the southwestern direction

The resultant displacement vector in the southwestern direction can be calculated as

$\begin{array}{c}{\stackrel{\to }{D}}_{R-southwest}={\stackrel{\to }{D}}_{south-west}\mathrm{sin}\theta \\ {\stackrel{\to }{D}}_{R-southwest}=\left(98\mathrm{km}\right)\mathrm{sin}\left(45°\right)\\ {\stackrel{\to }{D}}_{R-southwest}=69.3\mathrm{km}\end{array}$

## Step 6. Calculate the magnitude of the resultant displacement vector

The magnitude of the resultant displacement vector can be calculated as

$\begin{array}{c}{\stackrel{\to }{D}}_{R}=\sqrt{{\left({\stackrel{\to }{D}}_{R-west}\right)}^{2}+{\left({\stackrel{\to }{D}}_{R-southwest}\right)}^{2}}\\ {\stackrel{\to }{D}}_{R}=\sqrt{{\left(294.3\mathrm{km}\right)}^{2}+{\left(69.3\mathrm{km}\right)}^{2}}\\ {\stackrel{\to }{D}}_{R}=302.3\mathrm{km}\end{array}$

## Step 7. Calculate the direction for the resultant displacement vector

The direction for the resultant displacement vector can be calculated as

$\begin{array}{c}\theta ={\mathrm{tan}}^{-1}\left(\frac{{\stackrel{\to }{D}}_{R-southwest}}{{\stackrel{\to }{D}}_{R-west}}\right)\\ \theta ={\mathrm{tan}}^{-1}\left(\frac{69.3\mathrm{km}}{294.3\mathrm{km}}\right)\\ \theta =13.2°\end{array}$

Thus, the magnitude of the displacement of the car from the point of origin is 302.3 km, and the direction is $13.2°$south of the west. ### Want to see more solutions like these? 