Chapter 3, Problem 2P

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.

Solution

The final displacement of the truck from the point of origin.

Solution: $16.76\text{blocks}$ and ${72.6}^{\text{o}}$ South of East

Explanation:

Given:

The truck travels 21 blocks north, 16 blocks east and 26 blocks south.

The blocks are of equal length.

Formula used:

The magnitude of the resultant vector and direction can be obtained using the following formulae:

  $\begin{array}{l}R=\sqrt{P^2+Q^2+2PQ\cos \theta }\\ \alpha ={\tan }^{-1}\left(\frac{Q\sin \theta }{P+Q\cos \theta }\right)\end{array}$

Where, P and Q are vectors and $\theta$ is the angle between them.

Calculations:

As per question, there are three displacement vectors.

  $\begin{array}{l}A=21\text{blocks}\text{=21}\text{<mover accent="true"><mo>j</mo><mo>^</mo></mover>}\\ B=16\text{blocks}\text{= 16}\text{<mover accent="true"><mo>i</mo><mo>^</mo></mover>}\\ C=\text{26}\text{blocks}\text{= 26}\left(-\text{<mover accent="true"><mo>j</mo><mo>^</mo></mover>}\right)\end{array}$

Let’s represent it in a diagram:

  

  $\begin{array}{l}\overrightarrow{R}=\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}\\ \overrightarrow{R}=21\text{<mover accent="true"><mo>j</mo><mo>^</mo></mover>}+16\text{<mover accent="true"><mo>i</mo><mo>^</mo></mover>}-26\text{<mover accent="true"><mo>j</mo><mo>^</mo></mover>}\\ \overrightarrow{R}=16\text{<mover accent="true"><mo>i</mo><mo>^</mo></mover>}-5\text{<mover accent="true"><mo>j</mo><mo>^</mo></mover>}\end{array}$

Substitute the values and solve to find the magnitude of the resultant displacement.

  $\begin{array}{l}R=\sqrt{{16}^2+{\left(-5\right)}^2+2\left(16\right)\left(-5\right)\cos {90}^{\text{o}}}\\ R=16.76\text{blocks}\end{array}$

Now, substitute the values to find the direction:

  $\begin{array}{l}\alpha ={\tan }^{-1}\left(\frac{-5\sin 90}{16-5\cos 90}\right)\\ \alpha =-{17.4}^{\text{o}}\\ \beta ={90}^o-{17.4}^{\text{o}}={72.6}^{\text{o}}\end{array}$

It will be directed South of East.