Tag Archives: vector subtraction

College Physics by Openstax Chapter 3 Problem 13

Find the following for path C in Figure 3.56: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.

Figure 3.56 The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side.

Solution:

Part A

Looking at path C, it moves 1 block upward, 5 blocks to the right, 2 blocks downward, 1 block to the left, 1 block upward, and 3 blocks to the left. So, the total distance is

\begin{align*}
\text{distance} \  = \  &\left( 1\times 120 \ \text{m} \right)+\left( 5\times 120\ \text{m} \right)+\left( 2\times 120\ \text{m} \right)+\left( 1\times 120\ \text{m} \right) \\
& +\left( 1\times 120\ \text{m} \right)+\left( 3\times 120\ \text{m} \right) \\
= \ & 120\ \text{m}+600 \ \text{m}+240 \ \text{m} + 120 \ \text{m}+ 120 \ \text{m}+360\ \text{m} \\
= \ & 1560 \ \text{m} \ \qquad \ {\color{DarkOrange}\left( \text{Answer} \right) }
\end{align*}

Part B

It can be seen from the figure that the end of path C is just one block to the right from the starting point. Therefore, the magnitude of the displacement is 

\begin{align*}
\text{displacement} = 120\ \text{m} \ \qquad \ {\color{DarkOrange}\left( \text{Answer} \right) }
\end{align*}

The direction is to the right or is equivalent to 0° measured from the positive x-axis.

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College Physics by Openstax Chapter 3 Problem 7

(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction 40.0º north of east (which is equivalent to subtracting B from A —that is, to finding R’=A−B ). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction 40.0º south of west and then 12.0 m in a direction 20.0º east of south (which is equivalent to subtracting A from B —that is, to finding R”=B−A=−R’ ). Show that this is the case.

Solution:

Refer to this problem.

Part A

Consider Figure 3-7A

Figure 3-7A

First, we solve for the value of α using a simple geometry. 

\alpha = 40^\circ+\left( 90^\circ-20^\circ \right) = 110^\circ

We can solve for R using the cosine law.

\begin{align*}
R^2 & = A^2 +B^2-2AB \cos \alpha \\
R & = \sqrt{A^2 +B^2-2AB \cos \alpha} \\
R & = \sqrt{\left( 12.0\ \text{m} \right)^2+\left( 20.0\ \text{m} \right)^2-2\left(  12.0\ \text{m} \right)\left( 20.0\ \text{m} \right) \cos 110^\circ} \\
R & =  26.6115 \ \text{m} \\
R & = 26.6 \ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}
\end{align*}

Before we can solve for θ, we need to solve for 𝛽 first using the sine law.

\begin{align*}
\frac{\sin \beta}{B} & = \frac{\sin \alpha }{R} \\
\sin \beta & = \frac{B \  \sin \alpha}{R} \\
\beta & = \sin ^{-1} \left( \frac{B \  \sin \alpha}{R} \right) \\
\beta & = \sin ^{-1} \left( \frac{20.0 \ \text{m}\ \sin 110^{\circ}}{26.6115 \ \text{m}} \right) \\
\beta & = 44.9290^\circ
\end{align*}

Now, we can solve for θ. 

\begin{align*}
\theta & = \left( 90^\circ + 20^\circ \right)-\beta \\
\theta & = 110^\circ - 44.9290^\circ \\
\theta & = 65.071 \\
\theta & = 65.1 ^\circ
\end{align*}

Therefore, the compass reading is

65.1^\circ, \text{North of East} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}

Part B

Refer to Figure 3-7B

Figure 3-7B

First, we solve for the value of α using a simple geometry. 

\alpha = 40^\circ+\left( 90^\circ-20^\circ \right) = 110^\circ

We can solve for R using the cosine law. 

\begin{align*}
R^2 & = A^2 +B^2-2AB \cos \alpha \\
R & = \sqrt{A^2 +B^2-2AB \cos \alpha} \\
R & = \sqrt{\left( 12.0\ \text{m} \right)^2+\left( 20.0\ \text{m} \right)^2-2\left(  12.0\ \text{m} \right)\left( 20.0\ \text{m} \right) \cos 110^\circ} \\
R & =  26.6115 \ \text{m} \\
R & = 26.6 \ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}
\end{align*}

Before we can solve for θ, we need to solve for 𝛽 first using the sine law. 

\begin{align*}
\frac{\sin \beta}{A} & = \frac{\sin \alpha }{R} \\
\sin \beta & = \frac{A \  \sin \alpha}{R} \\
\beta & = \sin ^{-1} \left( \frac{A \  \sin \alpha}{R} \right) \\
\beta & = \sin ^{-1} \left( \frac{12.0 \ \text{m}\ \sin 110^{\circ}}{26.6115 \ \text{m}} \right) \\
\beta & = 25.0708^\circ
\end{align*}

Now, we can solve for θ. 

\begin{align*}
\theta & = 40^\circ + \beta \\
\theta & = 40^\circ +  25.0708^\circ\\
\theta & = 65.0708^\circ\\
\theta & = 65.1 ^\circ
\end{align*}

Therefore, the compass reading is 

65.1^\circ, \text{South of West} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}

This is consistent with Part A because (A-B) = -(B-A).

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College Physics by Openstax Chapter 3 Problem 3

Find the north and east components of the displacement for the hikers shown in Figure 3.50.

Figure 3.50

Solution:

Refer to Figure 3-3-A for the north and east components of the displacement s of the hikers.

Figure 3-3-A

Considering the right triangle formed. The north component is computed as

\begin{align*}
\text{s}_{\text{north}} & = \left( 5.00 \ \text{km}  \right)\sin 40^\circ  \\
\text{s}_{\text{north}} & = 3.21 \ \text{km} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}\\
\end{align*}

Using the same right triangle, the east component is computed as follows. 

\begin{align*}
\text{s}_{\text{east}} & = \left( 5.00 \ \text{km}  \right)\cos 40^\circ  \\
\text{s}_{\text{east}} & = 3.83 \ \text{km} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}\\
\end{align*}
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College Physics by Openstax Chapter 3 Problem 1

Find the following for path A in Figure 3.52:
(a) The total distance traveled, and
(b) The magnitude and direction of the displacement from start to finish.

Figure 3.54 The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side
Figure 3.52 The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side

Solution:

Part A

The total distance traveled is 

\begin{align*}

\text{d} & =\left(3\times 120\ \text{m}\right)+\left(1\times 120\:\text{m}\right) \\
\text{d} & =480\:\text{m}  \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}\\

\end{align*}

Part B

The magnitude of the displacement is 

\begin{align*}

\text{s }& =\sqrt{\left( s_x \right)^{2\:}+\left( s_y \right)^2} \\
\text{s }& = \sqrt{\left(1\times 120\:\text{m}\right)^2+\left(3\times 120\:\text{m}\right)^2} \\
\text{s }& = 379\ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}

\end{align*}

The direction is

\begin{align*}

 \theta & = \tan^{-1}\left(\frac{s_x}{s_y}\right) \\
\theta & = \tan^{-1}\left(\frac{1\times 120\:\text{m}}{3\times 120 \ \text{m}}\right) \\
\theta & =71.6^{\circ} ,\:\text{E of N} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}

\end{align*}
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