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.

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.