Question

A postal employee drives a delivery truck along the route shown in Fig. E1.25. Determine the magnitude and direction of the resultant displacement by drawing a scale diagram. (See also Exercise 1.32 for a different approach.)

Short answer

The displacement is 7.8 km, and the direction is 38 degrees north of east.

Step-by-step solution

 Vector and its direction

The vector's various coordinates are presented here. We know that in coordinates, the x component of a vector comes first, followed by the y component. To calculate the angle of the vector with respect to the x-axis, just multiply the y component by the x component and then take the inverse of that result.

This will tell you the vector's angle with the x-axis. tan^-1 (y/x) is one way to express it.

Finding the magnitude

Here, in\(\Delta DCG\)

CG =\({\rm{CD}}\cos {45^ \circ } = 3.1\;{\rm{km}} \times {\rm{.707}} = 2.1920\;{\rm{km}}\)

DG =\({\rm{CD}}\sin {45^ \circ } = 3.1\;{\rm{km}} \times {\rm{.707}} = 2.1920\;{\rm{km}}\)

\(\begin{aligned}{c}{\rm{AE}} &= {\rm{AF}} + {\rm{FE}}\\ &= {\rm{AF}} + {\rm{CG}}\\ &= {\rm{4}}\;{\rm{km}} + {\rm{2}}{\rm{.1920}}\;{\rm{km}}\\ &= 6.1920\;{\rm{km}}\end{aligned}\)

\(\begin{aligned}{c}{\rm{DE}} &= {\rm{DG}} + {\rm{GE}}\\ &= {\rm{DG}} + {\rm{AB}}\\ &= {\rm{2}}{\rm{.1920}}\;{\rm{km}} + {\rm{2}}{\rm{.6}}\;{\rm{km}}\\ &= 4.792\;{\rm{km}}\end{aligned}\)

If AE and ED are vectors and the resultant will be the displacement. The magnitude of displacement can be calculated as:

\(\begin{aligned}{c}{\rm{Displacement}}\left( {AD} \right) &= \sqrt {{\rm{A}}{{\rm{E}}^2} + {\rm{E}}{{\rm{D}}^2}} \\ &= \sqrt {{\rm{6}}{\rm{.192}}{{\rm{0}}^2} + {{4.792}^2}} \\ &= \sqrt {61.304} \\ &= 7.82\;{\rm{km}}\end{aligned}\)

Direction can be calculated as:

\(\begin{aligned}{c}\tan \theta &= \frac{{{\rm{DE}}}}{{{\rm{AE}}}}\\ &= \frac{{4.792}}{{6.1920}}\\ &= 0.7739\\\theta &= {\tan ^{ - 1}}0.7739\\\theta &= {37.73^ \circ }\\ &\sim {38^ \circ }\end{aligned}\)

Hence, the displacement is 7.8 km, and the direction is 38 degrees north of east.