Engineering Mathematics and Sciences

Your virtual study buddy in Mathematics, Engineering Sciences, and Civil Engineering

College Physics by Openstax Chapter 3 Problem 5

Suppose you first walk 12.0 m in a direction 20º west of north and then 20.0 m in a direction 40.0º south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B, as in Figure 3.54, then this problem finds their sum R=A+B.)

Solution:

Consider Figure 3.5A shown below.

Before we can use cosine law to solve for the magnitude of R, we need to solve for the interior angle 𝛽 first. The value of 𝛽 can be calculated by inspecting the figure and use simple knowledge on geometry. It is equal to the sum of 20° and the complement of 40°. That is

\beta = 20^\circ +\left( 90^\circ -40^\circ \right) = 70^\circ

We can use cosine law to solve for R.

\begin R^2 & =A^2+B^2 -2AB \cos \beta \\ R^2 & = \left( 12.0\ \text \right) ^2+\left( 20.0\ \text \right)^2-2 \left( 12.0\ \text \right) \left( 20.0\ \text \right) \cos 70^\circ \\ R & = \sqrt < \left( 12.0\ \text\right) ^2+\left( 20.0\ \text \right)^2-2 \left( 12.0\ \text \right) \left( 20.0\ \text \right) \cos 70^\circ> \\ R & =19.4892 \ \text \\ R & =19.5 \ \text \ \qquad \ \left( \text \right)> \end

We can solve for α using sine law.

\begin \frac & = \frac \\ \frac> & = \frac> \\ \sin \alpha & = \frac \\ \alpha & = \sin ^ \left( \frac \right) \\ \alpha & = 74.6488 ^\circ \end

Then we solve for the value of θ by subtracting 70° from α.

\theta=74.6488 ^\circ -70 ^\circ = 4.65^\circ

Therefore, the compass reading is

4.65^\circ, \text \ \qquad \ \left( \text \right)>