Vector Addition and Subtraction| Analytical Method| College Physics| Problem 3.16

Solve the following problem using analytical techniques: Suppose you walk 18.0 m straight west and then 25.0 m straight north. 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 3.60, then this problem asks you to find their sum R = A + B .) Figure3.60 The two displacements A and B add to give a total displacement R having magnitude R and direction θ.

Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique.

SOLUTION:

Consider the right triangle formed by the legs A, B, and R. We know that A is 18 m, B is 25 m, and we are solving for the magnitude of R. We can do this by using the Pythagorean Theorem. That is $R=\sqrt{A^2+B^2}$ $R=\sqrt{\left(18\:m\right)^2+\left(25\:m\right)^2}$ $R=\sqrt{324+625}$ $R=\sqrt{949}$ $R=30.8\:m$

So, the distance is about 30.8 meters from the starting point. To solve for the value of the unknown angle, θ, we can use the tangent function. That is $tan\:\theta =\frac{B}{A}$ $tan\:\theta =\frac{25\:m}{18\:m}$ $\theta =tan^{-1}\left(\frac{25}{18}\right)$ $\theta =54.25^{\circ}$

So, the compass reading, can be solved by taking the complimentary angle, $\phi$ as shown in the figure.  $\phi =90^{\circ} -54.25^{\circ}$ $\phi =35.75^{\circ}$

Therefore, the compass angle is $35.75^{\circ} \:West\:of\:North$